Henrywood and Agarwal, Equation (13)

Percentage Accurate: 23.5% → 55.3%

Time: 12.0s

Alternatives: 17

Speedup: 5.1×

• could not determine a ground truth

  1. c0 = -1.6503674271681922e+293
  2. w = -1.3551466427730245e-264
  3. h = -2.4425913715842252e-135
  4. D = 1.6538795319918418e-284
  5. d = -9.425343278259293e+306
  6. M = -1.5691715853786762e-229

Specification

Math

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

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: 23.5% accurate, 1.0× speedup

Math

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

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: 55.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot w\right) \cdot D\right) \cdot h}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (* (* d c0) (/ d (* (* (* D w) D) h))))
       (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 (pow (* (pow M 8.0) (pow M 8.0)) 0.25))))
      w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * c0) * (d / (((D * w) * D) * h));
	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(pow((pow(M, 8.0) * pow(M, 8.0)), 0.25)))) / 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 / (((D * w) * D) * h));
	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.pow((Math.pow(M, 8.0) * Math.pow(M, 8.0)), 0.25)))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d * c0) * (d / (((D * w) * D) * h))
	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.pow((math.pow(M, 8.0) * math.pow(M, 8.0)), 0.25)))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d * c0) * Float64(d / Float64(Float64(Float64(D * w) * D) * h)))
	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((Float64((M ^ 8.0) * (M ^ 8.0)) ^ 0.25)))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d * c0) * (d / (((D * w) * D) * h));
	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((((M ^ 8.0) * (M ^ 8.0)) ^ 0.25)))) / 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[(N[(D * w), $MachinePrecision] * D), $MachinePrecision] * h), $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[Power[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision], 0.25], $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 23.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 23.1%

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

      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 22.4%

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

   3. Applied rewrites 22.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 22.5%

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

      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 22.6%

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

   5. Applied rewrites 22.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\color{blue}{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 26.3%

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

      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 28.0%

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

   7. Applied rewrites 28.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot \color{blue}{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right)} - M \cdot M}\right) \]
   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. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 27.7%

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

   9. Applied rewrites 27.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 27.7%

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

   11. Applied rewrites 27.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 30.2%

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

   13. Applied rewrites 30.2%

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

   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 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. swap-sqr N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lift-*.f64 N/A

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

      19. pow-prod-down N/A

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

      20. pow-prod-up N/A

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

      21. lower-pow.f64 N/A

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

   8. Applied rewrites 39.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}}}{w} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{1}{4}\right)}}}}{w} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{\frac{1}{2}}{2}\right)}}}}{w} \]

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 40.7%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]

   10. Applied rewrites 40.7%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 55.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (* (* d c0) (/ d (* (* (* D D) w) h))))
       (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 (pow (* (pow M 8.0) (pow M 8.0)) 0.25))))
      w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * c0) * (d / (((D * D) * w) * h));
	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(pow((pow(M, 8.0) * pow(M, 8.0)), 0.25)))) / 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 / (((D * D) * w) * h));
	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.pow((Math.pow(M, 8.0) * Math.pow(M, 8.0)), 0.25)))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d * c0) * (d / (((D * D) * w) * h))
	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.pow((math.pow(M, 8.0) * math.pow(M, 8.0)), 0.25)))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d * c0) * Float64(d / Float64(Float64(Float64(D * D) * w) * h)))
	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((Float64((M ^ 8.0) * (M ^ 8.0)) ^ 0.25)))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d * c0) * (d / (((D * D) * w) * h));
	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((((M ^ 8.0) * (M ^ 8.0)) ^ 0.25)))) / 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[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $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[Power[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision], 0.25], $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 23.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 23.1%

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

      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 22.4%

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

   3. Applied rewrites 22.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 22.5%

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

      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 22.6%

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

   5. Applied rewrites 22.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} + \sqrt{\color{blue}{\left(\left(d \cdot c0\right) \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 26.3%

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

      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 28.0%

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

   7. Applied rewrites 28.0%

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

   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 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. swap-sqr N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lift-*.f64 N/A

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

      19. pow-prod-down N/A

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

      20. pow-prod-up N/A

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

      21. lower-pow.f64 N/A

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

   8. Applied rewrites 39.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}}}{w} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{1}{4}\right)}}}}{w} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{\frac{1}{2}}{2}\right)}}}}{w} \]

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 40.7%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]

   10. Applied rewrites 40.7%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 54.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot d\right) \cdot c0\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(t\_0 + \sqrt{{t\_0}^{2} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (* (* (/ d (* (* (* D D) w) h)) d) c0))
       (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 (pow (* (pow M 8.0) (pow M 8.0)) 0.25))))
      w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d / (((D * D) * w) * h)) * d) * c0;
	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(pow((pow(M, 8.0) * pow(M, 8.0)), 0.25)))) / 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)) * d) * c0;
	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.pow((Math.pow(M, 8.0) * Math.pow(M, 8.0)), 0.25)))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d / (((D * D) * w) * h)) * d) * c0
	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.pow((math.pow(M, 8.0) * math.pow(M, 8.0)), 0.25)))) / 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)) * d) * c0)
	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((Float64((M ^ 8.0) * (M ^ 8.0)) ^ 0.25)))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d / (((D * D) * w) * h)) * d) * c0;
	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((((M ^ 8.0) * (M ^ 8.0)) ^ 0.25)))) / 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] * d), $MachinePrecision] * c0), $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[Power[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision], 0.25], $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 23.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 23.1%

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

   3. Applied rewrites 23.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 23.3%

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

   5. Applied rewrites 23.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 32.9%

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

   7. Applied rewrites 32.9%

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

   9. Applied rewrites 28.2%

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

   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 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. swap-sqr N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lift-*.f64 N/A

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

      19. pow-prod-down N/A

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

      20. pow-prod-up N/A

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

      21. lower-pow.f64 N/A

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

   8. Applied rewrites 39.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}}}{w} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{1}{4}\right)}}}}{w} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{\frac{1}{2}}{2}\right)}}}}{w} \]

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 40.7%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]

   10. Applied rewrites 40.7%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 54.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{w + w} \cdot \mathsf{fma}\left(\frac{d}{\left(h \cdot D\right) \cdot w}, c0 \cdot \frac{d}{D}, \sqrt{{\left(\left(\frac{d}{\left(\left(D \cdot h\right) \cdot w\right) \cdot D} \cdot d\right) \cdot c0\right)}^{2} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w}\\ \end{array} \]

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)
    (*
     (/ c0 (+ w w))
     (fma
      (/ d (* (* h D) w))
      (* c0 (/ d D))
      (sqrt
       (- (pow (* (* (/ d (* (* (* D h) w) D)) d) c0) 2.0) (* M M)))))
    (*
     0.5
     (/
      (* c0 (sqrt (sqrt (pow (* (pow M 8.0) (pow M 8.0)) 0.25))))
      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 = (c0 / (w + w)) * fma((d / ((h * D) * w)), (c0 * (d / D)), sqrt((pow((((d / (((D * h) * w) * D)) * d) * c0), 2.0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(pow((pow(M, 8.0) * pow(M, 8.0)), 0.25)))) / 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(c0 / Float64(w + w)) * fma(Float64(d / Float64(Float64(h * D) * w)), Float64(c0 * Float64(d / D)), sqrt(Float64((Float64(Float64(Float64(d / Float64(Float64(Float64(D * h) * w) * D)) * d) * c0) ^ 2.0) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt((Float64((M ^ 8.0) * (M ^ 8.0)) ^ 0.25)))) / 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[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[(d / N[(N[(h * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * N[(c0 * N[(d / D), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(N[(N[(d / N[(N[(N[(D * h), $MachinePrecision] * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * d), $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[Power[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision], 0.25], $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 23.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 23.1%

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

   3. Applied rewrites 23.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 23.3%

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

   5. Applied rewrites 23.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 32.9%

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

   7. Applied rewrites 32.9%

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

   9. Applied rewrites 26.4%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. associate-*l* N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 29.1%

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 29.1%

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

   11. Applied rewrites 29.1%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lower-+.f64 29.1%

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

   13. Applied rewrites 29.1%

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

   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 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. swap-sqr N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lift-*.f64 N/A

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

      19. pow-prod-down N/A

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

      20. pow-prod-up N/A

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

      21. lower-pow.f64 N/A

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

   8. Applied rewrites 39.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}}}{w} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{1}{4}\right)}}}}{w} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{\frac{1}{2}}{2}\right)}}}}{w} \]

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 40.7%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]

   10. Applied rewrites 40.7%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 53.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(D \cdot D\right) \cdot w\right) \cdot h\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(\frac{d \cdot d}{t\_0}, c0, \sqrt{{\left(\frac{\left(d \cdot d\right) \cdot c0}{t\_0}\right)}^{2} - M \cdot M}\right)}{w + w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (* (* (* D D) w) h))
       (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
       (/ (* d d) t_0)
       c0
       (sqrt (- (pow (/ (* (* d d) c0) t_0) 2.0) (* M M))))
      (+ w w)))
    (*
     0.5
     (/
      (* c0 (sqrt (sqrt (pow (* (pow M 8.0) (pow M 8.0)) 0.25))))
      w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((D * D) * w) * h;
	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(((d * d) / t_0), c0, sqrt((pow((((d * d) * c0) / t_0), 2.0) - (M * M)))) / (w + w));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(pow((pow(M, 8.0) * pow(M, 8.0)), 0.25)))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(D * D) * w) * h)
	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(Float64(Float64(d * d) / t_0), c0, sqrt(Float64((Float64(Float64(Float64(d * d) * c0) / t_0) ^ 2.0) - Float64(M * M)))) / Float64(w + w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt((Float64((M ^ 8.0) * (M ^ 8.0)) ^ 0.25)))) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $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[(N[(N[(d * d), $MachinePrecision] / t$95$0), $MachinePrecision] * c0 + N[Sqrt[N[(N[Power[N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / t$95$0), $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[Power[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision], 0.25], $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 23.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 23.1%

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

   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 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. swap-sqr N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lift-*.f64 N/A

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

      19. pow-prod-down N/A

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

      20. pow-prod-up N/A

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

      21. lower-pow.f64 N/A

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

   8. Applied rewrites 39.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}}}{w} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{1}{4}\right)}}}}{w} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{\frac{1}{2}}{2}\right)}}}}{w} \]

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 40.7%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]

   10. Applied rewrites 40.7%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 53.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot d\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(t\_0, c0, \sqrt{{\left(t\_0 \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (* (/ d (* (* (* D D) w) h)) d))
       (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 c0 (sqrt (- (pow (* t_0 c0) 2.0) (* M M)))) (+ w w)))
    (*
     0.5
     (/
      (* c0 (sqrt (sqrt (pow (* (pow M 8.0) (pow M 8.0)) 0.25))))
      w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / (((D * D) * w) * h)) * d;
	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, c0, sqrt((pow((t_0 * c0), 2.0) - (M * M)))) / (w + w));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(pow((pow(M, 8.0) * pow(M, 8.0)), 0.25)))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / Float64(Float64(Float64(D * D) * w) * h)) * d)
	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, c0, sqrt(Float64((Float64(t_0 * c0) ^ 2.0) - Float64(M * M)))) / Float64(w + w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt((Float64((M ^ 8.0) * (M ^ 8.0)) ^ 0.25)))) / 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] * d), $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 * c0 + N[Sqrt[N[(N[Power[N[(t$95$0 * c0), $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[Power[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision], 0.25], $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 23.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 23.1%

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

   3. Applied rewrites 23.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 23.3%

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

   5. Applied rewrites 23.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 32.9%

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

   7. Applied rewrites 32.9%

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

   9. Applied rewrites 27.6%

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

   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 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. swap-sqr N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lift-*.f64 N/A

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

      19. pow-prod-down N/A

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

      20. pow-prod-up N/A

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

      21. lower-pow.f64 N/A

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

   8. Applied rewrites 39.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}}}{w} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{1}{4}\right)}}}}{w} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{\frac{1}{2}}{2}\right)}}}}{w} \]

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 40.7%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]

   10. Applied rewrites 40.7%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 53.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(D \cdot \left(D \cdot h\right)\right) \cdot w\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0 \cdot \frac{d}{t\_0}, d, \sqrt{{\left(\frac{d \cdot d}{t\_0} \cdot c0\right)}^{2} - M \cdot M}\right)}{w + w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (* (* D (* D h)) w))
       (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
       (* c0 (/ d t_0))
       d
       (sqrt (- (pow (* (/ (* d d) t_0) c0) 2.0) (* M M))))
      (+ w w)))
    (*
     0.5
     (/
      (* c0 (sqrt (sqrt (pow (* (pow M 8.0) (pow M 8.0)) 0.25))))
      w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (D * (D * h)) * w;
	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((c0 * (d / t_0)), d, sqrt((pow((((d * d) / t_0) * c0), 2.0) - (M * M)))) / (w + w));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(pow((pow(M, 8.0) * pow(M, 8.0)), 0.25)))) / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(D * Float64(D * h)) * w)
	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(Float64(c0 * Float64(d / t_0)), d, sqrt(Float64((Float64(Float64(Float64(d * d) / t_0) * c0) ^ 2.0) - Float64(M * M)))) / Float64(w + w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt((Float64((M ^ 8.0) * (M ^ 8.0)) ^ 0.25)))) / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * w), $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[(N[(c0 * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision] * d + N[Sqrt[N[(N[Power[N[(N[(N[(d * d), $MachinePrecision] / t$95$0), $MachinePrecision] * c0), $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[Power[N[(N[Power[M, 8.0], $MachinePrecision] * N[Power[M, 8.0], $MachinePrecision]), $MachinePrecision], 0.25], $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 23.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 23.1%

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

   3. Applied rewrites 23.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 23.3%

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

   5. Applied rewrites 23.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D} + \sqrt{\color{blue}{\left(\frac{d \cdot c0}{D \cdot \left(h \cdot w\right)} \cdot \frac{d}{D}\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 32.9%

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

   7. Applied rewrites 32.9%

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

   9. Applied rewrites 26.4%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. associate-*l* N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 29.1%

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 29.1%

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

   11. Applied rewrites 29.1%

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

   13. Applied rewrites 25.0%

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

   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 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. swap-sqr N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lift-*.f64 N/A

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

      19. pow-prod-down N/A

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

      20. pow-prod-up N/A

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

      21. lower-pow.f64 N/A

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

   8. Applied rewrites 39.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}}}{w} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{1}{4}\right)}}}}{w} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{\frac{1}{2}}{2}\right)}}}}{w} \]

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 40.7%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]

   10. Applied rewrites 40.7%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 51.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|M\right|\right)}^{8}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - \left|M\right| \cdot \left|M\right|}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(t\_2 + \sqrt{\left|M\right|} \cdot \sqrt{\left|\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right| - \left|M\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left(t\_0 \cdot t\_0\right)}^{0.25}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (pow (fabs M) 8.0))
       (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) (* (fabs M) (fabs M))))))
       INFINITY)
    (*
     t_1
     (+
      t_2
      (*
       (sqrt (fabs M))
       (sqrt
        (- (fabs (/ (* (* d d) c0) (* (* (* D D) w) h))) (fabs M))))))
    (* 0.5 (/ (* c0 (sqrt (sqrt (pow (* t_0 t_0) 0.25)))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow(fabs(M), 8.0);
	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) - (fabs(M) * fabs(M)))))) <= ((double) INFINITY)) {
		tmp = t_1 * (t_2 + (sqrt(fabs(M)) * sqrt((fabs((((d * d) * c0) / (((D * D) * w) * h))) - fabs(M)))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(pow((t_0 * t_0), 0.25)))) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = Math.pow(Math.abs(M), 8.0);
	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) - (Math.abs(M) * Math.abs(M)))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * (t_2 + (Math.sqrt(Math.abs(M)) * Math.sqrt((Math.abs((((d * d) * c0) / (((D * D) * w) * h))) - Math.abs(M)))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.pow((t_0 * t_0), 0.25)))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow(math.fabs(M), 8.0)
	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) - (math.fabs(M) * math.fabs(M)))))) <= math.inf:
		tmp = t_1 * (t_2 + (math.sqrt(math.fabs(M)) * math.sqrt((math.fabs((((d * d) * c0) / (((D * D) * w) * h))) - math.fabs(M)))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt(math.pow((t_0 * t_0), 0.25)))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = abs(M) ^ 8.0
	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(abs(M) * abs(M)))))) <= Inf)
		tmp = Float64(t_1 * Float64(t_2 + Float64(sqrt(abs(M)) * sqrt(Float64(abs(Float64(Float64(Float64(d * d) * c0) / Float64(Float64(Float64(D * D) * w) * h))) - abs(M))))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt((Float64(t_0 * t_0) ^ 0.25)))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = abs(M) ^ 8.0;
	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) - (abs(M) * abs(M)))))) <= Inf)
		tmp = t_1 * (t_2 + (sqrt(abs(M)) * sqrt((abs((((d * d) * c0) / (((D * D) * w) * h))) - abs(M)))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt(((t_0 * t_0) ^ 0.25)))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[Abs[M], $MachinePrecision], 8.0], $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[(N[Abs[M], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(t$95$2 + N[(N[Sqrt[N[Abs[M], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Abs[M], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $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 23.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-abs-rev N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. sqrt-prod N/A

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

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

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

   3. Applied rewrites 29.5%

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

   4. Taylor expanded in d around 0

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

   6. Applied rewrites 13.8%

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

   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 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. swap-sqr N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lift-*.f64 N/A

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

      19. pow-prod-down N/A

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

      20. pow-prod-up N/A

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

      21. lower-pow.f64 N/A

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

   8. Applied rewrites 39.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}}}{w} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{1}{4}\right)}}}}{w} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8}\right)}^{\left(\frac{\frac{1}{2}}{2} + \frac{\frac{1}{2}}{2}\right)}}}}{w} \]

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 40.7%

          \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]

   10. Applied rewrites 40.7%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{{\left({M}^{8} \cdot {M}^{8}\right)}^{0.25}}}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 50.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (log (fabs (fabs M))))
       (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) (* (fabs M) (fabs M))))))
       INFINITY)
    (*
     t_1
     (+
      t_2
      (*
       (sqrt (fabs M))
       (sqrt
        (- (fabs (/ (* (* d d) c0) (* (* (* D D) w) h))) (fabs M))))))
    (* 0.5 (/ (* c0 (+ (sinh t_0) (cosh t_0))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = log(fabs(fabs(M)));
	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) - (fabs(M) * fabs(M)))))) <= ((double) INFINITY)) {
		tmp = t_1 * (t_2 + (sqrt(fabs(M)) * sqrt((fabs((((d * d) * c0) / (((D * D) * w) * h))) - fabs(M)))));
	} else {
		tmp = 0.5 * ((c0 * (sinh(t_0) + cosh(t_0))) / w);
	}
	return tmp;
}

Java

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

Python

def code(c0, w, h, D, d, M):
	t_0 = math.log(math.fabs(math.fabs(M)))
	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) - (math.fabs(M) * math.fabs(M)))))) <= math.inf:
		tmp = t_1 * (t_2 + (math.sqrt(math.fabs(M)) * math.sqrt((math.fabs((((d * d) * c0) / (((D * D) * w) * h))) - math.fabs(M)))))
	else:
		tmp = 0.5 * ((c0 * (math.sinh(t_0) + math.cosh(t_0))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = log(abs(abs(M)))
	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(abs(M) * abs(M)))))) <= Inf)
		tmp = Float64(t_1 * Float64(t_2 + Float64(sqrt(abs(M)) * sqrt(Float64(abs(Float64(Float64(Float64(d * d) * c0) / Float64(Float64(Float64(D * D) * w) * h))) - abs(M))))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * Float64(sinh(t_0) + cosh(t_0))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = log(abs(abs(M)));
	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) - (abs(M) * abs(M)))))) <= Inf)
		tmp = t_1 * (t_2 + (sqrt(abs(M)) * sqrt((abs((((d * d) * c0) / (((D * D) * w) * h))) - abs(M)))));
	else
		tmp = 0.5 * ((c0 * (sinh(t_0) + cosh(t_0))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Log[N[Abs[N[Abs[M], $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[(N[Abs[M], $MachinePrecision] * N[Abs[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(t$95$2 + N[(N[Sqrt[N[Abs[M], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Abs[M], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[(N[Sinh[t$95$0], $MachinePrecision] + N[Cosh[t$95$0], $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 23.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-abs-rev N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. sqrt-prod N/A

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

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

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

   3. Applied rewrites 29.5%

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

   4. Taylor expanded in d around 0

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

   6. Applied rewrites 13.8%

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

   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 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. swap-sqr N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lift-*.f64 N/A

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

      19. pow-prod-down N/A

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

      20. pow-prod-up N/A

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

      21. lower-pow.f64 N/A

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

   8. Applied rewrites 39.5%

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

   10. Applied rewrites 36.1%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \left(\sinh \log \left(\left|M\right|\right) + \cosh \log \left(\left|M\right|\right)\right)}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 40.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \log \left(\left|M\right|\right)\\ \mathbf{if}\;M \cdot M \leq 5 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \left(\sinh t\_0 + \cosh t\_0\right)}{w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (log (fabs M))))
  (if (<= (* M M) 5e+84)
    (* 0.5 (/ (* c0 (+ (sinh t_0) (cosh t_0))) 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 = log(fabs(M));
	double tmp;
	if ((M * M) <= 5e+84) {
		tmp = 0.5 * ((c0 * (sinh(t_0) + cosh(t_0))) / w);
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt(pow(M, 8.0))))) / 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) :: t_0
    real(8) :: tmp
    t_0 = log(abs(m))
    if ((m * m) <= 5d+84) then
        tmp = 0.5d0 * ((c0 * (sinh(t_0) + cosh(t_0))) / w)
    else
        tmp = 0.5d0 * ((c0 * sqrt(sqrt(sqrt((m ** 8.0d0))))) / 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 t_0 = Math.log(Math.abs(M));
	double tmp;
	if ((M * M) <= 5e+84) {
		tmp = 0.5 * ((c0 * (Math.sinh(t_0) + Math.cosh(t_0))) / w);
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt(Math.sqrt(Math.pow(M, 8.0))))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.log(math.fabs(M))
	tmp = 0
	if (M * M) <= 5e+84:
		tmp = 0.5 * ((c0 * (math.sinh(t_0) + math.cosh(t_0))) / w)
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt(math.sqrt(math.pow(M, 8.0))))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = log(abs(M))
	tmp = 0.0
	if (Float64(M * M) <= 5e+84)
		tmp = Float64(0.5 * Float64(Float64(c0 * Float64(sinh(t_0) + cosh(t_0))) / w));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(sqrt((M ^ 8.0))))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = log(abs(M));
	tmp = 0.0;
	if ((M * M) <= 5e+84)
		tmp = 0.5 * ((c0 * (sinh(t_0) + cosh(t_0))) / w);
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt(sqrt((M ^ 8.0))))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Log[N[Abs[M], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(M * M), $MachinePrecision], 5e+84], N[(0.5 * N[(N[(c0 * N[(N[Sinh[t$95$0], $MachinePrecision] + N[Cosh[t$95$0], $MachinePrecision]), $MachinePrecision]), $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 M M) < 5.0000000000000001e84

   1. Initial program 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. swap-sqr N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lift-*.f64 N/A

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

      19. pow-prod-down N/A

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

      20. pow-prod-up N/A

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

      21. lower-pow.f64 N/A

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

   8. Applied rewrites 39.5%

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

   10. Applied rewrites 36.1%

       \[\leadsto 0.5 \cdot \frac{c0 \cdot \left(\sinh \log \left(\left|M\right|\right) + \cosh \log \left(\left|M\right|\right)\right)}{w} \]

   if 5.0000000000000001e84 < (*.f64 M M)

   1. Initial program 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. swap-sqr N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lift-*.f64 N/A

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

      19. pow-prod-down N/A

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

      20. pow-prod-up N/A

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

      21. lower-pow.f64 N/A

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

   8. Applied rewrites 39.5%

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

Alternative 11: 39.5% accurate, 2.4× speedup

Math

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

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 23.5%

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

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 14.5%

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

4. Applied rewrites 14.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
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. pow2 N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. swap-sqr N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. sqr-neg-rev N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. pow-prod-down N/A

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

   17. pow-prod-up N/A

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

   18. lift-*.f64 N/A

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

   19. pow-prod-down N/A

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

   20. pow-prod-up N/A

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

   21. lower-pow.f64 N/A

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

8. Applied rewrites 39.5%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\sqrt{{M}^{8}}}}}{w} \]
9. Add Preprocessing

Alternative 12: 37.2% accurate, 3.4× speedup

Math

\[0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(M \cdot M\right) \cdot M\right) \cdot M}}}{w} \]

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 23.5%

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

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 14.5%

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

4. Applied rewrites 14.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
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. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. sqr-neg-rev N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. cube-unmult N/A

       \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\sqrt{{M}^{3} \cdot M}}}{w} \]

   14. pow3 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-*.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 13: 31.9% accurate, 5.1× speedup

Math

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

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 23.5%

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

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 14.5%

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

4. Applied rewrites 14.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
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 31.9%

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

8. Applied rewrites 31.9%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{M \cdot M}}{w} \]
9. Add Preprocessing

Alternative 14: 23.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;d \cdot d \leq 7 \cdot 10^{-145}:\\ \;\;\;\;\left|M\right| \cdot \frac{c0}{w + w}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|M\right|}{w} \cdot \left(0.5 \cdot c0\right)\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (if (<= (* d d) 7e-145)
  (* (fabs M) (/ c0 (+ w w)))
  (* (/ (fabs M) w) (* 0.5 c0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 7e-145) {
		tmp = fabs(M) * (c0 / (w + w));
	} else {
		tmp = (fabs(M) / w) * (0.5 * c0);
	}
	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 ((d_1 * d_1) <= 7d-145) then
        tmp = abs(m) * (c0 / (w + w))
    else
        tmp = (abs(m) / w) * (0.5d0 * c0)
    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 ((d * d) <= 7e-145) {
		tmp = Math.abs(M) * (c0 / (w + w));
	} else {
		tmp = (Math.abs(M) / w) * (0.5 * c0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d * d) <= 7e-145:
		tmp = math.fabs(M) * (c0 / (w + w))
	else:
		tmp = (math.fabs(M) / w) * (0.5 * c0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(d * d) <= 7e-145)
		tmp = Float64(abs(M) * Float64(c0 / Float64(w + w)));
	else
		tmp = Float64(Float64(abs(M) / w) * Float64(0.5 * c0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d * d) <= 7e-145)
		tmp = abs(M) * (c0 / (w + w));
	else
		tmp = (abs(M) / w) * (0.5 * c0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(d * d), $MachinePrecision], 7e-145], N[(N[Abs[M], $MachinePrecision] * N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[M], $MachinePrecision] / w), $MachinePrecision] * N[(0.5 * c0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 d d) < 6.9999999999999999e-145

   1. Initial program 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

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

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   8. Applied rewrites 23.3%

      \[\leadsto \left|M\right| \cdot \color{blue}{\frac{c0}{w + w}} \]

   if 6.9999999999999999e-145 < (*.f64 d d)

   1. Initial program 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

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

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   8. Applied rewrites 23.3%

      \[\leadsto \frac{\left|M\right|}{w} \cdot \color{blue}{\left(0.5 \cdot c0\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 23.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;w \leq 1.35 \cdot 10^{+22}:\\ \;\;\;\;\left|M\right| \cdot \frac{c0}{w + w}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(c0 \cdot \frac{M}{w}\right)\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (if (<= w 1.35e+22)
  (* (fabs M) (/ c0 (+ w w)))
  (* -0.5 (* c0 (/ M w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= 1.35e+22) {
		tmp = fabs(M) * (c0 / (w + w));
	} else {
		tmp = -0.5 * (c0 * (M / 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 <= 1.35d+22) then
        tmp = abs(m) * (c0 / (w + w))
    else
        tmp = (-0.5d0) * (c0 * (m / 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 <= 1.35e+22) {
		tmp = Math.abs(M) * (c0 / (w + w));
	} else {
		tmp = -0.5 * (c0 * (M / w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if w <= 1.35e+22:
		tmp = math.fabs(M) * (c0 / (w + w))
	else:
		tmp = -0.5 * (c0 * (M / w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (w <= 1.35e+22)
		tmp = Float64(abs(M) * Float64(c0 / Float64(w + w)));
	else
		tmp = Float64(-0.5 * Float64(c0 * Float64(M / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (w <= 1.35e+22)
		tmp = abs(M) * (c0 / (w + w));
	else
		tmp = -0.5 * (c0 * (M / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[w, 1.35e+22], N[(N[Abs[M], $MachinePrecision] * N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c0 * N[(M / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1.3500000000000001e22

   1. Initial program 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

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

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{\left(\left(-M\right) \cdot M\right) \cdot \left(\left(-M\right) \cdot M\right)}}}{w} \]

   8. Applied rewrites 23.3%

      \[\leadsto \left|M\right| \cdot \color{blue}{\frac{c0}{w + w}} \]

   if 1.3500000000000001e22 < w

   1. Initial program 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 18.5%

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

   9. Applied rewrites 18.5%

      \[\leadsto -0.5 \cdot \color{blue}{\frac{M \cdot c0}{w}} \]
   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 18.2%

          \[\leadsto -0.5 \cdot \left(c0 \cdot \frac{M}{w}\right) \]

   11. Applied rewrites 18.2%

       \[\leadsto -0.5 \cdot \left(c0 \cdot \frac{M}{\color{blue}{w}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 22.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{-196}:\\ \;\;\;\;-0.5 \cdot \left(c0 \cdot \frac{M}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{M \cdot c0}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (if (<= M 3e-196) (* -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 (M <= 3e-196) {
		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 (m <= 3d-196) 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 (M <= 3e-196) {
		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 M <= 3e-196:
		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 (M <= 3e-196)
		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 (M <= 3e-196)
		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[M, 3e-196], 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 M < 2.9999999999999999e-196

   1. Initial program 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 18.5%

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

   9. Applied rewrites 18.5%

      \[\leadsto -0.5 \cdot \color{blue}{\frac{M \cdot c0}{w}} \]
   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 18.2%

          \[\leadsto -0.5 \cdot \left(c0 \cdot \frac{M}{w}\right) \]

   11. Applied rewrites 18.2%

       \[\leadsto -0.5 \cdot \left(c0 \cdot \frac{M}{\color{blue}{w}}\right) \]

   if 2.9999999999999999e-196 < M

   1. Initial program 23.5%

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

   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 14.5%

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

   4. Applied rewrites 14.5%

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

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

      1. lower-*.f64 18.2%

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

   9. Applied rewrites 18.2%

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

Alternative 17: 18.2% accurate, 7.4× speedup

Math

\[-0.5 \cdot \left(c0 \cdot \frac{M}{w}\right) \]

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 23.5%

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

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 14.5%

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

4. Applied rewrites 14.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
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 18.5%

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

9. Applied rewrites 18.5%

   \[\leadsto -0.5 \cdot \color{blue}{\frac{M \cdot c0}{w}} \]
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 18.2%

       \[\leadsto -0.5 \cdot \left(c0 \cdot \frac{M}{w}\right) \]

11. Applied rewrites 18.2%

    \[\leadsto -0.5 \cdot \left(c0 \cdot \frac{M}{\color{blue}{w}}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025214 
(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))))))