Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.5% → 57.4%

Time: 10.9s

Alternatives: 19

Speedup: 5.2×

• could not determine a ground truth

  1. c0 = -1.664509038240479e+297
  2. w = 3.7376625642681784e-308
  3. h = 9.071539872228413e-194
  4. D = 3.1856368592024782e-279
  5. d = 1.6566898690268682e+274
  6. M = 2.5611500883183843e+303

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 25.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 57.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ (* d c0) (* D (* h w))) (/ d D)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
     (*
      0.5
      (/ (* c0 (sqrt (sqrt (sqrt (sqrt (* (pow M 8.0) (pow M 8.0))))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * c0) / (D * (h * w))) * (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((pow(M, 8.0) * pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * c0) / (D * (h * w))) * (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.sqrt((Math.pow(M, 8.0) * Math.pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * c0) / (D * (h * w))) * (d / D)
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.sqrt((math.pow(M, 8.0) * math.pow(M, 8.0))))))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d * c0) / Float64(D * Float64(h * w))) * Float64(d / D))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt(sqrt(Float64((M ^ 8.0) * (M ^ 8.0))))))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d * c0) / (D * (h * w))) * (d / D);
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_1 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt(((M ^ 8.0) * (M ^ 8.0))))))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * c0), $MachinePrecision] / N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 25.0

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 25.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 25.2

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 25.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      18. lower-/.f64 35.1

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

   7. Applied rewrites 35.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. sqrt-fabs-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      14. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      16. sqrt-unprod N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      18. lower-unsound-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   6. Applied rewrites 37.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      7. swap-sqr N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      10. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      12. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot {M}^{2}\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      13. pow-prod-up N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      17. swap-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      20. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      21. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      22. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot {M}^{2}\right)}}}}{w} \]

   8. Applied rewrites 39.4%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8}} \cdot \sqrt{{M}^{8}}}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      4. lower-*.f64 40.5

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

   10. Applied rewrites 40.5%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 56.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := c0 \cdot \left(d \cdot d\right)\\ t_2 := \frac{t\_1}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := \frac{t\_1}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D}\\ \mathbf{if}\;t\_0 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (* c0 (* d d)))
        (t_2 (/ t_1 (* (* w h) (* D D))))
        (t_3 (/ t_1 (* (* D (* h w)) D))))
   (if (<= (* t_0 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_0 (+ t_3 (sqrt (- (* t_3 t_3) (* M M)))))
     (*
      0.5
      (/ (* c0 (sqrt (sqrt (sqrt (sqrt (* (pow M 8.0) (pow M 8.0))))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = c0 * (d * d);
	double t_2 = t_1 / ((w * h) * (D * D));
	double t_3 = t_1 / ((D * (h * w)) * D);
	double tmp;
	if ((t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((pow(M, 8.0) * pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = c0 * (d * d);
	double t_2 = t_1 / ((w * h) * (D * D));
	double t_3 = t_1 / ((D * (h * w)) * D);
	double tmp;
	if ((t_0 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.sqrt((Math.pow(M, 8.0) * Math.pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = c0 * (d * d)
	t_2 = t_1 / ((w * h) * (D * D))
	t_3 = t_1 / ((D * (h * w)) * D)
	tmp = 0
	if (t_0 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_0 * (t_3 + math.sqrt(((t_3 * t_3) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.sqrt((math.pow(M, 8.0) * math.pow(M, 8.0))))))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(c0 * Float64(d * d))
	t_2 = Float64(t_1 / Float64(Float64(w * h) * Float64(D * D)))
	t_3 = Float64(t_1 / Float64(Float64(D * Float64(h * w)) * D))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt(sqrt(Float64((M ^ 8.0) * (M ^ 8.0))))))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = c0 * (d * d);
	t_2 = t_1 / ((w * h) * (D * D));
	t_3 = t_1 / ((D * (h * w)) * D);
	tmp = 0.0;
	if ((t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_0 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt(((M ^ 8.0) * (M ^ 8.0))))))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot \left(w \cdot h\right)\right)} \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lower-*.f64 25.2

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot \left(w \cdot h\right)\right)} \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \color{blue}{\left(w \cdot h\right)}\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \color{blue}{\left(h \cdot w\right)}\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 25.2

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \color{blue}{\left(h \cdot w\right)}\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 25.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot \left(w \cdot h\right)\right)} \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lower-*.f64 25.3

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot \left(w \cdot h\right)\right)} \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \color{blue}{\left(w \cdot h\right)}\right) \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \color{blue}{\left(h \cdot w\right)}\right) \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 25.3

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \color{blue}{\left(h \cdot w\right)}\right) \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 25.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} - M \cdot M}\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot \left(w \cdot h\right)\right)} \cdot D} - M \cdot M}\right) \]

      6. lower-*.f64 28.3

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot \left(w \cdot h\right)\right)} \cdot D} - M \cdot M}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \color{blue}{\left(w \cdot h\right)}\right) \cdot D} - M \cdot M}\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \color{blue}{\left(h \cdot w\right)}\right) \cdot D} - M \cdot M}\right) \]

      9. lower-*.f64 28.3

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \color{blue}{\left(h \cdot w\right)}\right) \cdot D} - M \cdot M}\right) \]

   7. Applied rewrites 28.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D}} - M \cdot M}\right) \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. sqrt-fabs-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      14. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      16. sqrt-unprod N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      18. lower-unsound-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   6. Applied rewrites 37.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      7. swap-sqr N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      10. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      12. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot {M}^{2}\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      13. pow-prod-up N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      17. swap-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      20. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      21. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      22. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot {M}^{2}\right)}}}}{w} \]

   8. Applied rewrites 39.4%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8}} \cdot \sqrt{{M}^{8}}}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      4. lower-*.f64 40.5

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

   10. Applied rewrites 40.5%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 56.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* d c0) (/ d (* (* h D) (* D w)))))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
     (*
      0.5
      (/ (* c0 (sqrt (sqrt (sqrt (sqrt (* (pow M 8.0) (pow M 8.0))))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * c0) * (d / ((h * D) * (D * w)));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((pow(M, 8.0) * pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * c0) * (d / ((h * D) * (D * w)));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.sqrt((Math.pow(M, 8.0) * Math.pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d * c0) * (d / ((h * D) * (D * w)))
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.sqrt((math.pow(M, 8.0) * math.pow(M, 8.0))))))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d * c0) * Float64(d / Float64(Float64(h * D) * Float64(D * w))))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt(sqrt(Float64((M ^ 8.0) * (M ^ 8.0))))))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d * c0) * (d / ((h * D) * (D * w)));
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_1 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt(((M ^ 8.0) * (M ^ 8.0))))))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d * c0), $MachinePrecision] * N[(d / N[(N[(h * D), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(c0 \cdot d\right) \cdot \frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(c0 \cdot d\right) \cdot \frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot c0\right)} \cdot \frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot c0\right)} \cdot \frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-/.f64 25.1

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \color{blue}{\frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. associate-*r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lower-*.f64 24.4

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(\left(D \cdot D\right) \cdot w\right)} \cdot h} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 24.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\color{blue}{\left(\left(c0 \cdot d\right) \cdot \frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\color{blue}{\left(\left(c0 \cdot d\right) \cdot \frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\color{blue}{\left(d \cdot c0\right)} \cdot \frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\color{blue}{\left(d \cdot c0\right)} \cdot \frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-/.f64 24.5

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \color{blue}{\frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. associate-*r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lower-*.f64 24.6

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(\left(D \cdot D\right) \cdot w\right)} \cdot h}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 24.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\color{blue}{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \color{blue}{\left(\left(c0 \cdot d\right) \cdot \frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)} - M \cdot M}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \color{blue}{\left(\left(c0 \cdot d\right) \cdot \frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)} - M \cdot M}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\color{blue}{\left(d \cdot c0\right)} \cdot \frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right) - M \cdot M}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\color{blue}{\left(d \cdot c0\right)} \cdot \frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right) - M \cdot M}\right) \]

      9. lower-/.f64 28.6

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \color{blue}{\frac{d}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}\right) - M \cdot M}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}\right) - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}}\right) - M \cdot M}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(D \cdot D\right) \cdot \color{blue}{\left(w \cdot h\right)}}\right) - M \cdot M}\right) \]

      13. associate-*r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}}\right) - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}}\right) - M \cdot M}\right) \]

      15. lower-*.f64 30.1

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(\left(D \cdot D\right) \cdot w\right)} \cdot h}\right) - M \cdot M}\right) \]

   7. Applied rewrites 30.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \color{blue}{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right)} - M \cdot M}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{h \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot w\right)}} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{h \cdot \color{blue}{\left(D \cdot \left(D \cdot w\right)\right)}} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(D \cdot h\right)} \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(D \cdot h\right) \cdot \left(D \cdot w\right)}} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(h \cdot D\right)} \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(h \cdot D\right)} \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

      11. lower-*.f64 29.3

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \color{blue}{\left(D \cdot w\right)}} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

   9. Applied rewrites 29.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

       2. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{h \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot w\right)}}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

       5. associate-*l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{h \cdot \color{blue}{\left(D \cdot \left(D \cdot w\right)\right)}}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

       6. associate-*r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

       7. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(D \cdot h\right)} \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(D \cdot h\right) \cdot \left(D \cdot w\right)}}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

       9. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(h \cdot D\right)} \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(h \cdot D\right)} \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

       11. lower-*.f64 29.2

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \color{blue}{\left(D \cdot w\right)}}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]

   11. Applied rewrites 29.2%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) - M \cdot M}\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}}\right) - M \cdot M}\right) \]

       2. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot w\right)}}\right) - M \cdot M}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{h \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot w\right)}}\right) - M \cdot M}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot w\right)}\right) - M \cdot M}\right) \]

       5. associate-*l* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{h \cdot \color{blue}{\left(D \cdot \left(D \cdot w\right)\right)}}\right) - M \cdot M}\right) \]

       6. associate-*r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}}\right) - M \cdot M}\right) \]

       7. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(D \cdot h\right)} \cdot \left(D \cdot w\right)}\right) - M \cdot M}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(D \cdot h\right) \cdot \left(D \cdot w\right)}}\right) - M \cdot M}\right) \]

       9. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(h \cdot D\right)} \cdot \left(D \cdot w\right)}\right) - M \cdot M}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(h \cdot D\right)} \cdot \left(D \cdot w\right)}\right) - M \cdot M}\right) \]

       11. lower-*.f64 33.4

           \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \color{blue}{\left(D \cdot w\right)}}\right) - M \cdot M}\right) \]

   13. Applied rewrites 33.4%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}\right) \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\color{blue}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)}}\right) - M \cdot M}\right) \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. sqrt-fabs-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      14. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      16. sqrt-unprod N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      18. lower-unsound-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   6. Applied rewrites 37.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      7. swap-sqr N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      10. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      12. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot {M}^{2}\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      13. pow-prod-up N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      17. swap-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      20. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      21. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      22. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot {M}^{2}\right)}}}}{w} \]

   8. Applied rewrites 39.4%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8}} \cdot \sqrt{{M}^{8}}}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      4. lower-*.f64 40.5

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

   10. Applied rewrites 40.5%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 56.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(d \cdot d\right) \cdot {\left(\frac{c0}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)}^{2} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (*
      t_0
      (+
       (* (/ (* d c0) (* D (* h w))) (/ d D))
       (sqrt
        (- (* (* d d) (pow (* (/ c0 (* (* h w) D)) (/ d D)) 2.0)) (* M M)))))
     (*
      0.5
      (/ (* c0 (sqrt (sqrt (sqrt (sqrt (* (pow M 8.0) (pow M 8.0))))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * ((((d * c0) / (D * (h * w))) * (d / D)) + sqrt((((d * d) * pow(((c0 / ((h * w) * D)) * (d / D)), 2.0)) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((pow(M, 8.0) * pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((((d * c0) / (D * (h * w))) * (d / D)) + Math.sqrt((((d * d) * Math.pow(((c0 / ((h * w) * D)) * (d / D)), 2.0)) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.sqrt((Math.pow(M, 8.0) * Math.pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * ((((d * c0) / (D * (h * w))) * (d / D)) + math.sqrt((((d * d) * math.pow(((c0 / ((h * w) * D)) * (d / D)), 2.0)) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.sqrt((math.pow(M, 8.0) * math.pow(M, 8.0))))))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(d * c0) / Float64(D * Float64(h * w))) * Float64(d / D)) + sqrt(Float64(Float64(Float64(d * d) * (Float64(Float64(c0 / Float64(Float64(h * w) * D)) * Float64(d / D)) ^ 2.0)) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt(sqrt(Float64((M ^ 8.0) * (M ^ 8.0))))))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * ((((d * c0) / (D * (h * w))) * (d / D)) + sqrt((((d * d) * (((c0 / ((h * w) * D)) * (d / D)) ^ 2.0)) - (M * M))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt(((M ^ 8.0) * (M ^ 8.0))))))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(N[(N[(d * c0), $MachinePrecision] / N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(N[(d * d), $MachinePrecision] * N[Power[N[(N[(c0 / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 25.0

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 25.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 25.2

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 25.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      18. lower-/.f64 35.1

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

   7. Applied rewrites 35.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      2. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)}^{2}} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)}}^{2} - M \cdot M}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{{\left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}} \cdot \frac{d}{D}\right)}^{2} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{{\left(\frac{\color{blue}{d \cdot c0}}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)}^{2} - M \cdot M}\right) \]

      6. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{{\left(\color{blue}{\left(d \cdot \frac{c0}{D \cdot \left(h \cdot w\right)}\right)} \cdot \frac{d}{D}\right)}^{2} - M \cdot M}\right) \]

      7. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{{\color{blue}{\left(d \cdot \left(\frac{c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)\right)}}^{2} - M \cdot M}\right) \]

      8. unpow-prod-down N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{{d}^{2} \cdot {\left(\frac{c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)}^{2}} - M \cdot M}\right) \]

      9. lower-unsound-pow.f32 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{{d}^{2}} \cdot {\left(\frac{c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)}^{2} - M \cdot M}\right) \]

      10. lower-pow.f32 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{{d}^{2}} \cdot {\left(\frac{c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)}^{2} - M \cdot M}\right) \]

      11. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(d \cdot d\right)} \cdot {\left(\frac{c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)}^{2} - M \cdot M}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(d \cdot d\right)} \cdot {\left(\frac{c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)}^{2} - M \cdot M}\right) \]

      13. lower-unsound-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(d \cdot d\right) \cdot {\left(\frac{c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)}^{2}} - M \cdot M}\right) \]

      14. lower-unsound-pow.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(d \cdot d\right) \cdot \color{blue}{{\left(\frac{c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)}^{2}} - M \cdot M}\right) \]

   9. Applied rewrites 28.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(d \cdot d\right) \cdot {\left(\frac{c0}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)}^{2}} - M \cdot M}\right) \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. sqrt-fabs-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      14. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      16. sqrt-unprod N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      18. lower-unsound-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   6. Applied rewrites 37.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      7. swap-sqr N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      10. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      12. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot {M}^{2}\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      13. pow-prod-up N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      17. swap-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      20. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      21. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      22. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot {M}^{2}\right)}}}}{w} \]

   8. Applied rewrites 39.4%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8}} \cdot \sqrt{{M}^{8}}}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      4. lower-*.f64 40.5

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

   10. Applied rewrites 40.5%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 56.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0\right) \cdot d\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(t\_0 + \sqrt{{t\_0}^{2} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* (/ d (* (* (* D D) w) h)) c0) d))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (+ t_0 (sqrt (- (pow t_0 2.0) (* M M)))))
     (*
      0.5
      (/ (* c0 (sqrt (sqrt (sqrt (sqrt (* (pow M 8.0) (pow M 8.0))))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d / (((D * D) * w) * h)) * c0) * d;
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * (t_0 + sqrt((pow(t_0, 2.0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((pow(M, 8.0) * pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d / (((D * D) * w) * h)) * c0) * d;
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * (t_0 + Math.sqrt((Math.pow(t_0, 2.0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.sqrt((Math.pow(M, 8.0) * Math.pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d / (((D * D) * w) * h)) * c0) * d
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * (t_0 + math.sqrt((math.pow(t_0, 2.0) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.sqrt((math.pow(M, 8.0) * math.pow(M, 8.0))))))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d / Float64(Float64(Float64(D * D) * w) * h)) * c0) * d)
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(t_0 + sqrt(Float64((t_0 ^ 2.0) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt(sqrt(Float64((M ^ 8.0) * (M ^ 8.0))))))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d / (((D * D) * w) * h)) * c0) * d;
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_1 * (t_0 + sqrt(((t_0 ^ 2.0) - (M * M))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt(((M ^ 8.0) * (M ^ 8.0))))))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * d), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(t$95$0 + N[Sqrt[N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 25.0

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 25.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 25.2

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 25.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      18. lower-/.f64 35.1

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

   7. Applied rewrites 35.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

   9. Applied rewrites 30.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(\frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0\right) \cdot d + \sqrt{{\left(\left(\frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0\right) \cdot d\right)}^{2} - M \cdot M}\right)} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. sqrt-fabs-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      14. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      16. sqrt-unprod N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      18. lower-unsound-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   6. Applied rewrites 37.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      7. swap-sqr N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      10. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      12. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot {M}^{2}\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      13. pow-prod-up N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      17. swap-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      20. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      21. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      22. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot {M}^{2}\right)}}}}{w} \]

   8. Applied rewrites 39.4%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8}} \cdot \sqrt{{M}^{8}}}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      4. lower-*.f64 40.5

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

   10. Applied rewrites 40.5%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 55.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \left(\frac{0.5}{w} \cdot \left(t\_0 + \sqrt{{t\_0}^{2} - M \cdot M}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ (* d d) (* (* h D) (* D w))) c0))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (* c0 (* (/ 0.5 w) (+ t_0 (sqrt (- (pow t_0 2.0) (* M M))))))
     (*
      0.5
      (/ (* c0 (sqrt (sqrt (sqrt (sqrt (* (pow M 8.0) (pow M 8.0))))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * d) / ((h * D) * (D * w))) * c0;
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = c0 * ((0.5 / w) * (t_0 + sqrt((pow(t_0, 2.0) - (M * M)))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((pow(M, 8.0) * pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * d) / ((h * D) * (D * w))) * c0;
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = c0 * ((0.5 / w) * (t_0 + Math.sqrt((Math.pow(t_0, 2.0) - (M * M)))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.sqrt((Math.pow(M, 8.0) * Math.pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) / ((h * D) * (D * w))) * c0
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = c0 * ((0.5 / w) * (t_0 + math.sqrt((math.pow(t_0, 2.0) - (M * M)))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.sqrt((math.pow(M, 8.0) * math.pow(M, 8.0))))))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d * d) / Float64(Float64(h * D) * Float64(D * w))) * c0)
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(Float64(0.5 / w) * Float64(t_0 + sqrt(Float64((t_0 ^ 2.0) - Float64(M * M))))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt(sqrt(Float64((M ^ 8.0) * (M ^ 8.0))))))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d * d) / ((h * D) * (D * w))) * c0;
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = c0 * ((0.5 / w) * (t_0 + sqrt(((t_0 ^ 2.0) - (M * M)))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt(((M ^ 8.0) * (M ^ 8.0))))))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] / N[(N[(h * D), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(c0 * N[(N[(0.5 / w), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 25.0

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 25.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 25.2

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 25.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      18. lower-/.f64 35.1

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

   7. Applied rewrites 35.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

   9. Applied rewrites 28.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(c0 \cdot \frac{d}{D}, \frac{\frac{d}{D}}{h \cdot w}, \sqrt{{\left(\left(\frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0\right) \cdot d\right)}^{2} - M \cdot M}\right)} \]

   11. Applied rewrites 29.5%

       \[\leadsto \color{blue}{c0 \cdot \left(\frac{0.5}{w} \cdot \mathsf{fma}\left(\frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot d, c0, \sqrt{{\left(\left(\frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot d\right) \cdot c0\right)}^{2} - M \cdot M}\right)\right)} \]

   13. Applied rewrites 29.5%

       \[\leadsto c0 \cdot \left(\frac{0.5}{w} \cdot \color{blue}{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0 + \sqrt{{\left(\frac{d \cdot d}{\left(h \cdot D\right) \cdot \left(D \cdot w\right)} \cdot c0\right)}^{2} - M \cdot M}\right)}\right) \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. sqrt-fabs-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      14. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      16. sqrt-unprod N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      18. lower-unsound-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   6. Applied rewrites 37.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      7. swap-sqr N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      10. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      12. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot {M}^{2}\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      13. pow-prod-up N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      17. swap-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      20. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      21. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      22. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot {M}^{2}\right)}}}}{w} \]

   8. Applied rewrites 39.4%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8}} \cdot \sqrt{{M}^{8}}}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      4. lower-*.f64 40.5

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

   10. Applied rewrites 40.5%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 55.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \cdot c0\right)}^{2} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (*
      t_0
      (fma
       (/ (* d c0) (* D (* h w)))
       (/ d D)
       (sqrt (- (pow (* (/ (* d d) (* (* (* h w) D) D)) c0) 2.0) (* M M)))))
     (*
      0.5
      (/ (* c0 (sqrt (sqrt (sqrt (sqrt (* (pow M 8.0) (pow M 8.0))))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * fma(((d * c0) / (D * (h * w))), (d / D), sqrt((pow((((d * d) / (((h * w) * D) * D)) * c0), 2.0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((pow(M, 8.0) * pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * fma(Float64(Float64(d * c0) / Float64(D * Float64(h * w))), Float64(d / D), sqrt(Float64((Float64(Float64(Float64(d * d) / Float64(Float64(Float64(h * w) * D) * D)) * c0) ^ 2.0) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt(sqrt(Float64((M ^ 8.0) * (M ^ 8.0))))))) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(N[(d * c0), $MachinePrecision] / N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(N[(N[(d * d), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], 2.0], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}, \frac{d}{D}, \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]

   3. Applied rewrites 24.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0\right)}^{2} - M \cdot M}\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}} \cdot c0\right)}^{2} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(\left(D \cdot D\right) \cdot w\right)} \cdot h} \cdot c0\right)}^{2} - M \cdot M}\right) \]

      3. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}} \cdot c0\right)}^{2} - M \cdot M}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(D \cdot D\right)} \cdot \left(w \cdot h\right)} \cdot c0\right)}^{2} - M \cdot M}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot w\right)}} \cdot c0\right)}^{2} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot w\right)}} \cdot c0\right)}^{2} - M \cdot M}\right) \]

      7. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}} \cdot c0\right)}^{2} - M \cdot M}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{D \cdot \color{blue}{\left(D \cdot \left(h \cdot w\right)\right)}} \cdot c0\right)}^{2} - M \cdot M}\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D}} \cdot c0\right)}^{2} - M \cdot M}\right) \]

      10. lower-*.f64 27.3

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D}} \cdot c0\right)}^{2} - M \cdot M}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(D \cdot \left(h \cdot w\right)\right)} \cdot D} \cdot c0\right)}^{2} - M \cdot M}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(\left(h \cdot w\right) \cdot D\right)} \cdot D} \cdot c0\right)}^{2} - M \cdot M}\right) \]

      13. lower-*.f64 27.3

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(\left(h \cdot w\right) \cdot D\right)} \cdot D} \cdot c0\right)}^{2} - M \cdot M}\right) \]

   5. Applied rewrites 27.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)}, \frac{d}{D}, \sqrt{{\left(\frac{d \cdot d}{\color{blue}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \cdot c0\right)}^{2} - M \cdot M}\right) \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. sqrt-fabs-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      14. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      16. sqrt-unprod N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      18. lower-unsound-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   6. Applied rewrites 37.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      7. swap-sqr N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      10. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      12. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot {M}^{2}\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      13. pow-prod-up N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      17. swap-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      20. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      21. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      22. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot {M}^{2}\right)}}}}{w} \]

   8. Applied rewrites 39.4%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8}} \cdot \sqrt{{M}^{8}}}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      4. lower-*.f64 40.5

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

   10. Applied rewrites 40.5%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 55.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(t\_0, d, \sqrt{{\left(t\_0 \cdot d\right)}^{2} - M \cdot M}\right)}{w + w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d (* (* (* D D) w) h)) c0))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (* c0 (/ (fma t_0 d (sqrt (- (pow (* t_0 d) 2.0) (* M M)))) (+ w w)))
     (*
      0.5
      (/ (* c0 (sqrt (sqrt (sqrt (sqrt (* (pow M 8.0) (pow M 8.0))))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / (((D * D) * w) * h)) * c0;
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = c0 * (fma(t_0, d, sqrt((pow((t_0 * d), 2.0) - (M * M)))) / (w + w));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((pow(M, 8.0) * pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / Float64(Float64(Float64(D * D) * w) * h)) * c0)
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(fma(t_0, d, sqrt(Float64((Float64(t_0 * d) ^ 2.0) - Float64(M * M)))) / Float64(w + w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt(sqrt(Float64((M ^ 8.0) * (M ^ 8.0))))))) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(c0 * N[(N[(t$95$0 * d + N[Sqrt[N[(N[Power[N[(t$95$0 * d), $MachinePrecision], 2.0], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 25.0

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 25.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      18. lower-/.f64 25.2

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   5. Applied rewrites 25.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{c0 \cdot \left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot \color{blue}{\left(d \cdot d\right)}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\color{blue}{\left(c0 \cdot d\right) \cdot d}}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\left(w \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}} - M \cdot M}\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \frac{\left(c0 \cdot d\right) \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M \cdot M}\right) \]

      8. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\color{blue}{\frac{c0 \cdot d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{\color{blue}{d \cdot c0}}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{\color{blue}{D \cdot \left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(w \cdot h\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \color{blue}{\left(h \cdot w\right)}} \cdot \frac{d}{D}\right) - M \cdot M}\right) \]

      18. lower-/.f64 35.1

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \color{blue}{\frac{d}{D}}\right) - M \cdot M}\right) \]

   7. Applied rewrites 35.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right) \cdot \color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} - M \cdot M}\right) \]

   9. Applied rewrites 29.9%

      \[\leadsto \color{blue}{c0 \cdot \frac{\mathsf{fma}\left(\frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0, d, \sqrt{{\left(\left(\frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0\right) \cdot d\right)}^{2} - M \cdot M}\right)}{w + w}} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. sqrt-fabs-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      14. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      16. sqrt-unprod N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      18. lower-unsound-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   6. Applied rewrites 37.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      7. swap-sqr N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      10. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      12. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot {M}^{2}\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      13. pow-prod-up N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      17. swap-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      20. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      21. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      22. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot {M}^{2}\right)}}}}{w} \]

   8. Applied rewrites 39.4%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8}} \cdot \sqrt{{M}^{8}}}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      4. lower-*.f64 40.5

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

   10. Applied rewrites 40.5%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 53.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(\frac{d}{h \cdot w} \cdot d, c0, \left(\left|M\right| \cdot D\right) \cdot D\right) \cdot 0.5\right) \cdot c0}{D}}{D \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/
      (/ (* (* (fma (* (/ d (* h w)) d) c0 (* (* (fabs M) D) D)) 0.5) c0) D)
      (* D w))
     (*
      0.5
      (/ (* c0 (sqrt (sqrt (sqrt (sqrt (* (pow M 8.0) (pow M 8.0))))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (((fma(((d / (h * w)) * d), c0, ((fabs(M) * D) * D)) * 0.5) * c0) / D) / (D * w);
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(sqrt((pow(M, 8.0) * pow(M, 8.0))))))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(Float64(d / Float64(h * w)) * d), c0, Float64(Float64(abs(M) * D) * D)) * 0.5) * c0) / D) / Float64(D * w));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt(sqrt(Float64((M ^ 8.0) * (M ^ 8.0))))))) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(N[(N[(d / N[(h * w), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * c0 + N[(N[(N[Abs[M], $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * c0), $MachinePrecision] / D), $MachinePrecision] / N[(D * w), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Sqrt[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 14.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{c0}{w + w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{h \cdot w}\right), w, \left(D \cdot D\right) \cdot \left(\left(c0 \cdot 0.5\right) \cdot \sqrt{{\left(\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0\right)}^{2} - M \cdot M}\right)\right)}{\left(D \cdot D\right) \cdot w}} \]

   4. Taylor expanded in c0 around 0

      \[\leadsto \frac{\color{blue}{c0 \cdot \left(\frac{1}{2} \cdot \left({D}^{2} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}\right) + \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}}{\left(D \cdot D\right) \cdot w} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({D}^{2} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}\right) + \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}}{\left(D \cdot D\right) \cdot w} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{{D}^{2} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \color{blue}{\sqrt{\mathsf{neg}\left({M}^{2}\right)}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      12. lower-*.f64 8.4

          \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(0.5, {D}^{2} \cdot \sqrt{-{M}^{2}}, 0.5 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

   6. Applied rewrites 8.4%

      \[\leadsto \frac{\color{blue}{c0 \cdot \mathsf{fma}\left(0.5, {D}^{2} \cdot \sqrt{-{M}^{2}}, 0.5 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}}{\left(D \cdot D\right) \cdot w} \]

   8. Applied rewrites 40.1%

      \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{fma}\left(\frac{d}{h \cdot w} \cdot d, c0, \left(\left|M\right| \cdot D\right) \cdot D\right) \cdot 0.5\right) \cdot c0}{D}}{D \cdot w}} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. sqrt-fabs-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      14. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      16. sqrt-unprod N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      18. lower-unsound-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   6. Applied rewrites 37.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      7. swap-sqr N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      10. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      12. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot {M}^{2}\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      13. pow-prod-up N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      17. swap-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      20. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      21. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      22. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot {M}^{2}\right)}}}}{w} \]

   8. Applied rewrites 39.4%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8}} \cdot \sqrt{{M}^{8}}}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

      4. lower-*.f64 40.5

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]

   10. Applied rewrites 40.5%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\sqrt{{M}^{8} \cdot {M}^{8}}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 52.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(\frac{d}{h \cdot w} \cdot d, c0, \left(\left|M\right| \cdot D\right) \cdot D\right) \cdot 0.5\right) \cdot c0}{D}}{D \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/
      (/ (* (* (fma (* (/ d (* h w)) d) c0 (* (* (fabs M) D) D)) 0.5) c0) D)
      (* D w))
     (* 0.5 (/ (* c0 (sqrt (sqrt (sqrt (pow M 8.0))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (((fma(((d / (h * w)) * d), c0, ((fabs(M) * D) * D)) * 0.5) * c0) / D) / (D * w);
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(pow(M, 8.0))))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(Float64(d / Float64(h * w)) * d), c0, Float64(Float64(abs(M) * D) * D)) * 0.5) * c0) / D) / Float64(D * w));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt((M ^ 8.0))))) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(N[(N[(d / N[(h * w), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * c0 + N[(N[(N[Abs[M], $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * c0), $MachinePrecision] / D), $MachinePrecision] / N[(D * w), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Power[M, 8.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 14.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{c0}{w + w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{h \cdot w}\right), w, \left(D \cdot D\right) \cdot \left(\left(c0 \cdot 0.5\right) \cdot \sqrt{{\left(\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0\right)}^{2} - M \cdot M}\right)\right)}{\left(D \cdot D\right) \cdot w}} \]

   4. Taylor expanded in c0 around 0

      \[\leadsto \frac{\color{blue}{c0 \cdot \left(\frac{1}{2} \cdot \left({D}^{2} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}\right) + \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}}{\left(D \cdot D\right) \cdot w} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({D}^{2} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}\right) + \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}}{\left(D \cdot D\right) \cdot w} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{{D}^{2} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \color{blue}{\sqrt{\mathsf{neg}\left({M}^{2}\right)}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      12. lower-*.f64 8.4

          \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(0.5, {D}^{2} \cdot \sqrt{-{M}^{2}}, 0.5 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

   6. Applied rewrites 8.4%

      \[\leadsto \frac{\color{blue}{c0 \cdot \mathsf{fma}\left(0.5, {D}^{2} \cdot \sqrt{-{M}^{2}}, 0.5 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}}{\left(D \cdot D\right) \cdot w} \]

   8. Applied rewrites 40.1%

      \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{fma}\left(\frac{d}{h \cdot w} \cdot d, c0, \left(\left|M\right| \cdot D\right) \cdot D\right) \cdot 0.5\right) \cdot c0}{D}}{D \cdot w}} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. sqrt-fabs-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      14. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      16. sqrt-unprod N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      18. lower-unsound-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   6. Applied rewrites 37.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      7. swap-sqr N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      10. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      12. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot {M}^{2}\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      13. pow-prod-up N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      17. swap-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      20. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      21. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      22. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot {M}^{2}\right)}}}}{w} \]

   8. Applied rewrites 39.4%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 52.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{d}{h \cdot w} \cdot d, c0, \left(\left|M\right| \cdot D\right) \cdot D\right) \cdot 0.5\right) \cdot c0}{\left(D \cdot D\right) \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/
      (* (* (fma (* (/ d (* h w)) d) c0 (* (* (fabs M) D) D)) 0.5) c0)
      (* (* D D) w))
     (* 0.5 (/ (* c0 (sqrt (sqrt (sqrt (pow M 8.0))))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = ((fma(((d / (h * w)) * d), c0, ((fabs(M) * D) * D)) * 0.5) * c0) / ((D * D) * w);
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(pow(M, 8.0))))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(d / Float64(h * w)) * d), c0, Float64(Float64(abs(M) * D) * D)) * 0.5) * c0) / Float64(Float64(D * D) * w));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt((M ^ 8.0))))) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(N[(d / N[(h * w), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * c0 + N[(N[(N[Abs[M], $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Power[M, 8.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 14.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{c0}{w + w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{h \cdot w}\right), w, \left(D \cdot D\right) \cdot \left(\left(c0 \cdot 0.5\right) \cdot \sqrt{{\left(\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot c0\right)}^{2} - M \cdot M}\right)\right)}{\left(D \cdot D\right) \cdot w}} \]

   4. Taylor expanded in c0 around 0

      \[\leadsto \frac{\color{blue}{c0 \cdot \left(\frac{1}{2} \cdot \left({D}^{2} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}\right) + \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}}{\left(D \cdot D\right) \cdot w} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({D}^{2} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}\right) + \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}}{\left(D \cdot D\right) \cdot w} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{{D}^{2} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \color{blue}{\sqrt{\mathsf{neg}\left({M}^{2}\right)}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(\frac{1}{2}, {D}^{2} \cdot \sqrt{-{M}^{2}}, \frac{1}{2} \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

      12. lower-*.f64 8.4

          \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(0.5, {D}^{2} \cdot \sqrt{-{M}^{2}}, 0.5 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w} \]

   6. Applied rewrites 8.4%

      \[\leadsto \frac{\color{blue}{c0 \cdot \mathsf{fma}\left(0.5, {D}^{2} \cdot \sqrt{-{M}^{2}}, 0.5 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}}{\left(D \cdot D\right) \cdot w} \]

   8. Applied rewrites 33.6%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\frac{d}{h \cdot w} \cdot d, c0, \left(\left|M\right| \cdot D\right) \cdot D\right) \cdot 0.5\right) \cdot c0}{\left(D \cdot D\right) \cdot w}} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. sqrt-fabs-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      14. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      16. sqrt-unprod N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      18. lower-unsound-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   6. Applied rewrites 37.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      7. swap-sqr N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      10. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      11. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      12. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot {M}^{2}\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      13. pow-prod-up N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

      17. swap-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      20. sqr-neg-rev N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      21. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot \left(M \cdot M\right)\right)}}}}{w} \]

      22. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot {M}^{2}\right)}}}}{w} \]

   8. Applied rewrites 39.4%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 39.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* 0.5 (/ (* c0 (sqrt (sqrt (sqrt (pow M 8.0))))) w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * sqrt(sqrt(sqrt(pow(M, 8.0))))) / w);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.5d0 * ((c0 * sqrt(sqrt(sqrt((m ** 8.0d0))))) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.pow(M, 8.0))))) / w);
}

Python

def code(c0, w, h, D, d, M):
	return 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.pow(M, 8.0))))) / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt((M ^ 8.0))))) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt((M ^ 8.0))))) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Sqrt[N[Power[M, 8.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.5%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 14.7

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 14.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   2. sqrt-fabs-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

   4. rem-sqrt-square-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   7. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   9. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   12. lift-neg.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

   14. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

   16. sqrt-unprod N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   17. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   18. lower-unsound-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   19. lower-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   20. lower-unsound-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

6. Applied rewrites 37.2%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
7. Step-by-step derivation

   1. rem-square-sqrt N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

   2. sqrt-unprod N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   7. swap-sqr N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   8. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   9. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   10. sqr-neg-rev N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   11. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   12. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot {M}^{2}\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   13. pow-prod-up N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   17. swap-sqr N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

   18. lift-neg.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

   19. lift-neg.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

   20. sqr-neg-rev N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

   21. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot \left(M \cdot M\right)\right)}}}}{w} \]

   22. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot {M}^{2}\right)}}}}{w} \]

8. Applied rewrites 39.4%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
9. Add Preprocessing

Alternative 13: 37.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot M\right) \cdot M}}}{w} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* 0.5 (/ (* c0 (sqrt (sqrt (* (* (* M M) M) M)))) w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * sqrt(sqrt((((M * M) * M) * M)))) / w);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.5d0 * ((c0 * sqrt(sqrt((((m * m) * m) * m)))) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * Math.sqrt(Math.sqrt((((M * M) * M) * M)))) / w);
}

Python

def code(c0, w, h, D, d, M):
	return 0.5 * ((c0 * math.sqrt(math.sqrt((((M * M) * M) * M)))) / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(Float64(Float64(M * M) * M) * M)))) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.5 * ((c0 * sqrt(sqrt((((M * M) * M) * M)))) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[(N[(N[(M * M), $MachinePrecision] * M), $MachinePrecision] * M), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.5%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 14.7

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 14.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   2. sqrt-fabs-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

   4. rem-sqrt-square-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   7. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   9. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   12. lift-neg.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

   14. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

   16. sqrt-unprod N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   17. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   18. lower-unsound-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   19. lower-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   20. lower-unsound-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

6. Applied rewrites 37.2%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   4. swap-sqr N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)}}}{w} \]

   5. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)}}}{w} \]

   6. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)}}}{w} \]

   7. sqr-neg-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(M \cdot M\right) \cdot \left(M \cdot M\right)}}}{w} \]

   8. associate-*r* N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot M\right) \cdot M}}}{w} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot M\right) \cdot M}}}{w} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot M\right) \cdot M}}}{w} \]

   11. lift-*.f64 37.2

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot M\right) \cdot M}}}{w} \]

8. Applied rewrites 37.2%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot M\right) \cdot M}}}{w} \]
9. Add Preprocessing

Alternative 14: 32.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{c0 \cdot \sqrt{M \cdot M}}{w} \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* 0.5 (/ (* c0 (sqrt (* M M))) w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * sqrt((M * M))) / w);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.5d0 * ((c0 * sqrt((m * m))) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * Math.sqrt((M * M))) / w);
}

Python

def code(c0, w, h, D, d, M):
	return 0.5 * ((c0 * math.sqrt((M * M))) / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.5 * Float64(Float64(c0 * sqrt(Float64(M * M))) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.5 * ((c0 * sqrt((M * M))) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.5 * N[(N[(c0 * N[Sqrt[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.5%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 14.7

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 14.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   2. sqrt-fabs-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

   4. rem-sqrt-square-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   7. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   9. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   12. lift-neg.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

   14. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

   16. sqrt-unprod N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   17. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   18. lower-unsound-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   19. lower-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   20. lower-unsound-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

6. Applied rewrites 37.2%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
7. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   3. rem-sqrt-square N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\left|\left(-M\right) \cdot M\right|}}{w} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\left|\left(-M\right) \cdot M\right|}}{w} \]

   5. fabs-mul N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\left|-M\right| \cdot \left|M\right|}}{w} \]

   6. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\left|\mathsf{neg}\left(M\right)\right| \cdot \left|M\right|}}{w} \]

   7. neg-fabs N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\left|M\right| \cdot \left|M\right|}}{w} \]

   8. sqr-abs-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{M \cdot M}}{w} \]

   9. lift-*.f64 32.4

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{M \cdot M}}{w} \]

8. Applied rewrites 32.4%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{M \cdot M}}{w} \]
9. Add Preprocessing

Alternative 15: 24.2% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{c0}{\frac{1}{\left|M\right|} \cdot w} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* 0.5 (/ c0 (* (/ 1.0 (fabs M)) w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * (c0 / ((1.0 / fabs(M)) * w));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.5d0 * (c0 / ((1.0d0 / abs(m)) * w))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * (c0 / ((1.0 / Math.abs(M)) * w));
}

Python

def code(c0, w, h, D, d, M):
	return 0.5 * (c0 / ((1.0 / math.fabs(M)) * w))

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.5 * Float64(c0 / Float64(Float64(1.0 / abs(M)) * w)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.5 * (c0 / ((1.0 / abs(M)) * w));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.5 * N[(c0 / N[(N[(1.0 / N[Abs[M], $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.5%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 14.7

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 14.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   2. sqrt-fabs-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

   4. rem-sqrt-square-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   7. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   9. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   12. lift-neg.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

   14. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

   16. sqrt-unprod N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   17. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   18. lower-unsound-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   19. lower-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   20. lower-unsound-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

6. Applied rewrites 37.2%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
7. Step-by-step derivation

   1. rem-square-sqrt N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)} \cdot \sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}}{w} \]

   2. sqrt-unprod N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   7. swap-sqr N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   8. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   9. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   10. sqr-neg-rev N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   11. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot \left(M \cdot M\right)\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   12. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{\left({M}^{2} \cdot {M}^{2}\right) \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   13. pow-prod-up N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)\right)}}}}{w} \]

   17. swap-sqr N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(-M\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

   18. lift-neg.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(-M\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

   19. lift-neg.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(\left(\mathsf{neg}\left(M\right)\right) \cdot \left(\mathsf{neg}\left(M\right)\right)\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

   20. sqr-neg-rev N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left(\left(M \cdot M\right) \cdot \left(M \cdot M\right)\right)}}}}{w} \]

   21. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot \left(M \cdot M\right)\right)}}}}{w} \]

   22. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{\left(2 + 2\right)} \cdot \left({M}^{2} \cdot {M}^{2}\right)}}}}{w} \]

8. Applied rewrites 39.4%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]

10. Applied rewrites 24.2%

    \[\leadsto 0.5 \cdot \frac{c0}{\color{blue}{\frac{1}{\left|M\right|} \cdot w}} \]
11. Add Preprocessing

Alternative 16: 24.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \frac{\left|M\right|}{w} \cdot \left(0.5 \cdot c0\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* (/ (fabs M) w) (* 0.5 c0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (fabs(M) / w) * (0.5 * c0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (abs(m) / w) * (0.5d0 * c0)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (Math.abs(M) / w) * (0.5 * c0);
}

Python

def code(c0, w, h, D, d, M):
	return (math.fabs(M) / w) * (0.5 * c0)

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(abs(M) / w) * Float64(0.5 * c0))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (abs(M) / w) * (0.5 * c0);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[Abs[M], $MachinePrecision] / w), $MachinePrecision] * N[(0.5 * c0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.5%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 14.7

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 14.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   2. sqrt-fabs-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

   4. rem-sqrt-square-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   7. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   9. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   12. lift-neg.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

   14. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

   16. sqrt-unprod N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   17. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   18. lower-unsound-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   19. lower-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   20. lower-unsound-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

6. Applied rewrites 37.2%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{\color{blue}{w}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   4. associate-/l* N/A

      \[\leadsto \frac{1}{2} \cdot \left(c0 \cdot \color{blue}{\frac{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w}}\right) \]

   5. associate-*r* N/A

      \[\leadsto \left(\frac{1}{2} \cdot c0\right) \cdot \color{blue}{\frac{\sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w}} \]

8. Applied rewrites 24.0%

   \[\leadsto \frac{\left|M\right|}{w} \cdot \color{blue}{\left(0.5 \cdot c0\right)} \]
9. Add Preprocessing

Alternative 17: 19.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{M \cdot c0}{w} \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* 0.5 (/ (* M c0) w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((M * c0) / w);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.5d0 * ((m * c0) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((M * c0) / w);
}

Python

def code(c0, w, h, D, d, M):
	return 0.5 * ((M * c0) / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.5 * Float64(Float64(M * c0) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.5 * ((M * c0) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.5 * N[(N[(M * c0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.5%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 14.7

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 14.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   2. sqrt-fabs-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

   4. rem-sqrt-square-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   7. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   9. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   12. lift-neg.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

   14. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

   16. sqrt-unprod N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   17. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   18. lower-unsound-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   19. lower-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   20. lower-unsound-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

6. Applied rewrites 37.2%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

7. Taylor expanded in M around 0

   \[\leadsto \frac{1}{2} \cdot \frac{M \cdot c0}{w} \]
8. Step-by-step derivation

   1. lower-*.f64 19.0

      \[\leadsto 0.5 \cdot \frac{M \cdot c0}{w} \]

9. Applied rewrites 19.0%

   \[\leadsto 0.5 \cdot \frac{M \cdot c0}{w} \]
10. Add Preprocessing

Alternative 18: 19.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -3 \cdot 10^{+67}:\\ \;\;\;\;-0.5 \cdot \left(c0 \cdot \frac{M}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{M \cdot c0}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= w -3e+67) (* -0.5 (* c0 (/ M w))) (* -0.5 (/ (* M c0) w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -3e+67) {
		tmp = -0.5 * (c0 * (M / w));
	} else {
		tmp = -0.5 * ((M * c0) / w);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (w <= (-3d+67)) then
        tmp = (-0.5d0) * (c0 * (m / w))
    else
        tmp = (-0.5d0) * ((m * c0) / w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -3e+67) {
		tmp = -0.5 * (c0 * (M / w));
	} else {
		tmp = -0.5 * ((M * c0) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if w <= -3e+67:
		tmp = -0.5 * (c0 * (M / w))
	else:
		tmp = -0.5 * ((M * c0) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (w <= -3e+67)
		tmp = Float64(-0.5 * Float64(c0 * Float64(M / w)));
	else
		tmp = Float64(-0.5 * Float64(Float64(M * c0) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (w <= -3e+67)
		tmp = -0.5 * (c0 * (M / w));
	else
		tmp = -0.5 * ((M * c0) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[w, -3e+67], N[(-0.5 * N[(c0 * N[(M / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(M * c0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -3.0000000000000001e67

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. sqrt-fabs-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      14. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      16. sqrt-unprod N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      18. lower-unsound-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   6. Applied rewrites 37.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   7. Taylor expanded in M around -inf

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{M \cdot c0}{w}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{\color{blue}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{w} \]

      3. lower-*.f64 18.8

         \[\leadsto -0.5 \cdot \frac{M \cdot c0}{w} \]

   9. Applied rewrites 18.8%

      \[\leadsto -0.5 \cdot \color{blue}{\frac{M \cdot c0}{w}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{w} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{w} \]

       3. *-commutative N/A

          \[\leadsto \frac{-1}{2} \cdot \frac{c0 \cdot M}{w} \]

       4. associate-/l* N/A

          \[\leadsto \frac{-1}{2} \cdot \left(c0 \cdot \frac{M}{\color{blue}{w}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \left(c0 \cdot \frac{M}{\color{blue}{w}}\right) \]

       6. lower-/.f64 18.7

          \[\leadsto -0.5 \cdot \left(c0 \cdot \frac{M}{w}\right) \]

   11. Applied rewrites 18.7%

       \[\leadsto -0.5 \cdot \left(c0 \cdot \frac{M}{\color{blue}{w}}\right) \]

   if -3.0000000000000001e67 < w

   1. Initial program 25.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. lower-pow.f64 14.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   4. Applied rewrites 14.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      2. sqrt-fabs-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

      14. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

      16. sqrt-unprod N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      18. lower-unsound-*.f32 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

      20. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   6. Applied rewrites 37.2%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   7. Taylor expanded in M around -inf

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{M \cdot c0}{w}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{\color{blue}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{w} \]

      3. lower-*.f64 18.8

         \[\leadsto -0.5 \cdot \frac{M \cdot c0}{w} \]

   9. Applied rewrites 18.8%

      \[\leadsto -0.5 \cdot \color{blue}{\frac{M \cdot c0}{w}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 18.7% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \left(c0 \cdot \frac{M}{w}\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* -0.5 (* c0 (/ M w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return -0.5 * (c0 * (M / w));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (-0.5d0) * (c0 * (m / w))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return -0.5 * (c0 * (M / w));
}

Python

def code(c0, w, h, D, d, M):
	return -0.5 * (c0 * (M / w))

Julia

function code(c0, w, h, D, d, M)
	return Float64(-0.5 * Float64(c0 * Float64(M / w)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = -0.5 * (c0 * (M / w));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(-0.5 * N[(c0 * N[(M / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.5%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. lower-pow.f64 14.7

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

4. Applied rewrites 14.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   2. sqrt-fabs-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \left|\sqrt{-{M}^{2}}\right|}{w} \]

   4. rem-sqrt-square-rev N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{-{M}^{2}} \cdot \sqrt{-{M}^{2}}}}{w} \]

   7. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   9. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{-{M}^{2}}}}{w} \]

   12. lift-neg.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}}{w} \]

   14. pow2 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\mathsf{neg}\left(M \cdot M\right)} \cdot \sqrt{\mathsf{neg}\left(M \cdot M\right)}}}{w} \]

   16. sqrt-unprod N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   17. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   18. lower-unsound-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   19. lower-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

   20. lower-unsound-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\mathsf{neg}\left(M \cdot M\right)\right) \cdot \left(\mathsf{neg}\left(M \cdot M\right)\right)}}}{w} \]

6. Applied rewrites 37.2%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

7. Taylor expanded in M around -inf

   \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{M \cdot c0}{w}} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{\color{blue}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{w} \]

   3. lower-*.f64 18.8

      \[\leadsto -0.5 \cdot \frac{M \cdot c0}{w} \]

9. Applied rewrites 18.8%

   \[\leadsto -0.5 \cdot \color{blue}{\frac{M \cdot c0}{w}} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{w} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{-1}{2} \cdot \frac{M \cdot c0}{w} \]

    3. *-commutative N/A

       \[\leadsto \frac{-1}{2} \cdot \frac{c0 \cdot M}{w} \]

    4. associate-/l* N/A

       \[\leadsto \frac{-1}{2} \cdot \left(c0 \cdot \frac{M}{\color{blue}{w}}\right) \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{-1}{2} \cdot \left(c0 \cdot \frac{M}{\color{blue}{w}}\right) \]

    6. lower-/.f64 18.7

       \[\leadsto -0.5 \cdot \left(c0 \cdot \frac{M}{w}\right) \]

11. Applied rewrites 18.7%

    \[\leadsto -0.5 \cdot \left(c0 \cdot \frac{M}{\color{blue}{w}}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))