Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.1% → 54.5%

Time: 40.4s

Alternatives: 9

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = -7.403078903324848e+300
  2. w = 5.845596391768551e-304
  3. h = 4.1476070772566856e-131
  4. D = 1.4386566202160685e-245
  5. d = 5.151898746903322e+186
  6. M = -4.147817076162456e-142

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: 25.1% 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: 54.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/
      (* c0 (pow (* (* (sqrt (/ c0 (* w h))) (/ d D)) (sqrt 2.0)) 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 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 * pow(((sqrt((c0 / (w * h))) * (d / D)) * sqrt(2.0)), 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 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * Math.pow(((Math.sqrt((c0 / (w * h))) * (d / D)) * Math.sqrt(2.0)), 2.0)) / (2.0 * w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = (c0 * math.pow(((math.sqrt((c0 / (w * h))) * (d / D)) * math.sqrt(2.0)), 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(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 * (Float64(Float64(sqrt(Float64(c0 / Float64(w * h))) * Float64(d / D)) * sqrt(2.0)) ^ 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 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (c0 * (((sqrt((c0 / (w * h))) * (d / D)) * sqrt(2.0)) ^ 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[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * N[Power[N[(N[(N[Sqrt[N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision], 0.0]]

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 77.0%

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

      1. +-commutative 77.0%

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

      2. +-commutative 77.0%

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

      3. times-frac 74.2%

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

      4. fma-neg 74.2%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in c0 around inf 79.0%

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

      1. *-commutative 79.0%

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

      2. associate-/l/ 80.1%

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

      3. associate-*l/ 80.0%

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

      4. associate-/l* 78.8%

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

      5. *-commutative 78.8%

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

   7. Simplified 78.8%

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

      1. expm1-log1p-u 46.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)\right)} \]

      2. expm1-udef 49.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)} - 1} \]

   9. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 46.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)\right)} \]

       2. expm1-log1p 77.3%

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

       3. *-commutative 77.3%

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

       4. associate-/l* 77.3%

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

       5. associate-/r* 80.0%

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

   11. Simplified 80.0%

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

       1. associate-*l/ 82.4%

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

       2. associate-/r/ 82.4%

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

       3. associate-/l/ 81.2%

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

       4. *-commutative 81.2%

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

   13. Applied egg-rr 81.2%

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

   15. Applied egg-rr 84.3%

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

   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. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 0.7%

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

   5. Taylor expanded in c0 around -inf 2.2%

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

      1. mul-1-neg 2.2%

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

      2. distribute-lft-in 0.4%

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

   7. Simplified 42.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   8. Taylor expanded in c0 around 0 47.0%

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

4. Final simplification 58.2%

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

Alternative 2: 53.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/
      (* c0 (/ (* 2.0 (/ (* c0 (pow d 2.0)) (pow D 2.0))) (* w h)))
      (* 2.0 w))
     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 tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 * ((2.0 * ((c0 * pow(d, 2.0)) / pow(D, 2.0))) / (w * h))) / (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 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * ((2.0 * ((c0 * Math.pow(d, 2.0)) / Math.pow(D, 2.0))) / (w * h))) / (2.0 * w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = (c0 * ((2.0 * ((c0 * math.pow(d, 2.0)) / math.pow(D, 2.0))) / (w * h))) / (2.0 * w)
	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)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 * Float64(Float64(2.0 * Float64(Float64(c0 * (d ^ 2.0)) / (D ^ 2.0))) / Float64(w * h))) / 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 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (c0 * ((2.0 * ((c0 * (d ^ 2.0)) / (D ^ 2.0))) / (w * h))) / (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[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * N[(N[(2.0 * N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision], 0.0]]

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 77.0%

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

      1. +-commutative 77.0%

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

      2. +-commutative 77.0%

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

      3. times-frac 74.2%

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

      4. fma-neg 74.2%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in c0 around inf 79.0%

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

      1. *-commutative 79.0%

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

      2. associate-/l/ 80.1%

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

      3. associate-*l/ 80.0%

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

      4. associate-/l* 78.8%

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

      5. *-commutative 78.8%

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

   7. Simplified 78.8%

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

      1. expm1-log1p-u 46.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)\right)} \]

      2. expm1-udef 49.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)} - 1} \]

   9. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 46.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)\right)} \]

       2. expm1-log1p 77.3%

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

       3. *-commutative 77.3%

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

       4. associate-/l* 77.3%

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

       5. associate-/r* 80.0%

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

   11. Simplified 80.0%

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

       1. associate-*l/ 82.4%

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

       2. associate-/r/ 82.4%

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

       3. associate-/l/ 81.2%

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

       4. *-commutative 81.2%

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

   13. Applied egg-rr 81.2%

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

   14. Taylor expanded in d around 0 81.6%

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

       1. associate-/r* 84.0%

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

       2. *-commutative 84.0%

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

       3. associate-*r/ 84.0%

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

       4. *-commutative 84.0%

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

   16. Simplified 84.0%

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

   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. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 0.7%

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

   5. Taylor expanded in c0 around -inf 2.2%

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

      1. mul-1-neg 2.2%

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

      2. distribute-lft-in 0.4%

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

   7. Simplified 42.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   8. Taylor expanded in c0 around 0 47.0%

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

4. Final simplification 58.1%

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

Alternative 3: 54.9% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (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 (pow (* (sqrt (/ c0 (* w h))) (/ d D)) 2.0)))
     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 * pow((sqrt((c0 / (w * h))) * (d / D)), 2.0));
	} 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 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (2.0 * Math.pow((Math.sqrt((c0 / (w * h))) * (d / D)), 2.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * (2.0 * math.pow((math.sqrt((c0 / (w * h))) * (d / D)), 2.0))
	else:
		tmp = 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(sqrt(Float64(c0 / Float64(w * h))) * Float64(d / D)) ^ 2.0)));
	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);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * (2.0 * ((sqrt((c0 / (w * h))) * (d / D)) ^ 2.0));
	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]}, 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[Power[N[(N[Sqrt[N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

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 77.0%

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

      1. +-commutative 77.0%

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

      2. +-commutative 77.0%

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

      3. times-frac 74.2%

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

      4. fma-neg 74.2%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in c0 around inf 79.0%

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

      1. *-commutative 79.0%

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

      2. associate-/l/ 80.1%

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

      3. associate-*l/ 80.0%

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

      4. associate-/l* 78.8%

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

      5. *-commutative 78.8%

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

   7. Simplified 78.8%

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

      1. add-cbrt-cube 77.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{\sqrt[3]{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{D}^{2}}{{d}^{2}}}}}\right) \]

      2. pow1/3 77.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{{\left(\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{D}^{2}}{{d}^{2}}\right)}^{0.3333333333333333}}}\right) \]

      3. pow3 77.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\color{blue}{\left({\left(\frac{{D}^{2}}{{d}^{2}}\right)}^{3}\right)}}^{0.3333333333333333}}\right) \]

      4. clear-num 77.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\color{blue}{\left(\frac{1}{\frac{{d}^{2}}{{D}^{2}}}\right)}}^{3}\right)}^{0.3333333333333333}}\right) \]

      5. unpow2 77.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left(\frac{1}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

      6. unpow2 77.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left(\frac{1}{\frac{d \cdot d}{\color{blue}{D \cdot D}}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

      7. frac-times 77.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left(\frac{1}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

      8. pow2 77.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left(\frac{1}{\color{blue}{{\left(\frac{d}{D}\right)}^{2}}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

      9. pow-flip 77.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\color{blue}{\left({\left(\frac{d}{D}\right)}^{\left(-2\right)}\right)}}^{3}\right)}^{0.3333333333333333}}\right) \]

      10. metadata-eval 77.0%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left({\left(\frac{d}{D}\right)}^{\color{blue}{-2}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

   9. Applied egg-rr 77.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{{\left({\left({\left(\frac{d}{D}\right)}^{-2}\right)}^{3}\right)}^{0.3333333333333333}}}\right) \]

   11. Applied egg-rr 83.2%

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

   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. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 0.7%

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

   5. Taylor expanded in c0 around -inf 2.2%

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

      1. mul-1-neg 2.2%

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

      2. distribute-lft-in 0.4%

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

   7. Simplified 42.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   8. Taylor expanded in c0 around 0 47.0%

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

4. Final simplification 57.9%

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

Alternative 4: 53.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(c0 \cdot \frac{2}{\frac{{\left(\frac{d}{D}\right)}^{-2}}{\frac{c0}{w \cdot h}}}\right) \cdot \frac{1}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (* (* c0 (/ 2.0 (/ (pow (/ d D) -2.0) (/ c0 (* w h))))) (/ 1.0 (* 2.0 w)))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 * (2.0 / (pow((d / D), -2.0) / (c0 / (w * h))))) * (1.0 / (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 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * (2.0 / (Math.pow((d / D), -2.0) / (c0 / (w * h))))) * (1.0 / (2.0 * w));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = (c0 * (2.0 / (math.pow((d / D), -2.0) / (c0 / (w * h))))) * (1.0 / (2.0 * w))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 * Float64(2.0 / Float64((Float64(d / D) ^ -2.0) / Float64(c0 / Float64(w * h))))) * Float64(1.0 / 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 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (c0 * (2.0 / (((d / D) ^ -2.0) / (c0 / (w * h))))) * (1.0 / (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[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * N[(2.0 / N[(N[Power[N[(d / D), $MachinePrecision], -2.0], $MachinePrecision] / N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

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 77.0%

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

      1. +-commutative 77.0%

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

      2. +-commutative 77.0%

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

      3. times-frac 74.2%

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

      4. fma-neg 74.2%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in c0 around inf 79.0%

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

      1. *-commutative 79.0%

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

      2. associate-/l/ 80.1%

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

      3. associate-*l/ 80.0%

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

      4. associate-/l* 78.8%

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

      5. *-commutative 78.8%

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

   7. Simplified 78.8%

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

      1. expm1-log1p-u 46.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)\right)} \]

      2. expm1-udef 49.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)} - 1} \]

   9. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 46.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)\right)} \]

       2. expm1-log1p 77.3%

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

       3. *-commutative 77.3%

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

       4. associate-/l* 77.3%

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

       5. associate-/r* 80.0%

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

   11. Simplified 80.0%

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

       1. associate-*l/ 82.4%

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

       2. associate-/r/ 82.4%

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

       3. associate-/l/ 81.2%

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

       4. *-commutative 81.2%

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

   13. Applied egg-rr 81.2%

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

       1. div-inv 81.2%

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

       2. *-commutative 81.2%

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

       3. associate-/l/ 82.4%

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

       4. associate-/r/ 82.4%

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

       5. *-commutative 82.4%

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

   15. Applied egg-rr 82.4%

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

   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. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 0.7%

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

   5. Taylor expanded in c0 around -inf 2.2%

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

      1. mul-1-neg 2.2%

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

      2. distribute-lft-in 0.4%

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

   7. Simplified 42.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   8. Taylor expanded in c0 around 0 47.0%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(c0 \cdot \frac{2}{\frac{{\left(\frac{d}{D}\right)}^{-2}}{\frac{c0}{w \cdot h}}}\right) \cdot \frac{1}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{d}{D}}\\ \mathbf{if}\;w \leq -2.6 \cdot 10^{-52}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 1.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{t\_0 \cdot t\_0}{\frac{\frac{c0}{w}}{h}}}\\ \mathbf{elif}\;w \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 2.9 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{c0}{2} \cdot \left(\frac{c0}{w} \cdot \left(2 \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)}{w \cdot h}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ d D))))
   (if (<= w -2.6e-52)
     0.0
     (if (<= w 1.6e-73)
       (* (/ c0 (* 2.0 w)) (/ 2.0 (/ (* t_0 t_0) (/ (/ c0 w) h))))
       (if (<= w 5.8e-14)
         0.0
         (if (<= w 2.9e+65)
           (/ (* (/ c0 2.0) (* (/ c0 w) (* 2.0 (pow (/ d D) 2.0)))) (* w h))
           0.0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = 1.0 / (d / D);
	double tmp;
	if (w <= -2.6e-52) {
		tmp = 0.0;
	} else if (w <= 1.6e-73) {
		tmp = (c0 / (2.0 * w)) * (2.0 / ((t_0 * t_0) / ((c0 / w) / h)));
	} else if (w <= 5.8e-14) {
		tmp = 0.0;
	} else if (w <= 2.9e+65) {
		tmp = ((c0 / 2.0) * ((c0 / w) * (2.0 * pow((d / D), 2.0)))) / (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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (d_1 / d)
    if (w <= (-2.6d-52)) then
        tmp = 0.0d0
    else if (w <= 1.6d-73) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 / ((t_0 * t_0) / ((c0 / w) / h)))
    else if (w <= 5.8d-14) then
        tmp = 0.0d0
    else if (w <= 2.9d+65) then
        tmp = ((c0 / 2.0d0) * ((c0 / w) * (2.0d0 * ((d_1 / d) ** 2.0d0)))) / (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 t_0 = 1.0 / (d / D);
	double tmp;
	if (w <= -2.6e-52) {
		tmp = 0.0;
	} else if (w <= 1.6e-73) {
		tmp = (c0 / (2.0 * w)) * (2.0 / ((t_0 * t_0) / ((c0 / w) / h)));
	} else if (w <= 5.8e-14) {
		tmp = 0.0;
	} else if (w <= 2.9e+65) {
		tmp = ((c0 / 2.0) * ((c0 / w) * (2.0 * Math.pow((d / D), 2.0)))) / (w * h);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = 1.0 / (d / D)
	tmp = 0
	if w <= -2.6e-52:
		tmp = 0.0
	elif w <= 1.6e-73:
		tmp = (c0 / (2.0 * w)) * (2.0 / ((t_0 * t_0) / ((c0 / w) / h)))
	elif w <= 5.8e-14:
		tmp = 0.0
	elif w <= 2.9e+65:
		tmp = ((c0 / 2.0) * ((c0 / w) * (2.0 * math.pow((d / D), 2.0)))) / (w * h)
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(1.0 / Float64(d / D))
	tmp = 0.0
	if (w <= -2.6e-52)
		tmp = 0.0;
	elseif (w <= 1.6e-73)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 / Float64(Float64(t_0 * t_0) / Float64(Float64(c0 / w) / h))));
	elseif (w <= 5.8e-14)
		tmp = 0.0;
	elseif (w <= 2.9e+65)
		tmp = Float64(Float64(Float64(c0 / 2.0) * Float64(Float64(c0 / w) * Float64(2.0 * (Float64(d / D) ^ 2.0)))) / Float64(w * h));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = 1.0 / (d / D);
	tmp = 0.0;
	if (w <= -2.6e-52)
		tmp = 0.0;
	elseif (w <= 1.6e-73)
		tmp = (c0 / (2.0 * w)) * (2.0 / ((t_0 * t_0) / ((c0 / w) / h)));
	elseif (w <= 5.8e-14)
		tmp = 0.0;
	elseif (w <= 2.9e+65)
		tmp = ((c0 / 2.0) * ((c0 / w) * (2.0 * ((d / D) ^ 2.0)))) / (w * h);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(1.0 / N[(d / D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -2.6e-52], 0.0, If[LessEqual[w, 1.6e-73], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 5.8e-14], 0.0, If[LessEqual[w, 2.9e+65], N[(N[(N[(c0 / 2.0), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] * N[(2.0 * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -2.5999999999999999e-52 or 1.59999999999999993e-73 < w < 5.8000000000000005e-14 or 2.9e65 < w

   1. Initial program 13.4%

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

      1. +-commutative 13.4%

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

      2. +-commutative 13.4%

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

      3. times-frac 12.3%

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

      4. fma-neg 12.3%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in c0 around -inf 6.0%

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

      1. mul-1-neg 6.0%

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

      2. distribute-lft-in 2.8%

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

   7. Simplified 52.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   8. Taylor expanded in c0 around 0 52.1%

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

   if -2.5999999999999999e-52 < w < 1.59999999999999993e-73

   1. Initial program 29.2%

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

      1. +-commutative 29.2%

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

      2. +-commutative 29.2%

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

      3. times-frac 28.5%

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

      4. fma-neg 28.5%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in c0 around inf 42.2%

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

      1. *-commutative 42.2%

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

      2. associate-/l/ 42.2%

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

      3. associate-*l/ 42.2%

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

      4. associate-/l* 41.5%

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

      5. *-commutative 41.5%

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

   7. Simplified 41.5%

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

      1. expm1-log1p-u 21.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)\right)} \]

      2. expm1-udef 21.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)} - 1} \]

   9. Applied egg-rr 25.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 27.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)\right)} \]

       2. expm1-log1p 49.2%

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

       3. *-commutative 49.2%

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

       4. associate-/l* 49.2%

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

       5. associate-/r* 50.8%

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

   11. Simplified 50.8%

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

       1. metadata-eval 50.8%

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

       2. pow-prod-up 50.8%

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

       3. unpow-1 50.8%

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

       4. unpow-1 50.8%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \color{blue}{\frac{1}{\frac{d}{D}}}}{\frac{c0}{h \cdot w}}} \]

   13. Applied egg-rr 50.8%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\color{blue}{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}}{\frac{c0}{h \cdot w}}} \]
   14. Step-by-step derivation

       1. associate-/l/ 52.3%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\color{blue}{\frac{\frac{c0}{w}}{h}}}} \]

       2. div-inv 52.3%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\color{blue}{\frac{c0}{w} \cdot \frac{1}{h}}}} \]

   15. Applied egg-rr 52.3%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\color{blue}{\frac{c0}{w} \cdot \frac{1}{h}}}} \]
   16. Step-by-step derivation

       1. associate-*r/ 52.3%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\color{blue}{\frac{\frac{c0}{w} \cdot 1}{h}}}} \]

       2. *-rgt-identity 52.3%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\frac{\color{blue}{\frac{c0}{w}}}{h}}} \]

   17. Simplified 52.3%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\color{blue}{\frac{\frac{c0}{w}}{h}}}} \]

   if 5.8000000000000005e-14 < w < 2.9e65

   1. Initial program 33.8%

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

      1. +-commutative 33.8%

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

      2. +-commutative 33.8%

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

      3. times-frac 33.8%

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

      4. fma-neg 33.8%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in c0 around inf 39.7%

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

      1. *-commutative 39.7%

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

      2. associate-/l/ 50.3%

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

      3. associate-*l/ 50.3%

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

      4. associate-/l* 50.6%

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

      5. *-commutative 50.6%

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

   7. Simplified 50.6%

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

      1. expm1-log1p-u 28.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)\right)} \]

      2. expm1-udef 28.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)} - 1} \]

   9. Applied egg-rr 41.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 34.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)\right)} \]

       2. expm1-log1p 62.3%

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

       3. *-commutative 62.3%

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

       4. associate-/l* 62.4%

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

       5. associate-/r* 62.4%

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

   11. Simplified 62.4%

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

       1. metadata-eval 62.4%

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

       2. pow-prod-up 62.2%

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

       3. unpow-1 62.2%

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

       4. unpow-1 62.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \color{blue}{\frac{1}{\frac{d}{D}}}}{\frac{c0}{h \cdot w}}} \]

   13. Applied egg-rr 62.2%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\color{blue}{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}}{\frac{c0}{h \cdot w}}} \]
   14. Step-by-step derivation

       1. associate-/r* 62.2%

          \[\leadsto \color{blue}{\frac{\frac{c0}{2}}{w}} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\frac{c0}{h \cdot w}}} \]

       2. pow2 62.2%

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

       3. inv-pow 62.2%

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

       4. pow-pow 62.4%

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

       5. metadata-eval 62.4%

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

       6. pow-to-exp 28.4%

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

       7. *-commutative 28.4%

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

       8. associate-/r/ 28.4%

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

       9. *-commutative 28.4%

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

       10. pow-to-exp 62.4%

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

       11. associate-/l/ 62.4%

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

       12. associate-*r/ 62.4%

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

       13. frac-times 69.6%

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

   15. Applied egg-rr 69.5%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.6 \cdot 10^{-52}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 1.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\frac{\frac{c0}{w}}{h}}}\\ \mathbf{elif}\;w \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 2.9 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{c0}{2} \cdot \left(\frac{c0}{w} \cdot \left(2 \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)}{w \cdot h}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.9 \cdot 10^{-52}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 3.1 \cdot 10^{-73} \lor \neg \left(w \leq 2.2 \cdot 10^{-16}\right) \land w \leq 6.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{w \cdot h}}{\frac{D \cdot \frac{1}{d}}{\frac{d}{D}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= w -1.9e-52)
   0.0
   (if (or (<= w 3.1e-73) (and (not (<= w 2.2e-16)) (<= w 6.5e+65)))
     (*
      (/ c0 (* 2.0 w))
      (* 2.0 (/ (/ c0 (* w h)) (/ (* D (/ 1.0 d)) (/ d D)))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -1.9e-52) {
		tmp = 0.0;
	} else if ((w <= 3.1e-73) || (!(w <= 2.2e-16) && (w <= 6.5e+65))) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) / ((D * (1.0 / 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) :: tmp
    if (w <= (-1.9d-52)) then
        tmp = 0.0d0
    else if ((w <= 3.1d-73) .or. (.not. (w <= 2.2d-16)) .and. (w <= 6.5d+65)) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * ((c0 / (w * h)) / ((d * (1.0d0 / 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 tmp;
	if (w <= -1.9e-52) {
		tmp = 0.0;
	} else if ((w <= 3.1e-73) || (!(w <= 2.2e-16) && (w <= 6.5e+65))) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) / ((D * (1.0 / d)) / (d / D))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if w <= -1.9e-52:
		tmp = 0.0
	elif (w <= 3.1e-73) or (not (w <= 2.2e-16) and (w <= 6.5e+65)):
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) / ((D * (1.0 / d)) / (d / D))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (w <= -1.9e-52)
		tmp = 0.0;
	elseif ((w <= 3.1e-73) || (!(w <= 2.2e-16) && (w <= 6.5e+65)))
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(c0 / Float64(w * h)) / Float64(Float64(D * Float64(1.0 / d)) / Float64(d / D)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (w <= -1.9e-52)
		tmp = 0.0;
	elseif ((w <= 3.1e-73) || (~((w <= 2.2e-16)) && (w <= 6.5e+65)))
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) / ((D * (1.0 / d)) / (d / D))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[w, -1.9e-52], 0.0, If[Or[LessEqual[w, 3.1e-73], And[N[Not[LessEqual[w, 2.2e-16]], $MachinePrecision], LessEqual[w, 6.5e+65]]], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] / N[(N[(D * N[(1.0 / d), $MachinePrecision]), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if w < -1.9000000000000002e-52 or 3.09999999999999969e-73 < w < 2.2e-16 or 6.5000000000000003e65 < w

   1. Initial program 13.4%

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

      1. +-commutative 13.4%

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

      2. +-commutative 13.4%

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

      3. times-frac 12.3%

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

      4. fma-neg 12.3%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in c0 around -inf 6.0%

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

      1. mul-1-neg 6.0%

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

      2. distribute-lft-in 2.8%

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

   7. Simplified 52.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   8. Taylor expanded in c0 around 0 52.1%

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

   if -1.9000000000000002e-52 < w < 3.09999999999999969e-73 or 2.2e-16 < w < 6.5000000000000003e65

   1. Initial program 29.7%

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

      1. +-commutative 29.7%

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

      2. +-commutative 29.7%

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

      3. times-frac 29.1%

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

      4. fma-neg 29.1%

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

   3. Simplified 30.4%

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

   5. Taylor expanded in c0 around inf 41.9%

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

      1. *-commutative 41.9%

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

      2. associate-/l/ 43.2%

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

      3. associate-*l/ 43.2%

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

      4. associate-/l* 42.6%

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

      5. *-commutative 42.6%

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

   7. Simplified 42.6%

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

      1. add-cbrt-cube 41.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{\sqrt[3]{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{D}^{2}}{{d}^{2}}}}}\right) \]

      2. pow1/3 41.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{{\left(\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{D}^{2}}{{d}^{2}}\right)}^{0.3333333333333333}}}\right) \]

      3. pow3 41.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\color{blue}{\left({\left(\frac{{D}^{2}}{{d}^{2}}\right)}^{3}\right)}}^{0.3333333333333333}}\right) \]

      4. clear-num 41.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\color{blue}{\left(\frac{1}{\frac{{d}^{2}}{{D}^{2}}}\right)}}^{3}\right)}^{0.3333333333333333}}\right) \]

      5. unpow2 41.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left(\frac{1}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

      6. unpow2 41.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left(\frac{1}{\frac{d \cdot d}{\color{blue}{D \cdot D}}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

      7. frac-times 46.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left(\frac{1}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

      8. pow2 46.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left(\frac{1}{\color{blue}{{\left(\frac{d}{D}\right)}^{2}}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

      9. pow-flip 46.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\color{blue}{\left({\left(\frac{d}{D}\right)}^{\left(-2\right)}\right)}}^{3}\right)}^{0.3333333333333333}}\right) \]

      10. metadata-eval 46.4%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left({\left(\frac{d}{D}\right)}^{\color{blue}{-2}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

   9. Applied egg-rr 46.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{{\left({\left({\left(\frac{d}{D}\right)}^{-2}\right)}^{3}\right)}^{0.3333333333333333}}}\right) \]
   10. Step-by-step derivation

       1. unpow1/3 46.7%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{\sqrt[3]{{\left({\left(\frac{d}{D}\right)}^{-2}\right)}^{3}}}}\right) \]

       2. rem-cbrt-cube 52.2%

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

       3. metadata-eval 52.2%

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

       4. pow-div 52.2%

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

       5. inv-pow 52.2%

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

       6. pow1 52.2%

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

       7. associate-/r/ 52.2%

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

   11. Applied egg-rr 52.2%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{\frac{\frac{1}{d} \cdot D}{\frac{d}{D}}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.9 \cdot 10^{-52}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 3.1 \cdot 10^{-73} \lor \neg \left(w \leq 2.2 \cdot 10^{-16}\right) \land w \leq 6.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{w \cdot h}}{\frac{D \cdot \frac{1}{d}}{\frac{d}{D}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;w \leq -4.8 \cdot 10^{-53}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 2.1 \cdot 10^{-73}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \frac{t\_0}{\frac{D \cdot \frac{1}{d}}{\frac{d}{D}}}\right)\\ \mathbf{elif}\;w \leq 1.2 \cdot 10^{-9}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 1.1 \cdot 10^{+65}:\\ \;\;\;\;t\_1 \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{D}{d}}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h))) (t_1 (/ c0 (* 2.0 w))))
   (if (<= w -4.8e-53)
     0.0
     (if (<= w 2.1e-73)
       (* t_1 (* 2.0 (/ t_0 (/ (* D (/ 1.0 d)) (/ d D)))))
       (if (<= w 1.2e-9)
         0.0
         (if (<= w 1.1e+65)
           (* t_1 (/ 2.0 (/ (* (/ 1.0 (/ d D)) (/ D d)) t_0)))
           0.0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = c0 / (2.0 * w);
	double tmp;
	if (w <= -4.8e-53) {
		tmp = 0.0;
	} else if (w <= 2.1e-73) {
		tmp = t_1 * (2.0 * (t_0 / ((D * (1.0 / d)) / (d / D))));
	} else if (w <= 1.2e-9) {
		tmp = 0.0;
	} else if (w <= 1.1e+65) {
		tmp = t_1 * (2.0 / (((1.0 / (d / D)) * (D / d)) / t_0));
	} 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) :: t_1
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = c0 / (2.0d0 * w)
    if (w <= (-4.8d-53)) then
        tmp = 0.0d0
    else if (w <= 2.1d-73) then
        tmp = t_1 * (2.0d0 * (t_0 / ((d * (1.0d0 / d_1)) / (d_1 / d))))
    else if (w <= 1.2d-9) then
        tmp = 0.0d0
    else if (w <= 1.1d+65) then
        tmp = t_1 * (2.0d0 / (((1.0d0 / (d_1 / d)) * (d / d_1)) / t_0))
    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 / (w * h);
	double t_1 = c0 / (2.0 * w);
	double tmp;
	if (w <= -4.8e-53) {
		tmp = 0.0;
	} else if (w <= 2.1e-73) {
		tmp = t_1 * (2.0 * (t_0 / ((D * (1.0 / d)) / (d / D))));
	} else if (w <= 1.2e-9) {
		tmp = 0.0;
	} else if (w <= 1.1e+65) {
		tmp = t_1 * (2.0 / (((1.0 / (d / D)) * (D / d)) / t_0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w * h)
	t_1 = c0 / (2.0 * w)
	tmp = 0
	if w <= -4.8e-53:
		tmp = 0.0
	elif w <= 2.1e-73:
		tmp = t_1 * (2.0 * (t_0 / ((D * (1.0 / d)) / (d / D))))
	elif w <= 1.2e-9:
		tmp = 0.0
	elif w <= 1.1e+65:
		tmp = t_1 * (2.0 / (((1.0 / (d / D)) * (D / d)) / t_0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (w <= -4.8e-53)
		tmp = 0.0;
	elseif (w <= 2.1e-73)
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_0 / Float64(Float64(D * Float64(1.0 / d)) / Float64(d / D)))));
	elseif (w <= 1.2e-9)
		tmp = 0.0;
	elseif (w <= 1.1e+65)
		tmp = Float64(t_1 * Float64(2.0 / Float64(Float64(Float64(1.0 / Float64(d / D)) * Float64(D / d)) / t_0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w * h);
	t_1 = c0 / (2.0 * w);
	tmp = 0.0;
	if (w <= -4.8e-53)
		tmp = 0.0;
	elseif (w <= 2.1e-73)
		tmp = t_1 * (2.0 * (t_0 / ((D * (1.0 / d)) / (d / D))));
	elseif (w <= 1.2e-9)
		tmp = 0.0;
	elseif (w <= 1.1e+65)
		tmp = t_1 * (2.0 / (((1.0 / (d / D)) * (D / d)) / t_0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -4.8e-53], 0.0, If[LessEqual[w, 2.1e-73], N[(t$95$1 * N[(2.0 * N[(t$95$0 / N[(N[(D * N[(1.0 / d), $MachinePrecision]), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 1.2e-9], 0.0, If[LessEqual[w, 1.1e+65], N[(t$95$1 * N[(2.0 / N[(N[(N[(1.0 / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -4.80000000000000015e-53 or 2.0999999999999999e-73 < w < 1.2e-9 or 1.0999999999999999e65 < w

   1. Initial program 13.4%

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

      1. +-commutative 13.4%

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

      2. +-commutative 13.4%

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

      3. times-frac 12.3%

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

      4. fma-neg 12.3%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in c0 around -inf 6.0%

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

      1. mul-1-neg 6.0%

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

      2. distribute-lft-in 2.8%

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

   7. Simplified 52.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   8. Taylor expanded in c0 around 0 52.1%

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

   if -4.80000000000000015e-53 < w < 2.0999999999999999e-73

   1. Initial program 29.2%

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

      1. +-commutative 29.2%

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

      2. +-commutative 29.2%

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

      3. times-frac 28.5%

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

      4. fma-neg 28.5%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in c0 around inf 42.2%

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

      1. *-commutative 42.2%

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

      2. associate-/l/ 42.2%

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

      3. associate-*l/ 42.2%

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

      4. associate-/l* 41.5%

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

      5. *-commutative 41.5%

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

   7. Simplified 41.5%

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

      1. add-cbrt-cube 40.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{\sqrt[3]{\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{D}^{2}}{{d}^{2}}}}}\right) \]

      2. pow1/3 40.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{{\left(\left(\frac{{D}^{2}}{{d}^{2}} \cdot \frac{{D}^{2}}{{d}^{2}}\right) \cdot \frac{{D}^{2}}{{d}^{2}}\right)}^{0.3333333333333333}}}\right) \]

      3. pow3 40.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\color{blue}{\left({\left(\frac{{D}^{2}}{{d}^{2}}\right)}^{3}\right)}}^{0.3333333333333333}}\right) \]

      4. clear-num 40.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\color{blue}{\left(\frac{1}{\frac{{d}^{2}}{{D}^{2}}}\right)}}^{3}\right)}^{0.3333333333333333}}\right) \]

      5. unpow2 40.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left(\frac{1}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

      6. unpow2 40.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left(\frac{1}{\frac{d \cdot d}{\color{blue}{D \cdot D}}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

      7. frac-times 45.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left(\frac{1}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

      8. pow2 45.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left(\frac{1}{\color{blue}{{\left(\frac{d}{D}\right)}^{2}}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

      9. pow-flip 45.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\color{blue}{\left({\left(\frac{d}{D}\right)}^{\left(-2\right)}\right)}}^{3}\right)}^{0.3333333333333333}}\right) \]

      10. metadata-eval 45.1%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{{\left({\left({\left(\frac{d}{D}\right)}^{\color{blue}{-2}}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

   9. Applied egg-rr 45.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{{\left({\left({\left(\frac{d}{D}\right)}^{-2}\right)}^{3}\right)}^{0.3333333333333333}}}\right) \]
   10. Step-by-step derivation

       1. unpow1/3 45.4%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{\sqrt[3]{{\left({\left(\frac{d}{D}\right)}^{-2}\right)}^{3}}}}\right) \]

       2. rem-cbrt-cube 50.8%

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

       3. metadata-eval 50.8%

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

       4. pow-div 50.8%

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

       5. inv-pow 50.8%

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

       6. pow1 50.8%

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

       7. associate-/r/ 50.8%

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

   11. Applied egg-rr 50.8%

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

   if 1.2e-9 < w < 1.0999999999999999e65

   1. Initial program 33.8%

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

      1. +-commutative 33.8%

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

      2. +-commutative 33.8%

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

      3. times-frac 33.8%

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

      4. fma-neg 33.8%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in c0 around inf 39.7%

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

      1. *-commutative 39.7%

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

      2. associate-/l/ 50.3%

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

      3. associate-*l/ 50.3%

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

      4. associate-/l* 50.6%

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

      5. *-commutative 50.6%

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

   7. Simplified 50.6%

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

      1. expm1-log1p-u 28.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)\right)} \]

      2. expm1-udef 28.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)} - 1} \]

   9. Applied egg-rr 41.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 34.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)\right)} \]

       2. expm1-log1p 62.3%

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

       3. *-commutative 62.3%

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

       4. associate-/l* 62.4%

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

       5. associate-/r* 62.4%

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

   11. Simplified 62.4%

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

       1. metadata-eval 62.4%

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

       2. pow-prod-up 62.2%

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

       3. unpow-1 62.2%

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

       4. unpow-1 62.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \color{blue}{\frac{1}{\frac{d}{D}}}}{\frac{c0}{h \cdot w}}} \]

   13. Applied egg-rr 62.2%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\color{blue}{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}}{\frac{c0}{h \cdot w}}} \]

   14. Taylor expanded in d around 0 62.4%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\color{blue}{\frac{D}{d}} \cdot \frac{1}{\frac{d}{D}}}{\frac{c0}{h \cdot w}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -4.8 \cdot 10^{-53}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 2.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{w \cdot h}}{\frac{D \cdot \frac{1}{d}}{\frac{d}{D}}}\right)\\ \mathbf{elif}\;w \leq 1.2 \cdot 10^{-9}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 1.1 \cdot 10^{+65}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{D}{d}}{\frac{c0}{w \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{d}{D}}\\ t_1 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;w \leq -2.1 \cdot 10^{-53}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 2.9 \cdot 10^{-73}:\\ \;\;\;\;t\_1 \cdot \frac{2}{\frac{t\_0 \cdot t\_0}{\frac{\frac{c0}{w}}{h}}}\\ \mathbf{elif}\;w \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 2.6 \cdot 10^{+64}:\\ \;\;\;\;t\_1 \cdot \frac{2}{\frac{t\_0 \cdot \frac{D}{d}}{\frac{c0}{w \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ d D))) (t_1 (/ c0 (* 2.0 w))))
   (if (<= w -2.1e-53)
     0.0
     (if (<= w 2.9e-73)
       (* t_1 (/ 2.0 (/ (* t_0 t_0) (/ (/ c0 w) h))))
       (if (<= w 6.2e-12)
         0.0
         (if (<= w 2.6e+64)
           (* t_1 (/ 2.0 (/ (* t_0 (/ D d)) (/ c0 (* w h)))))
           0.0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = 1.0 / (d / D);
	double t_1 = c0 / (2.0 * w);
	double tmp;
	if (w <= -2.1e-53) {
		tmp = 0.0;
	} else if (w <= 2.9e-73) {
		tmp = t_1 * (2.0 / ((t_0 * t_0) / ((c0 / w) / h)));
	} else if (w <= 6.2e-12) {
		tmp = 0.0;
	} else if (w <= 2.6e+64) {
		tmp = t_1 * (2.0 / ((t_0 * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (d_1 / d)
    t_1 = c0 / (2.0d0 * w)
    if (w <= (-2.1d-53)) then
        tmp = 0.0d0
    else if (w <= 2.9d-73) then
        tmp = t_1 * (2.0d0 / ((t_0 * t_0) / ((c0 / w) / h)))
    else if (w <= 6.2d-12) then
        tmp = 0.0d0
    else if (w <= 2.6d+64) then
        tmp = t_1 * (2.0d0 / ((t_0 * (d / d_1)) / (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 t_0 = 1.0 / (d / D);
	double t_1 = c0 / (2.0 * w);
	double tmp;
	if (w <= -2.1e-53) {
		tmp = 0.0;
	} else if (w <= 2.9e-73) {
		tmp = t_1 * (2.0 / ((t_0 * t_0) / ((c0 / w) / h)));
	} else if (w <= 6.2e-12) {
		tmp = 0.0;
	} else if (w <= 2.6e+64) {
		tmp = t_1 * (2.0 / ((t_0 * (D / d)) / (c0 / (w * h))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = 1.0 / (d / D)
	t_1 = c0 / (2.0 * w)
	tmp = 0
	if w <= -2.1e-53:
		tmp = 0.0
	elif w <= 2.9e-73:
		tmp = t_1 * (2.0 / ((t_0 * t_0) / ((c0 / w) / h)))
	elif w <= 6.2e-12:
		tmp = 0.0
	elif w <= 2.6e+64:
		tmp = t_1 * (2.0 / ((t_0 * (D / d)) / (c0 / (w * h))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(1.0 / Float64(d / D))
	t_1 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (w <= -2.1e-53)
		tmp = 0.0;
	elseif (w <= 2.9e-73)
		tmp = Float64(t_1 * Float64(2.0 / Float64(Float64(t_0 * t_0) / Float64(Float64(c0 / w) / h))));
	elseif (w <= 6.2e-12)
		tmp = 0.0;
	elseif (w <= 2.6e+64)
		tmp = Float64(t_1 * Float64(2.0 / Float64(Float64(t_0 * 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)
	t_0 = 1.0 / (d / D);
	t_1 = c0 / (2.0 * w);
	tmp = 0.0;
	if (w <= -2.1e-53)
		tmp = 0.0;
	elseif (w <= 2.9e-73)
		tmp = t_1 * (2.0 / ((t_0 * t_0) / ((c0 / w) / h)));
	elseif (w <= 6.2e-12)
		tmp = 0.0;
	elseif (w <= 2.6e+64)
		tmp = t_1 * (2.0 / ((t_0 * (D / d)) / (c0 / (w * h))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(1.0 / N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -2.1e-53], 0.0, If[LessEqual[w, 2.9e-73], N[(t$95$1 * N[(2.0 / N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 6.2e-12], 0.0, If[LessEqual[w, 2.6e+64], N[(t$95$1 * N[(2.0 / N[(N[(t$95$0 * N[(D / d), $MachinePrecision]), $MachinePrecision] / N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -2.09999999999999977e-53 or 2.9e-73 < w < 6.2000000000000002e-12 or 2.59999999999999997e64 < w

   1. Initial program 13.4%

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

      1. +-commutative 13.4%

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

      2. +-commutative 13.4%

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

      3. times-frac 12.3%

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

      4. fma-neg 12.3%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in c0 around -inf 6.0%

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

      1. mul-1-neg 6.0%

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

      2. distribute-lft-in 2.8%

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

   7. Simplified 52.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   8. Taylor expanded in c0 around 0 52.1%

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

   if -2.09999999999999977e-53 < w < 2.9e-73

   1. Initial program 29.2%

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

      1. +-commutative 29.2%

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

      2. +-commutative 29.2%

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

      3. times-frac 28.5%

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

      4. fma-neg 28.5%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in c0 around inf 42.2%

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

      1. *-commutative 42.2%

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

      2. associate-/l/ 42.2%

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

      3. associate-*l/ 42.2%

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

      4. associate-/l* 41.5%

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

      5. *-commutative 41.5%

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

   7. Simplified 41.5%

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

      1. expm1-log1p-u 21.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)\right)} \]

      2. expm1-udef 21.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)} - 1} \]

   9. Applied egg-rr 25.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 27.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)\right)} \]

       2. expm1-log1p 49.2%

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

       3. *-commutative 49.2%

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

       4. associate-/l* 49.2%

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

       5. associate-/r* 50.8%

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

   11. Simplified 50.8%

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

       1. metadata-eval 50.8%

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

       2. pow-prod-up 50.8%

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

       3. unpow-1 50.8%

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

       4. unpow-1 50.8%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \color{blue}{\frac{1}{\frac{d}{D}}}}{\frac{c0}{h \cdot w}}} \]

   13. Applied egg-rr 50.8%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\color{blue}{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}}{\frac{c0}{h \cdot w}}} \]
   14. Step-by-step derivation

       1. associate-/l/ 52.3%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\color{blue}{\frac{\frac{c0}{w}}{h}}}} \]

       2. div-inv 52.3%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\color{blue}{\frac{c0}{w} \cdot \frac{1}{h}}}} \]

   15. Applied egg-rr 52.3%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\color{blue}{\frac{c0}{w} \cdot \frac{1}{h}}}} \]
   16. Step-by-step derivation

       1. associate-*r/ 52.3%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\color{blue}{\frac{\frac{c0}{w} \cdot 1}{h}}}} \]

       2. *-rgt-identity 52.3%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\frac{\color{blue}{\frac{c0}{w}}}{h}}} \]

   17. Simplified 52.3%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\color{blue}{\frac{\frac{c0}{w}}{h}}}} \]

   if 6.2000000000000002e-12 < w < 2.59999999999999997e64

   1. Initial program 33.8%

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

      1. +-commutative 33.8%

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

      2. +-commutative 33.8%

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

      3. times-frac 33.8%

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

      4. fma-neg 33.8%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in c0 around inf 39.7%

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

      1. *-commutative 39.7%

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

      2. associate-/l/ 50.3%

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

      3. associate-*l/ 50.3%

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

      4. associate-/l* 50.6%

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

      5. *-commutative 50.6%

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

   7. Simplified 50.6%

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

      1. expm1-log1p-u 28.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)\right)} \]

      2. expm1-udef 28.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\frac{{D}^{2}}{{d}^{2}}}\right)\right)} - 1} \]

   9. Applied egg-rr 41.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 34.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{w \cdot 2} \cdot \frac{2 \cdot \frac{\frac{c0}{h}}{w}}{{\left(\frac{d}{D}\right)}^{-2}}\right)\right)} \]

       2. expm1-log1p 62.3%

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

       3. *-commutative 62.3%

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

       4. associate-/l* 62.4%

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

       5. associate-/r* 62.4%

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

   11. Simplified 62.4%

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

       1. metadata-eval 62.4%

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

       2. pow-prod-up 62.2%

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

       3. unpow-1 62.2%

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

       4. unpow-1 62.2%

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \color{blue}{\frac{1}{\frac{d}{D}}}}{\frac{c0}{h \cdot w}}} \]

   13. Applied egg-rr 62.2%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\color{blue}{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}}{\frac{c0}{h \cdot w}}} \]

   14. Taylor expanded in d around 0 62.4%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\color{blue}{\frac{D}{d}} \cdot \frac{1}{\frac{d}{D}}}{\frac{c0}{h \cdot w}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.1 \cdot 10^{-53}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 2.9 \cdot 10^{-73}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}{\frac{\frac{c0}{w}}{h}}}\\ \mathbf{elif}\;w \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 2.6 \cdot 10^{+64}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2}{\frac{\frac{1}{\frac{d}{D}} \cdot \frac{D}{d}}{\frac{c0}{w \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.2% 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 23.1%

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

   1. +-commutative 23.1%

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

   2. +-commutative 23.1%

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

   3. times-frac 22.3%

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

   4. fma-neg 22.3%

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

3. Simplified 23.5%

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

5. Taylor expanded in c0 around -inf 3.3%

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

   1. mul-1-neg 3.3%

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

   2. distribute-lft-in 2.0%

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

7. Simplified 31.4%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

8. Taylor expanded in c0 around 0 35.0%

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

9. Final simplification 35.0%

   \[\leadsto 0 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(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))))))