Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.6% → 55.9%

Time: 10.4s

Alternatives: 13

Speedup: 3.1×

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: 25.6% 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: 55.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (* (/ (/ (* c0 (/ d D)) w) h) (/ d D)))
       (t_1 (/ c0 (* 2.0 w)))
       (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D))))
       (t_3 (* (- M) M)))
  (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 (* t_3 t_3)))) 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 t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = -M * M;
	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((t_3 * t_3)))) / w);
	}
	return tmp;
}

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);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = -M * M;
	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((t_3 * t_3)))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (((c0 * (d / D)) / w) / h) * (d / D)
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_3 = -M * M
	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((t_3 * t_3)))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(Float64(c0 * Float64(d / D)) / w) / h) * Float64(d / D))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_3 = Float64(Float64(-M) * M)
	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(t_3 * t_3)))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (((c0 * (d / D)) / w) / h) * (d / D);
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_3 = -M * M;
	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((t_3 * t_3)))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 25.0%

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

   3. Applied rewrites 25.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 25.2%

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

   5. Applied rewrites 25.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 35.3%

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

   7. Applied rewrites 35.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 33.9%

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

   9. Applied rewrites 33.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-/r* N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-/l* N/A

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

       12. lift-/.f64 N/A

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

       13. lower-*.f64 34.0%

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

   11. Applied rewrites 34.0%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-/r* N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-/l* N/A

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

       12. lift-/.f64 N/A

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

       13. lower-*.f64 39.2%

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

   13. Applied rewrites 39.2%

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

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

   1. Initial program 25.6%

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

   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 15.2%

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

   4. Applied rewrites 15.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 37.8%

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

Alternative 2: 55.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D))))
       (t_1 (* (/ (* d c0) (* D (* h w))) (/ d D)))
       (t_2 (* (- M) M)))
  (if (<=
       (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
       INFINITY)
    (* (/ c0 (+ w w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
    (* 0.5 (/ (* c0 (sqrt (sqrt (* t_2 t_2)))) 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 t_1 = ((d * c0) / (D * (h * w))) * (d / D);
	double t_2 = -M * M;
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 / (w + w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_2 * t_2)))) / w);
	}
	return tmp;
}

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

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_1 = ((d * c0) / (D * (h * w))) * (d / D)
	t_2 = -M * M
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = (c0 / (w + w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt((t_2 * t_2)))) / 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)))
	t_1 = Float64(Float64(Float64(d * c0) / Float64(D * Float64(h * w))) * Float64(d / D))
	t_2 = Float64(Float64(-M) * M)
	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)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_2 * t_2)))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_1 = ((d * c0) / (D * (h * w))) * (d / D);
	t_2 = -M * M;
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (c0 / (w + w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_2 * t_2)))) / w);
	end
	tmp_2 = 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]}, Block[{t$95$1 = N[(N[(N[(d * c0), $MachinePrecision] / N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-M) * M), $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[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[(t$95$2 * t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 25.0%

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

   3. Applied rewrites 25.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 25.2%

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

   5. Applied rewrites 25.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 35.3%

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

   7. Applied rewrites 35.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 35.3%

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

   9. Applied rewrites 35.3%

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

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

   1. Initial program 25.6%

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

   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 15.2%

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

   4. Applied rewrites 15.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 37.8%

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

Alternative 3: 54.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \left(-M\right) \cdot M\\ t_2 := c0 \cdot \left(d \cdot d\right)\\ t_3 := \frac{t\_2}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_4 := \frac{t\_2}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D}\\ \mathbf{if}\;t\_0 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(t\_4 + \sqrt{t\_4 \cdot t\_4 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{t\_1 \cdot t\_1}}}{w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (/ c0 (* 2.0 w)))
       (t_1 (* (- M) M))
       (t_2 (* c0 (* d d)))
       (t_3 (/ t_2 (* (* w h) (* D D))))
       (t_4 (/ t_2 (* (* D (* h w)) D))))
  (if (<= (* t_0 (+ t_3 (sqrt (- (* t_3 t_3) (* M M))))) INFINITY)
    (* t_0 (+ t_4 (sqrt (- (* t_4 t_4) (* M M)))))
    (* 0.5 (/ (* c0 (sqrt (sqrt (* t_1 t_1)))) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = -M * M;
	double t_2 = c0 * (d * d);
	double t_3 = t_2 / ((w * h) * (D * D));
	double t_4 = t_2 / ((D * (h * w)) * D);
	double tmp;
	if ((t_0 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (t_4 + sqrt(((t_4 * t_4) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_1 * t_1)))) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = -M * M;
	double t_2 = c0 * (d * d);
	double t_3 = t_2 / ((w * h) * (D * D));
	double t_4 = t_2 / ((D * (h * w)) * D);
	double tmp;
	if ((t_0 * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (t_4 + Math.sqrt(((t_4 * t_4) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.sqrt(Math.sqrt((t_1 * t_1)))) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = -M * M
	t_2 = c0 * (d * d)
	t_3 = t_2 / ((w * h) * (D * D))
	t_4 = t_2 / ((D * (h * w)) * D)
	tmp = 0
	if (t_0 * (t_3 + math.sqrt(((t_3 * t_3) - (M * M))))) <= math.inf:
		tmp = t_0 * (t_4 + math.sqrt(((t_4 * t_4) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.sqrt(math.sqrt((t_1 * t_1)))) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(-M) * M)
	t_2 = Float64(c0 * Float64(d * d))
	t_3 = Float64(t_2 / Float64(Float64(w * h) * Float64(D * D)))
	t_4 = Float64(t_2 / Float64(Float64(D * Float64(h * w)) * D))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(t_4 + sqrt(Float64(Float64(t_4 * t_4) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_1 * t_1)))) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = -M * M;
	t_2 = c0 * (d * d);
	t_3 = t_2 / ((w * h) * (D * D));
	t_4 = t_2 / ((D * (h * w)) * D);
	tmp = 0.0;
	if ((t_0 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= Inf)
		tmp = t_0 * (t_4 + sqrt(((t_4 * t_4) - (M * M))));
	else
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_1 * t_1)))) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-M) * M), $MachinePrecision]}, Block[{t$95$2 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / N[(N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(t$95$4 + N[Sqrt[N[(N[(t$95$4 * t$95$4), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 25.3%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 25.3%

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

   3. Applied rewrites 25.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 25.4%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 25.4%

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

   5. Applied rewrites 25.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.3%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 28.3%

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

   7. Applied rewrites 28.3%

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

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

   1. Initial program 25.6%

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

   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 15.2%

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

   4. Applied rewrites 15.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 37.8%

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

Alternative 4: 54.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 25.6%

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

   3. Applied rewrites 25.6%

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

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

   1. Initial program 25.6%

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

   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 15.2%

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

   4. Applied rewrites 15.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 37.8%

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

Alternative 5: 54.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.6%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 24.9%

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

   3. Applied rewrites 26.4%

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

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

   1. Initial program 25.6%

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

   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 15.2%

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

   4. Applied rewrites 15.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 37.8%

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

Alternative 6: 53.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 24.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 24.6%

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

   5. Applied rewrites 24.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 26.6%

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

   7. Applied rewrites 26.6%

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

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

   1. Initial program 25.6%

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

   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 15.2%

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

   4. Applied rewrites 15.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 37.8%

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

Alternative 7: 53.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
  :precision binary64
  (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D))))
       (t_1 (* (* (* D D) w) h))
       (t_2 (* (- M) M)))
  (if (<=
       (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
       INFINITY)
    (*
     (fma
      (/ (* d d) t_1)
      c0
      (sqrt (- (pow (/ (* (* d d) c0) t_1) 2.0) (* M M))))
     (/ c0 (+ w w)))
    (* 0.5 (/ (* c0 (sqrt (sqrt (* t_2 t_2)))) 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 t_1 = ((D * D) * w) * h;
	double t_2 = -M * M;
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = fma(((d * d) / t_1), c0, sqrt((pow((((d * d) * c0) / t_1), 2.0) - (M * M)))) * (c0 / (w + w));
	} else {
		tmp = 0.5 * ((c0 * sqrt(sqrt((t_2 * t_2)))) / 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)))
	t_1 = Float64(Float64(Float64(D * D) * w) * h)
	t_2 = Float64(Float64(-M) * M)
	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(fma(Float64(Float64(d * d) / t_1), c0, sqrt(Float64((Float64(Float64(Float64(d * d) * c0) / t_1) ^ 2.0) - Float64(M * M)))) * Float64(c0 / Float64(w + w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_2 * t_2)))) / 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]}, Block[{t$95$1 = N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]}, Block[{t$95$2 = N[((-M) * M), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(d * d), $MachinePrecision] / t$95$1), $MachinePrecision] * c0 + N[Sqrt[N[(N[Power[N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / t$95$1), $MachinePrecision], 2.0], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[(t$95$2 * t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.6%

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

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

      3. lower-*.f64 25.6%

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

   3. Applied rewrites 25.2%

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

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

   1. Initial program 25.6%

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

   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 15.2%

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

   4. Applied rewrites 15.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 37.8%

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

Alternative 8: 53.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(-M) * M)
	t_1 = Float64(Float64(Float64(D * D) * w) * h)
	t_2 = 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_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(fma(Float64(Float64(d * d) / t_1), c0, sqrt(Float64((Float64(Float64(Float64(d * d) * c0) / t_1) ^ 2.0) - Float64(M * M)))) * c0) / Float64(w + w));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_0 * t_0)))) / w));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.6%

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

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

   3. Applied rewrites 25.3%

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

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

   1. Initial program 25.6%

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

   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 15.2%

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

   4. Applied rewrites 15.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 37.8%

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

Alternative 9: 53.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(-M) * M)
	t_1 = Float64(Float64(Float64(D * D) * w) * h)
	t_2 = 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_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(fma(Float64(Float64(d * d) / t_1), c0, sqrt(Float64((Float64(Float64(Float64(d * d) * c0) / t_1) ^ 2.0) - Float64(M * M)))) / Float64(w + w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_0 * t_0)))) / w));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 25.4%

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

   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 25.6%

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

   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 15.2%

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

   4. Applied rewrites 15.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 37.8%

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

Alternative 10: 52.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(-M) * M)
	t_1 = Float64(Float64(Float64(D * D) * w) * h)
	t_2 = 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_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(fma(Float64(Float64(d / t_1) * d), c0, sqrt(Float64((Float64(Float64(c0 / t_1) * Float64(d * d)) ^ 2.0) - Float64(M * M)))) / Float64(w + w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_0 * t_0)))) / w));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 25.0%

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

   3. Applied rewrites 25.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 25.2%

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

   5. Applied rewrites 25.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 35.3%

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

   7. Applied rewrites 35.3%

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

   9. Applied rewrites 25.8%

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

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

   1. Initial program 25.6%

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

   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 15.2%

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

   4. Applied rewrites 15.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
   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 37.8%

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

Alternative 11: 37.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} t_0 := \left(-M\right) \cdot M\\ 0.5 \cdot \frac{c0 \cdot \sqrt{\sqrt{t\_0 \cdot t\_0}}}{w} \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = -M * M;
	return 0.5 * ((c0 * sqrt(sqrt((t_0 * t_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
    real(8) :: t_0
    t_0 = -m * m
    code = 0.5d0 * ((c0 * sqrt(sqrt((t_0 * t_0)))) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = -M * M;
	return 0.5 * ((c0 * Math.sqrt(Math.sqrt((t_0 * t_0)))) / w);
}

Python

def code(c0, w, h, D, d, M):
	t_0 = -M * M
	return 0.5 * ((c0 * math.sqrt(math.sqrt((t_0 * t_0)))) / w)

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(-M) * M)
	return Float64(0.5 * Float64(Float64(c0 * sqrt(sqrt(Float64(t_0 * t_0)))) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = -M * M;
	tmp = 0.5 * ((c0 * sqrt(sqrt((t_0 * t_0)))) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[((-M) * M), $MachinePrecision]}, N[(0.5 * N[(N[(c0 * N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 25.6%

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

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 15.2%

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

4. Applied rewrites 15.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w}} \]
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 37.8%

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

Alternative 12: 15.2% accurate, 4.9× speedup

Math

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

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (sqrt((-M * M)) * c0) / (w + 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 = (sqrt((-m * m)) * c0) / (w + w)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[Sqrt[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.6%

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

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 15.2%

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

4. Applied rewrites 15.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 15.2%

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

   9. lift-neg.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. pow2 N/A

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

   12. distribute-lft-neg-out N/A

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

   13. lower-*.f64 N/A

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

   14. lower-neg.f64 15.2%

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

6. Applied rewrites 15.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. mult-flip N/A

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

   10. associate-/l* N/A

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

   11. associate-/r* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lower-*.f64 13.3%

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

   15. lift-*.f64 N/A

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

   16. count-2-rev N/A

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

   17. lower-+.f64 13.3%

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

8. Applied rewrites 13.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 15.2%

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

10. Applied rewrites 15.2%

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

Alternative 13: 13.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return sqrt((-M * M)) * (c0 / (w + 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 = sqrt((-m * m)) * (c0 / (w + w))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[Sqrt[N[((-M) * M), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.6%

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

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 15.2%

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

4. Applied rewrites 15.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 15.2%

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

   9. lift-neg.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. pow2 N/A

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

   12. distribute-lft-neg-out N/A

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

   13. lower-*.f64 N/A

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

   14. lower-neg.f64 15.2%

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

6. Applied rewrites 15.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. mult-flip N/A

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

   10. associate-/l* N/A

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

   11. associate-/r* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lower-*.f64 13.3%

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

   15. lift-*.f64 N/A

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

   16. count-2-rev N/A

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

   17. lower-+.f64 13.3%

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

8. Applied rewrites 13.3%

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

Reproduce

herbie shell --seed 2025212 
(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))))))