Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.6% → 54.6%

Time: 31.2s

Alternatives: 5

Speedup: 151.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    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: 24.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    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: 54.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{t\_0 + \sqrt{{t\_0}^{2} - {M}^{2}}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* w h)) (pow (/ d D) 2.0)))
        (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 (/ (+ t_0 (sqrt (- (pow t_0 2.0) (pow M 2.0)))) (* 2.0 w)))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * h)) * pow((d / D), 2.0);
	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 * ((t_0 + sqrt((pow(t_0, 2.0) - pow(M, 2.0)))) / (2.0 * w));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * h)) * Math.pow((d / D), 2.0);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = c0 * ((t_0 + Math.sqrt((Math.pow(t_0, 2.0) - Math.pow(M, 2.0)))) / (2.0 * w));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / (w * h)) * math.pow((d / D), 2.0)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = c0 * ((t_0 + math.sqrt((math.pow(t_0, 2.0) - math.pow(M, 2.0)))) / (2.0 * w))
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(c0 * N[(N[(t$95$0 + N[Sqrt[N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

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 73.1%

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

   3. Simplified 64.8%

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

      1. fma-undefine 71.7%

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

      2. associate-*r/ 69.3%

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

      3. *-commutative 69.3%

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

      4. associate-*r* 68.2%

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

      5. associate-*r* 68.1%

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

      6. associate-/l* 64.9%

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

      7. frac-times 63.5%

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

      8. times-frac 65.8%

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

      9. pow2 65.8%

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

   6. Applied egg-rr 74.2%

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

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

   1. Initial program 0.0%

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

   3. Simplified 2.3%

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

   6. Applied egg-rr 25.4%

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

      1. associate-/r* 25.5%

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

      2. associate-*l/ 24.4%

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

      3. associate-/l* 24.9%

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

      4. associate-*l/ 23.8%

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

      5. associate-/l* 24.9%

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

   8. Simplified 24.9%

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

   9. Taylor expanded in c0 around inf 14.3%

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

   10. Taylor expanded in d around -inf 1.9%

       \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left({d}^{2} \cdot \left(-1 \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)}{w}} \]
   11. Step-by-step derivation

       1. associate-/l* 1.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(c0 \cdot \frac{{d}^{2} \cdot \left(-1 \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right)} \]

       2. distribute-lft1-in 1.9%

          \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}}{w}\right) \]

       3. metadata-eval 1.9%

          \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \left(\color{blue}{0} \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

       4. mul0-lft 22.1%

          \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \color{blue}{0}}{w}\right) \]

   12. Simplified 22.1%

       \[\leadsto \color{blue}{0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot 0}{w}\right)} \]

   13. Taylor expanded in c0 around 0 43.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 55.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

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_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 INFINITY) t_1 0.0)))

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)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 0.0;
	}
	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)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 0.0;
	}
	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_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = 0.0
	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(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = 0.0;
	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_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = 0.0;
	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[(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]}, If[LessEqual[t$95$1, Infinity], t$95$1, 0.0]]]

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 73.1%

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

   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 0.0%

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

   3. Simplified 2.3%

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

   6. Applied egg-rr 25.4%

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

      1. associate-/r* 25.5%

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

      2. associate-*l/ 24.4%

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

      3. associate-/l* 24.9%

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

      4. associate-*l/ 23.8%

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

      5. associate-/l* 24.9%

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

   8. Simplified 24.9%

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

   9. Taylor expanded in c0 around inf 14.3%

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

   10. Taylor expanded in d around -inf 1.9%

       \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left({d}^{2} \cdot \left(-1 \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)}{w}} \]
   11. Step-by-step derivation

       1. associate-/l* 1.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(c0 \cdot \frac{{d}^{2} \cdot \left(-1 \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right)} \]

       2. distribute-lft1-in 1.9%

          \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}}{w}\right) \]

       3. metadata-eval 1.9%

          \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \left(\color{blue}{0} \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

       4. mul0-lft 22.1%

          \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \color{blue}{0}}{w}\right) \]

   12. Simplified 22.1%

       \[\leadsto \color{blue}{0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot 0}{w}\right)} \]

   13. Taylor expanded in c0 around 0 43.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 34.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 5.8 \cdot 10^{+160}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{d}{D} \cdot \sqrt{\frac{c0 \cdot M}{w \cdot h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 5.8e+160)
   0.0
   (* (/ c0 (* 2.0 w)) (* (/ d D) (sqrt (/ (* c0 M) (* w h)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 5.8e+160) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((d / D) * sqrt(((c0 * M) / (w * h))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 5.8d+160) then
        tmp = 0.0d0
    else
        tmp = (c0 / (2.0d0 * w)) * ((d_1 / d) * sqrt(((c0 * m) / (w * h))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 5.8e+160) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((d / D) * Math.sqrt(((c0 * M) / (w * h))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 5.8e+160:
		tmp = 0.0
	else:
		tmp = (c0 / (2.0 * w)) * ((d / D) * math.sqrt(((c0 * M) / (w * h))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 5.8e+160)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(d / D) * sqrt(Float64(Float64(c0 * M) / Float64(w * h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 5.8e+160)
		tmp = 0.0;
	else
		tmp = (c0 / (2.0 * w)) * ((d / D) * sqrt(((c0 * M) / (w * h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 5.8e+160], 0.0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[Sqrt[N[(N[(c0 * M), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.7999999999999998e160

   1. Initial program 24.8%

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

   3. Simplified 25.7%

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

   6. Applied egg-rr 38.7%

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

      1. associate-/r* 38.8%

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

      2. associate-*l/ 37.2%

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

      3. associate-/l* 38.0%

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

      4. associate-*l/ 36.3%

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

      5. associate-/l* 38.0%

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

   8. Simplified 38.0%

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

   9. Taylor expanded in c0 around inf 19.6%

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

   10. Taylor expanded in d around -inf 7.2%

       \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left({d}^{2} \cdot \left(-1 \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)}{w}} \]
   11. Step-by-step derivation

       1. associate-/l* 7.2%

          \[\leadsto 0.5 \cdot \color{blue}{\left(c0 \cdot \frac{{d}^{2} \cdot \left(-1 \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right)} \]

       2. distribute-lft1-in 7.2%

          \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}}{w}\right) \]

       3. metadata-eval 7.2%

          \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \left(\color{blue}{0} \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

       4. mul0-lft 21.8%

          \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \color{blue}{0}}{w}\right) \]

   12. Simplified 21.8%

       \[\leadsto \color{blue}{0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot 0}{w}\right)} \]

   13. Taylor expanded in c0 around 0 37.4%

       \[\leadsto \color{blue}{0} \]

   if 5.7999999999999998e160 < M

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   6. Applied egg-rr 34.8%

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

      1. associate-/r* 34.8%

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

      2. associate-*l/ 34.8%

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

      3. associate-/l* 34.8%

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

      4. associate-*l/ 34.8%

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

      5. associate-/l* 34.8%

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

   8. Simplified 34.8%

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

      1. associate-*r/ 34.8%

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

   10. Applied egg-rr 34.8%

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

       1. times-frac 34.8%

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

   12. Simplified 34.8%

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

   13. Taylor expanded in c0 around inf 30.4%

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

       1. *-commutative 30.4%

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

       2. *-commutative 30.4%

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

       3. associate-/r* 30.4%

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

   15. Simplified 30.4%

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

   16. Taylor expanded in c0 around 0 30.6%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \sqrt{\frac{M \cdot c0}{h \cdot w}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.8 \cdot 10^{+160}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{d}{D} \cdot \sqrt{\frac{c0 \cdot M}{w \cdot h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 34.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.5 \cdot 10^{+161}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{d}{D} \cdot \sqrt{M \cdot \frac{c0}{w \cdot h}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 1.5e+161)
   0.0
   (* (/ c0 (* 2.0 w)) (* (/ d D) (sqrt (* M (/ c0 (* w h))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.5e+161) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((d / D) * sqrt((M * (c0 / (w * h)))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.5d+161) then
        tmp = 0.0d0
    else
        tmp = (c0 / (2.0d0 * w)) * ((d_1 / d) * sqrt((m * (c0 / (w * h)))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.5e+161) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((d / D) * Math.sqrt((M * (c0 / (w * h)))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 1.5e+161:
		tmp = 0.0
	else:
		tmp = (c0 / (2.0 * w)) * ((d / D) * math.sqrt((M * (c0 / (w * h)))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 1.5e+161)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(d / D) * sqrt(Float64(M * Float64(c0 / Float64(w * h))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 1.5e+161)
		tmp = 0.0;
	else
		tmp = (c0 / (2.0 * w)) * ((d / D) * sqrt((M * (c0 / (w * h)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 1.5e+161], 0.0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[Sqrt[N[(M * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.50000000000000006e161

   1. Initial program 24.8%

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

   3. Simplified 25.7%

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

   6. Applied egg-rr 38.7%

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

      1. associate-/r* 38.8%

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

      2. associate-*l/ 37.2%

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

      3. associate-/l* 38.0%

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

      4. associate-*l/ 36.3%

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

      5. associate-/l* 38.0%

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

   8. Simplified 38.0%

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

   9. Taylor expanded in c0 around inf 19.6%

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

   10. Taylor expanded in d around -inf 7.2%

       \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left({d}^{2} \cdot \left(-1 \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)}{w}} \]
   11. Step-by-step derivation

       1. associate-/l* 7.2%

          \[\leadsto 0.5 \cdot \color{blue}{\left(c0 \cdot \frac{{d}^{2} \cdot \left(-1 \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right)} \]

       2. distribute-lft1-in 7.2%

          \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}}{w}\right) \]

       3. metadata-eval 7.2%

          \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \left(\color{blue}{0} \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

       4. mul0-lft 21.8%

          \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \color{blue}{0}}{w}\right) \]

   12. Simplified 21.8%

       \[\leadsto \color{blue}{0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot 0}{w}\right)} \]

   13. Taylor expanded in c0 around 0 37.4%

       \[\leadsto \color{blue}{0} \]

   if 1.50000000000000006e161 < M

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   6. Applied egg-rr 34.8%

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

      1. associate-/r* 34.8%

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

      2. associate-*l/ 34.8%

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

      3. associate-/l* 34.8%

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

      4. associate-*l/ 34.8%

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

      5. associate-/l* 34.8%

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

   8. Simplified 34.8%

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

      1. associate-*r/ 34.8%

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

   10. Applied egg-rr 34.8%

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

       1. times-frac 34.8%

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

   12. Simplified 34.8%

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

   13. Taylor expanded in c0 around inf 30.4%

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

       1. *-commutative 30.4%

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

       2. *-commutative 30.4%

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

       3. associate-/r* 30.4%

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

   15. Simplified 30.4%

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

   16. Taylor expanded in c0 around 0 30.6%

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

       1. associate-/l* 30.6%

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

   18. Simplified 30.6%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{d}{D} \cdot \sqrt{M \cdot \frac{c0}{h \cdot w}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.5 \cdot 10^{+161}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{d}{D} \cdot \sqrt{M \cdot \frac{c0}{w \cdot h}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.3% accurate, 151.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 0.0)

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.0d0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Python

def code(c0, w, h, D, d, M):
	return 0.0

Julia

function code(c0, w, h, D, d, M)
	return 0.0
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.0;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 22.6%

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

3. Simplified 23.4%

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

6. Applied egg-rr 38.4%

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

   1. associate-/r* 38.4%

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

   2. associate-*l/ 37.0%

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

   3. associate-/l* 37.7%

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

   4. associate-*l/ 36.2%

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

   5. associate-/l* 37.7%

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

8. Simplified 37.7%

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

9. Taylor expanded in c0 around inf 20.5%

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

10. Taylor expanded in d around -inf 6.5%

    \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left({d}^{2} \cdot \left(-1 \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)}{w}} \]
11. Step-by-step derivation

    1. associate-/l* 6.5%

       \[\leadsto 0.5 \cdot \color{blue}{\left(c0 \cdot \frac{{d}^{2} \cdot \left(-1 \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right)} \]

    2. distribute-lft1-in 6.5%

       \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}}{w}\right) \]

    3. metadata-eval 6.5%

       \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \left(\color{blue}{0} \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

    4. mul0-lft 20.7%

       \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot \color{blue}{0}}{w}\right) \]

12. Simplified 20.7%

    \[\leadsto \color{blue}{0.5 \cdot \left(c0 \cdot \frac{{d}^{2} \cdot 0}{w}\right)} \]

13. Taylor expanded in c0 around 0 35.7%

    \[\leadsto \color{blue}{0} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024129 
(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))))))