Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.5% → 53.3%

Time: 40.2s

Alternatives: 7

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = -4.869390608779283e+268
  2. w = -5.750776146378652e-306
  3. h = -4.764410543227509e-215
  4. D = 2.6778836838704175e-223
  5. d = -4.3194934209168056e+278
  6. M = -3.803630488696596e+222

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: 24.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 53.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(\frac{c0}{\left(w \cdot h\right) \cdot D} \cdot \frac{d \cdot d}{D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(0.5, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}{c0}, c0 \cdot 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (* 2.0 (* (/ c0 (* (* w h) D)) (/ (* d d) D))))
     (*
      t_0
      (fma
       0.5
       (* (* (/ D d) (/ D d)) (/ (* w (* h (* M M))) c0))
       (* c0 0.0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (2.0 * ((c0 / ((w * h) * D)) * ((d * d) / D)));
	} else {
		tmp = t_0 * fma(0.5, (((D / d) * (D / d)) * ((w * (h * (M * M))) / c0)), (c0 * 0.0));
	}
	return tmp;
}

Julia

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(2.0 * N[(N[(c0 / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.5 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision] + N[(c0 * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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 76.2%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in c0 around inf 78.7%

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

      1. times-frac 75.7%

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

      2. unpow2 75.7%

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

      3. unpow2 75.7%

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

   6. Simplified 75.7%

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

      1. frac-times 78.7%

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

      2. *-commutative 78.7%

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

      3. *-commutative 78.7%

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

      4. associate-*r* 78.8%

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

      5. *-commutative 78.8%

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

      6. *-commutative 78.8%

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

   8. Applied egg-rr 78.8%

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

      1. times-frac 82.5%

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

      2. *-commutative 82.5%

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

   10. Applied egg-rr 82.5%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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%

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

      1. div-inv 0.0%

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

      2. associate-*l* 0.0%

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

   3. Applied egg-rr 0.0%

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

   4. Taylor expanded in c0 around -inf 4.0%

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

      1. fma-def 4.0%

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

      2. times-frac 4.5%

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

      3. unpow2 4.5%

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

      4. unpow2 4.5%

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

      5. times-frac 4.6%

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

      6. *-commutative 4.6%

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

      7. unpow2 4.6%

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

      8. associate-*r* 4.6%

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

   6. Simplified 44.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}{c0}, c0 \cdot 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{\left(w \cdot h\right) \cdot D} \cdot \frac{d \cdot d}{D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}{c0}, c0 \cdot 0\right)\\ \end{array} \]

Alternative 2: 44.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -2.4 \cdot 10^{+124}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 2.2 \cdot 10^{-39} \lor \neg \left(w \leq 0.095\right) \land w \leq 3.3 \cdot 10^{+101}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= w -2.4e+124)
   0.0
   (if (or (<= w 2.2e-39) (and (not (<= w 0.095)) (<= w 3.3e+101)))
     (* (/ c0 (* 2.0 w)) (* 2.0 (* (* (/ d D) (/ d D)) (/ c0 (* w h)))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -2.4e+124) {
		tmp = 0.0;
	} else if ((w <= 2.2e-39) || (!(w <= 0.095) && (w <= 3.3e+101))) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d / D) * (d / D)) * (c0 / (w * h))));
	} else {
		tmp = 0.0;
	}
	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 (w <= (-2.4d+124)) then
        tmp = 0.0d0
    else if ((w <= 2.2d-39) .or. (.not. (w <= 0.095d0)) .and. (w <= 3.3d+101)) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (((d_1 / d) * (d_1 / d)) * (c0 / (w * h))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -2.4e+124) {
		tmp = 0.0;
	} else if ((w <= 2.2e-39) || (!(w <= 0.095) && (w <= 3.3e+101))) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d / D) * (d / D)) * (c0 / (w * h))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if w <= -2.4e+124:
		tmp = 0.0
	elif (w <= 2.2e-39) or (not (w <= 0.095) and (w <= 3.3e+101)):
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d / D) * (d / D)) * (c0 / (w * h))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (w <= -2.4e+124)
		tmp = 0.0;
	elseif ((w <= 2.2e-39) || (!(w <= 0.095) && (w <= 3.3e+101)))
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(c0 / Float64(w * h)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (w <= -2.4e+124)
		tmp = 0.0;
	elseif ((w <= 2.2e-39) || (~((w <= 0.095)) && (w <= 3.3e+101)))
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d / D) * (d / D)) * (c0 / (w * h))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[w, -2.4e+124], 0.0, If[Or[LessEqual[w, 2.2e-39], And[N[Not[LessEqual[w, 0.095]], $MachinePrecision], LessEqual[w, 3.3e+101]]], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if w < -2.40000000000000006e124 or 2.20000000000000001e-39 < w < 0.095000000000000001 or 3.30000000000000011e101 < w

   1. Initial program 12.2%

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

   3. Simplified 12.2%

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

   4. Taylor expanded in c0 around -inf 2.0%

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

      1. associate-*r* 2.0%

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

      2. distribute-rgt1-in 2.0%

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

      3. metadata-eval 2.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

      4. mul0-lft 45.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 45.2%

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

      6. mul0-lft 3.5%

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

      7. metadata-eval 3.5%

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

      8. distribute-lft1-in 3.5%

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

      9. *-commutative 3.5%

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

      10. distribute-lft1-in 3.5%

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

      11. metadata-eval 3.5%

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

      12. mul0-lft 45.2%

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

   6. Simplified 45.2%

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

   7. Taylor expanded in c0 around 0 45.2%

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

   if -2.40000000000000006e124 < w < 2.20000000000000001e-39 or 0.095000000000000001 < w < 3.30000000000000011e101

   1. Initial program 26.3%

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

   3. Simplified 27.2%

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

   4. Taylor expanded in c0 around inf 39.7%

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

      1. times-frac 40.1%

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

      2. unpow2 40.1%

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

      3. unpow2 40.1%

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

   6. Simplified 40.1%

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

      1. frac-times 51.4%

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

   8. Applied egg-rr 51.4%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.4 \cdot 10^{+124}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 2.2 \cdot 10^{-39} \lor \neg \left(w \leq 0.095\right) \land w \leq 3.3 \cdot 10^{+101}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3: 36.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{h \cdot \left(\left(w \cdot D\right) \cdot \left(w \cdot D\right)\right)}\\ \mathbf{if}\;M \leq 2 \cdot 10^{-273}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 2.9 \cdot 10^{-224}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq 7.2 \cdot 10^{-152}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.02 \cdot 10^{-87}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \frac{\frac{d \cdot d}{D \cdot D}}{w \cdot h}\right)\right)\\ \mathbf{elif}\;M \leq 6.1 \cdot 10^{+66}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* d (* d (* c0 c0))) (* h (* (* w D) (* w D))))))
   (if (<= M 2e-273)
     0.0
     (if (<= M 2.9e-224)
       t_0
       (if (<= M 7.2e-152)
         0.0
         (if (<= M 1.02e-87)
           (* (/ c0 (* 2.0 w)) (* 2.0 (* c0 (/ (/ (* d d) (* D D)) (* w h)))))
           (if (<= M 6.1e+66) 0.0 t_0)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * (d * (c0 * c0))) / (h * ((w * D) * (w * D)));
	double tmp;
	if (M <= 2e-273) {
		tmp = 0.0;
	} else if (M <= 2.9e-224) {
		tmp = t_0;
	} else if (M <= 7.2e-152) {
		tmp = 0.0;
	} else if (M <= 1.02e-87) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 * (((d * d) / (D * D)) / (w * h))));
	} else if (M <= 6.1e+66) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (d_1 * (d_1 * (c0 * c0))) / (h * ((w * d) * (w * d)))
    if (m <= 2d-273) then
        tmp = 0.0d0
    else if (m <= 2.9d-224) then
        tmp = t_0
    else if (m <= 7.2d-152) then
        tmp = 0.0d0
    else if (m <= 1.02d-87) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (c0 * (((d_1 * d_1) / (d * d)) / (w * h))))
    else if (m <= 6.1d+66) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * (d * (c0 * c0))) / (h * ((w * D) * (w * D)));
	double tmp;
	if (M <= 2e-273) {
		tmp = 0.0;
	} else if (M <= 2.9e-224) {
		tmp = t_0;
	} else if (M <= 7.2e-152) {
		tmp = 0.0;
	} else if (M <= 1.02e-87) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 * (((d * d) / (D * D)) / (w * h))));
	} else if (M <= 6.1e+66) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d * (d * (c0 * c0))) / (h * ((w * D) * (w * D)))
	tmp = 0
	if M <= 2e-273:
		tmp = 0.0
	elif M <= 2.9e-224:
		tmp = t_0
	elif M <= 7.2e-152:
		tmp = 0.0
	elif M <= 1.02e-87:
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 * (((d * d) / (D * D)) / (w * h))))
	elif M <= 6.1e+66:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d * Float64(d * Float64(c0 * c0))) / Float64(h * Float64(Float64(w * D) * Float64(w * D))))
	tmp = 0.0
	if (M <= 2e-273)
		tmp = 0.0;
	elseif (M <= 2.9e-224)
		tmp = t_0;
	elseif (M <= 7.2e-152)
		tmp = 0.0;
	elseif (M <= 1.02e-87)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(c0 * Float64(Float64(Float64(d * d) / Float64(D * D)) / Float64(w * h)))));
	elseif (M <= 6.1e+66)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d * (d * (c0 * c0))) / (h * ((w * D) * (w * D)));
	tmp = 0.0;
	if (M <= 2e-273)
		tmp = 0.0;
	elseif (M <= 2.9e-224)
		tmp = t_0;
	elseif (M <= 7.2e-152)
		tmp = 0.0;
	elseif (M <= 1.02e-87)
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 * (((d * d) / (D * D)) / (w * h))));
	elseif (M <= 6.1e+66)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d * N[(d * N[(c0 * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(h * N[(N[(w * D), $MachinePrecision] * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 2e-273], 0.0, If[LessEqual[M, 2.9e-224], t$95$0, If[LessEqual[M, 7.2e-152], 0.0, If[LessEqual[M, 1.02e-87], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(c0 * N[(N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 6.1e+66], 0.0, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 2e-273 or 2.9e-224 < M < 7.2e-152 or 1.02000000000000009e-87 < M < 6.10000000000000021e66

   1. Initial program 22.2%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in c0 around -inf 1.9%

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

      1. associate-*r* 1.9%

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

      2. distribute-rgt1-in 1.9%

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

      3. metadata-eval 1.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

      4. mul0-lft 31.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 31.9%

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

      6. mul0-lft 2.9%

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

      7. metadata-eval 2.9%

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

      8. distribute-lft1-in 2.9%

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

      9. *-commutative 2.9%

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

      10. distribute-lft1-in 2.9%

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

      11. metadata-eval 2.9%

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

      12. mul0-lft 31.9%

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

   6. Simplified 31.9%

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

   7. Taylor expanded in c0 around 0 36.3%

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

   if 2e-273 < M < 2.9e-224 or 6.10000000000000021e66 < M

   1. Initial program 21.2%

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

   3. Simplified 22.5%

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

   4. Taylor expanded in c0 around inf 42.2%

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

      1. times-frac 41.7%

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

      2. unpow2 41.7%

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

      3. unpow2 41.7%

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

   6. Simplified 41.7%

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

   7. Taylor expanded in c0 around 0 35.7%

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

      1. unpow2 35.7%

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

      2. unpow2 35.7%

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

      3. associate-*r* 35.7%

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

      4. unpow2 35.7%

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

      5. unpow2 35.7%

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

      6. unswap-sqr 45.4%

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

   9. Simplified 45.4%

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

   10. Taylor expanded in d around 0 45.4%

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

       1. unpow2 45.4%

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

       2. unpow2 45.4%

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

       3. associate-*r* 53.1%

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

   12. Simplified 53.1%

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

   if 7.2e-152 < M < 1.02000000000000009e-87

   1. Initial program 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in c0 around inf 45.3%

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

      1. *-rgt-identity 45.3%

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

      2. *-commutative 45.3%

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

      3. associate-*r* 36.6%

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

      4. unpow2 36.6%

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

      5. associate-*r/ 36.6%

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

      6. *-commutative 36.6%

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

      7. associate-*l* 36.9%

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

      8. associate-*r/ 36.9%

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

      9. *-rgt-identity 36.9%

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

      10. unpow2 36.9%

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

      11. associate-*r* 36.9%

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

      12. *-commutative 36.9%

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

      13. associate-/r* 37.4%

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

      14. unpow2 37.4%

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

      15. unpow2 37.4%

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

   6. Simplified 37.4%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-273}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 2.9 \cdot 10^{-224}:\\ \;\;\;\;\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{h \cdot \left(\left(w \cdot D\right) \cdot \left(w \cdot D\right)\right)}\\ \mathbf{elif}\;M \leq 7.2 \cdot 10^{-152}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.02 \cdot 10^{-87}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \frac{\frac{d \cdot d}{D \cdot D}}{w \cdot h}\right)\right)\\ \mathbf{elif}\;M \leq 6.1 \cdot 10^{+66}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{h \cdot \left(\left(w \cdot D\right) \cdot \left(w \cdot D\right)\right)}\\ \end{array} \]

Alternative 4: 44.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;w \leq -2.4 \cdot 10^{+123}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 5.8 \cdot 10^{-40}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right)\right)\\ \mathbf{elif}\;w \leq 0.0235:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 8.5 \cdot 10^{+131}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(\frac{c0}{\left(w \cdot h\right) \cdot D} \cdot \frac{d \cdot d}{D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))))
   (if (<= w -2.4e+123)
     0.0
     (if (<= w 5.8e-40)
       (* t_0 (* 2.0 (* (* (/ d D) (/ d D)) (/ c0 (* w h)))))
       (if (<= w 0.0235)
         0.0
         (if (<= w 8.5e+131)
           (* t_0 (* 2.0 (* (/ c0 (* (* w h) D)) (/ (* d d) D))))
           0.0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double tmp;
	if (w <= -2.4e+123) {
		tmp = 0.0;
	} else if (w <= 5.8e-40) {
		tmp = t_0 * (2.0 * (((d / D) * (d / D)) * (c0 / (w * h))));
	} else if (w <= 0.0235) {
		tmp = 0.0;
	} else if (w <= 8.5e+131) {
		tmp = t_0 * (2.0 * ((c0 / ((w * h) * D)) * ((d * d) / D)));
	} else {
		tmp = 0.0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    if (w <= (-2.4d+123)) then
        tmp = 0.0d0
    else if (w <= 5.8d-40) then
        tmp = t_0 * (2.0d0 * (((d_1 / d) * (d_1 / d)) * (c0 / (w * h))))
    else if (w <= 0.0235d0) then
        tmp = 0.0d0
    else if (w <= 8.5d+131) then
        tmp = t_0 * (2.0d0 * ((c0 / ((w * h) * d)) * ((d_1 * d_1) / d)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double tmp;
	if (w <= -2.4e+123) {
		tmp = 0.0;
	} else if (w <= 5.8e-40) {
		tmp = t_0 * (2.0 * (((d / D) * (d / D)) * (c0 / (w * h))));
	} else if (w <= 0.0235) {
		tmp = 0.0;
	} else if (w <= 8.5e+131) {
		tmp = t_0 * (2.0 * ((c0 / ((w * h) * D)) * ((d * d) / D)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	tmp = 0
	if w <= -2.4e+123:
		tmp = 0.0
	elif w <= 5.8e-40:
		tmp = t_0 * (2.0 * (((d / D) * (d / D)) * (c0 / (w * h))))
	elif w <= 0.0235:
		tmp = 0.0
	elif w <= 8.5e+131:
		tmp = t_0 * (2.0 * ((c0 / ((w * h) * D)) * ((d * d) / D)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (w <= -2.4e+123)
		tmp = 0.0;
	elseif (w <= 5.8e-40)
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(c0 / Float64(w * h)))));
	elseif (w <= 0.0235)
		tmp = 0.0;
	elseif (w <= 8.5e+131)
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(c0 / Float64(Float64(w * h) * D)) * Float64(Float64(d * d) / D))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	tmp = 0.0;
	if (w <= -2.4e+123)
		tmp = 0.0;
	elseif (w <= 5.8e-40)
		tmp = t_0 * (2.0 * (((d / D) * (d / D)) * (c0 / (w * h))));
	elseif (w <= 0.0235)
		tmp = 0.0;
	elseif (w <= 8.5e+131)
		tmp = t_0 * (2.0 * ((c0 / ((w * h) * D)) * ((d * d) / D)));
	else
		tmp = 0.0;
	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]}, If[LessEqual[w, -2.4e+123], 0.0, If[LessEqual[w, 5.8e-40], N[(t$95$0 * N[(2.0 * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.0235], 0.0, If[LessEqual[w, 8.5e+131], N[(t$95$0 * N[(2.0 * N[(N[(c0 / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -2.39999999999999989e123 or 5.7999999999999998e-40 < w < 0.0235 or 8.50000000000000063e131 < w

   1. Initial program 11.5%

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

   3. Simplified 11.6%

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

   4. Taylor expanded in c0 around -inf 2.2%

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

      1. associate-*r* 2.2%

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

      2. distribute-rgt1-in 2.2%

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

      3. metadata-eval 2.2%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

      4. mul0-lft 47.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 47.0%

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

      6. mul0-lft 3.8%

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

      7. metadata-eval 3.8%

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

      8. distribute-lft1-in 3.8%

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

      9. *-commutative 3.8%

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

      10. distribute-lft1-in 3.8%

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

      11. metadata-eval 3.8%

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

      12. mul0-lft 47.0%

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

   6. Simplified 47.0%

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

   7. Taylor expanded in c0 around 0 47.0%

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

   if -2.39999999999999989e123 < w < 5.7999999999999998e-40

   1. Initial program 26.7%

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

   3. Simplified 27.2%

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

   4. Taylor expanded in c0 around inf 39.4%

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

      1. times-frac 39.9%

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

      2. unpow2 39.9%

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

      3. unpow2 39.9%

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

   6. Simplified 39.9%

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

      1. frac-times 52.0%

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

   8. Applied egg-rr 52.0%

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

   if 0.0235 < w < 8.50000000000000063e131

   1. Initial program 22.7%

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

   3. Simplified 26.3%

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

   4. Taylor expanded in c0 around inf 38.0%

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

      1. times-frac 37.7%

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

      2. unpow2 37.7%

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

      3. unpow2 37.7%

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

   6. Simplified 37.7%

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

      1. frac-times 38.0%

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

      2. *-commutative 38.0%

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

      3. *-commutative 38.0%

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

      4. associate-*r* 38.3%

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

      5. *-commutative 38.3%

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

      6. *-commutative 38.3%

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

   8. Applied egg-rr 38.3%

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

      1. times-frac 45.9%

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

      2. *-commutative 45.9%

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

   10. Applied egg-rr 45.9%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.4 \cdot 10^{+123}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 5.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right)\right)\\ \mathbf{elif}\;w \leq 0.0235:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 8.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{\left(w \cdot h\right) \cdot D} \cdot \frac{d \cdot d}{D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 37.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 9.5 \cdot 10^{-274}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.9 \cdot 10^{-224} \lor \neg \left(M \leq 2.5 \cdot 10^{-95} \lor \neg \left(M \leq 1.65 \cdot 10^{-87}\right) \land M \leq 1.86 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{h \cdot \left(\left(w \cdot D\right) \cdot \left(w \cdot D\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 9.5e-274)
   0.0
   (if (or (<= M 1.9e-224)
           (not
            (or (<= M 2.5e-95) (and (not (<= M 1.65e-87)) (<= M 1.86e+68)))))
     (/ (* d (* d (* c0 c0))) (* h (* (* w D) (* w D))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 9.5e-274) {
		tmp = 0.0;
	} else if ((M <= 1.9e-224) || !((M <= 2.5e-95) || (!(M <= 1.65e-87) && (M <= 1.86e+68)))) {
		tmp = (d * (d * (c0 * c0))) / (h * ((w * D) * (w * D)));
	} else {
		tmp = 0.0;
	}
	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 <= 9.5d-274) then
        tmp = 0.0d0
    else if ((m <= 1.9d-224) .or. (.not. (m <= 2.5d-95) .or. (.not. (m <= 1.65d-87)) .and. (m <= 1.86d+68))) then
        tmp = (d_1 * (d_1 * (c0 * c0))) / (h * ((w * d) * (w * d)))
    else
        tmp = 0.0d0
    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 <= 9.5e-274) {
		tmp = 0.0;
	} else if ((M <= 1.9e-224) || !((M <= 2.5e-95) || (!(M <= 1.65e-87) && (M <= 1.86e+68)))) {
		tmp = (d * (d * (c0 * c0))) / (h * ((w * D) * (w * D)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 9.5e-274:
		tmp = 0.0
	elif (M <= 1.9e-224) or not ((M <= 2.5e-95) or (not (M <= 1.65e-87) and (M <= 1.86e+68))):
		tmp = (d * (d * (c0 * c0))) / (h * ((w * D) * (w * D)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 9.5e-274)
		tmp = 0.0;
	elseif ((M <= 1.9e-224) || !((M <= 2.5e-95) || (!(M <= 1.65e-87) && (M <= 1.86e+68))))
		tmp = Float64(Float64(d * Float64(d * Float64(c0 * c0))) / Float64(h * Float64(Float64(w * D) * Float64(w * D))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 9.5e-274)
		tmp = 0.0;
	elseif ((M <= 1.9e-224) || ~(((M <= 2.5e-95) || (~((M <= 1.65e-87)) && (M <= 1.86e+68)))))
		tmp = (d * (d * (c0 * c0))) / (h * ((w * D) * (w * D)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 9.5e-274], 0.0, If[Or[LessEqual[M, 1.9e-224], N[Not[Or[LessEqual[M, 2.5e-95], And[N[Not[LessEqual[M, 1.65e-87]], $MachinePrecision], LessEqual[M, 1.86e+68]]]], $MachinePrecision]], N[(N[(d * N[(d * N[(c0 * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(h * N[(N[(w * D), $MachinePrecision] * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if M < 9.5000000000000009e-274 or 1.90000000000000001e-224 < M < 2.4999999999999999e-95 or 1.65e-87 < M < 1.8600000000000001e68

   1. Initial program 22.5%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in c0 around -inf 1.8%

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

      1. associate-*r* 1.8%

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

      2. distribute-rgt1-in 1.8%

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

      3. metadata-eval 1.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

      4. mul0-lft 31.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 31.5%

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

      6. mul0-lft 2.8%

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

      7. metadata-eval 2.8%

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

      8. distribute-lft1-in 2.8%

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

      9. *-commutative 2.8%

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

      10. distribute-lft1-in 2.8%

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

      11. metadata-eval 2.8%

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

      12. mul0-lft 31.5%

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

   6. Simplified 31.5%

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

   7. Taylor expanded in c0 around 0 35.8%

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

   if 9.5000000000000009e-274 < M < 1.90000000000000001e-224 or 2.4999999999999999e-95 < M < 1.65e-87 or 1.8600000000000001e68 < M

   1. Initial program 22.9%

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

   3. Simplified 24.0%

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

   4. Taylor expanded in c0 around inf 43.7%

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

      1. times-frac 43.2%

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

      2. unpow2 43.2%

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

      3. unpow2 43.2%

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

   6. Simplified 43.2%

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

   7. Taylor expanded in c0 around 0 36.8%

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

      1. unpow2 36.8%

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

      2. unpow2 36.8%

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

      3. associate-*r* 36.8%

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

      4. unpow2 36.8%

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

      5. unpow2 36.8%

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

      6. unswap-sqr 45.7%

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

   9. Simplified 45.7%

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

   10. Taylor expanded in d around 0 45.7%

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

       1. unpow2 45.7%

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

       2. unpow2 45.7%

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

       3. associate-*r* 52.7%

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

   12. Simplified 52.7%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9.5 \cdot 10^{-274}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.9 \cdot 10^{-224} \lor \neg \left(M \leq 2.5 \cdot 10^{-95} \lor \neg \left(M \leq 1.65 \cdot 10^{-87}\right) \land M \leq 1.86 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{h \cdot \left(\left(w \cdot D\right) \cdot \left(w \cdot D\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 45.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -2.85 \cdot 10^{+224}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 2.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= w -2.85e+224)
   0.0
   (if (<= w 2.6e+101)
     (* (/ c0 (* 2.0 w)) (* 2.0 (/ (* (/ c0 w) (* (/ d D) (/ d D))) h)))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -2.85e+224) {
		tmp = 0.0;
	} else if (w <= 2.6e+101) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (((c0 / w) * ((d / D) * (d / D))) / h));
	} else {
		tmp = 0.0;
	}
	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 (w <= (-2.85d+224)) then
        tmp = 0.0d0
    else if (w <= 2.6d+101) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (((c0 / w) * ((d_1 / d) * (d_1 / d))) / h))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -2.85e+224) {
		tmp = 0.0;
	} else if (w <= 2.6e+101) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (((c0 / w) * ((d / D) * (d / D))) / h));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if w <= -2.85e+224:
		tmp = 0.0
	elif w <= 2.6e+101:
		tmp = (c0 / (2.0 * w)) * (2.0 * (((c0 / w) * ((d / D) * (d / D))) / h))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (w <= -2.85e+224)
		tmp = 0.0;
	elseif (w <= 2.6e+101)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(Float64(c0 / w) * Float64(Float64(d / D) * Float64(d / D))) / h)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (w <= -2.85e+224)
		tmp = 0.0;
	elseif (w <= 2.6e+101)
		tmp = (c0 / (2.0 * w)) * (2.0 * (((c0 / w) * ((d / D) * (d / D))) / h));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[w, -2.85e+224], 0.0, If[LessEqual[w, 2.6e+101], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[(c0 / w), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if w < -2.84999999999999999e224 or 2.6e101 < w

   1. Initial program 16.7%

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

   3. Simplified 16.9%

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

   4. Taylor expanded in c0 around -inf 3.0%

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

      1. associate-*r* 3.0%

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

      2. distribute-rgt1-in 3.0%

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

      3. metadata-eval 3.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

      4. mul0-lft 51.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 51.6%

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

      6. mul0-lft 5.4%

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

      7. metadata-eval 5.4%

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

      8. distribute-lft1-in 5.4%

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

      9. *-commutative 5.4%

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

      10. distribute-lft1-in 5.4%

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

      11. metadata-eval 5.4%

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

      12. mul0-lft 51.6%

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

   6. Simplified 51.6%

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

   7. Taylor expanded in c0 around 0 51.6%

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

   if -2.84999999999999999e224 < w < 2.6e101

   1. Initial program 23.8%

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

   3. Simplified 24.6%

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

   4. Taylor expanded in c0 around inf 35.7%

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

      1. times-frac 36.0%

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

      2. unpow2 36.0%

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

      3. unpow2 36.0%

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

   6. Simplified 36.0%

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

      1. frac-times 47.4%

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

      2. *-commutative 47.4%

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

      3. associate-/r* 49.8%

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

      4. associate-*l/ 50.0%

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

      5. pow2 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. unpow2 50.0%

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

   10. Applied egg-rr 50.0%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.85 \cdot 10^{+224}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 2.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 34.2% 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 23.3%

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

4. Taylor expanded in c0 around -inf 1.5%

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

   1. associate-*r* 1.5%

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

   2. distribute-rgt1-in 1.5%

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

   3. metadata-eval 1.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

   4. mul0-lft 26.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

   5. metadata-eval 26.6%

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

   6. mul0-lft 2.3%

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

   7. metadata-eval 2.3%

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

   8. distribute-lft1-in 2.3%

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

   9. *-commutative 2.3%

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

   10. distribute-lft1-in 2.3%

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

   11. metadata-eval 2.3%

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

   12. mul0-lft 26.6%

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

6. Simplified 26.6%

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

7. Taylor expanded in c0 around 0 30.4%

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

8. Final simplification 30.4%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023274 
(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))))))