Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.3% → 54.3%

Time: 23.4s

Alternatives: 6

Speedup: 21.6×

• could not determine a ground truth

  1. c0 = 3.963337777373968e+242
  2. w = 3.3637458133512446e-304
  3. h = -5.912281085489983e-199
  4. D = -9.14124892420494e-307
  5. d = -7.94087797825461e+273
  6. M = 2.037844172402592e-38

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

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

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

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

Initial Program: 24.3% 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: 54.3% accurate, 0.4× speedup

Math

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

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 0.5)
      (/ w (* 2.0 (/ (* c0 (pow d 2.0)) (* (* w h) (pow D 2.0))))))
     (/ (* c0 0.5) (/ 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 * 0.5) / (w / (2.0 * ((c0 * pow(d, 2.0)) / ((w * h) * pow(D, 2.0)))));
	} else {
		tmp = (c0 * 0.5) / (w / 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 * 0.5) / (w / (2.0 * ((c0 * Math.pow(d, 2.0)) / ((w * h) * Math.pow(D, 2.0)))));
	} else {
		tmp = (c0 * 0.5) / (w / 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 * 0.5) / (w / (2.0 * ((c0 * math.pow(d, 2.0)) / ((w * h) * math.pow(D, 2.0)))))
	else:
		tmp = (c0 * 0.5) / (w / 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 * 0.5) / Float64(w / Float64(2.0 * Float64(Float64(c0 * (d ^ 2.0)) / Float64(Float64(w * h) * (D ^ 2.0))))));
	else
		tmp = Float64(Float64(c0 * 0.5) / Float64(w / 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 * 0.5) / (w / (2.0 * ((c0 * (d ^ 2.0)) / ((w * h) * (D ^ 2.0)))));
	else
		tmp = (c0 * 0.5) / (w / 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 * 0.5), $MachinePrecision] / N[(w / N[(2.0 * N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * 0.5), $MachinePrecision] / N[(w / 0.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.0%

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

      1. times-frac 76.7%

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

   3. Simplified 76.7%

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

   5. Applied egg-rr 79.0%

      \[\leadsto \color{blue}{\frac{\left(c0 \cdot 0.5\right) \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)}^{2} - {M}^{2}}\right)}{w}} \]
   6. Step-by-step derivation

      1. associate-/l* 79.5%

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

      2. associate-/r* 78.4%

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

      3. unpow2 78.4%

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

      4. swap-sqr 67.8%

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

      5. unpow2 67.8%

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

      6. pow-sqr 67.8%

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

      7. metadata-eval 67.8%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0}{w \cdot h}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{\color{blue}{4}} - {M}^{2}}\right)}} \]

      8. associate-/r* 67.8%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\color{blue}{\left(\frac{\frac{c0}{w}}{h}\right)}}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}\right)}} \]

   7. Simplified 67.8%

      \[\leadsto \color{blue}{\frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{\frac{c0}{w}}{h}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}\right)}}} \]

   8. Taylor expanded in c0 around inf 85.0%

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

   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. times-frac 0.0%

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

   3. Simplified 1.3%

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

   5. Applied egg-rr 10.1%

      \[\leadsto \color{blue}{\frac{\left(c0 \cdot 0.5\right) \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)}^{2} - {M}^{2}}\right)}{w}} \]
   6. Step-by-step derivation

      1. associate-/l* 10.6%

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

      2. associate-/r* 10.5%

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

      3. unpow2 10.5%

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

      4. swap-sqr 7.3%

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

      5. unpow2 7.3%

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

      6. pow-sqr 7.3%

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

      7. metadata-eval 7.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0}{w \cdot h}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{\color{blue}{4}} - {M}^{2}}\right)}} \]

      8. associate-/r* 7.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\color{blue}{\left(\frac{\frac{c0}{w}}{h}\right)}}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}\right)}} \]

   7. Simplified 7.3%

      \[\leadsto \color{blue}{\frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{\frac{c0}{w}}{h}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}\right)}}} \]

   8. Taylor expanded in c0 around -inf 2.4%

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

      1. associate-*r* 2.4%

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

      2. mul-1-neg 2.4%

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

      3. distribute-lft1-in 2.4%

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

      4. metadata-eval 2.4%

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

      5. mul0-lft 46.3%

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

      6. distribute-lft-neg-in 46.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{-c0 \cdot 0}}} \]

      7. mul0-rgt 46.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{-\color{blue}{0}}} \]

      8. metadata-eval 46.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{0}}} \]

   10. Simplified 46.3%

       \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.9%

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

Alternative 2: 54.2% accurate, 0.4× speedup

Math

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

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 (/ c0 (/ (* (* w h) (pow D 2.0)) (pow d 2.0)))))
     (/ (* c0 0.5) (/ w 0.0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (2.0 * (c0 / (((w * h) * pow(D, 2.0)) / pow(d, 2.0))));
	} else {
		tmp = (c0 * 0.5) / (w / 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 * (c0 / (((w * h) * Math.pow(D, 2.0)) / Math.pow(d, 2.0))));
	} else {
		tmp = (c0 * 0.5) / (w / 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 * (c0 / (((w * h) * math.pow(D, 2.0)) / math.pow(d, 2.0))))
	else:
		tmp = (c0 * 0.5) / (w / 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(c0 / Float64(Float64(Float64(w * h) * (D ^ 2.0)) / (d ^ 2.0)))));
	else
		tmp = Float64(Float64(c0 * 0.5) / Float64(w / 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 * (c0 / (((w * h) * (D ^ 2.0)) / (d ^ 2.0))));
	else
		tmp = (c0 * 0.5) / (w / 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[(c0 / N[(N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * 0.5), $MachinePrecision] / N[(w / 0.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.0%

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

      1. times-frac 76.7%

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

   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. Taylor expanded in c0 around inf 83.9%

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

      1. associate-/l* 82.6%

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

   6. Simplified 82.6%

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

   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. times-frac 0.0%

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

   3. Simplified 1.3%

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

   5. Applied egg-rr 10.1%

      \[\leadsto \color{blue}{\frac{\left(c0 \cdot 0.5\right) \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)}^{2} - {M}^{2}}\right)}{w}} \]
   6. Step-by-step derivation

      1. associate-/l* 10.6%

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

      2. associate-/r* 10.5%

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

      3. unpow2 10.5%

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

      4. swap-sqr 7.3%

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

      5. unpow2 7.3%

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

      6. pow-sqr 7.3%

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

      7. metadata-eval 7.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0}{w \cdot h}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{\color{blue}{4}} - {M}^{2}}\right)}} \]

      8. associate-/r* 7.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\color{blue}{\left(\frac{\frac{c0}{w}}{h}\right)}}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}\right)}} \]

   7. Simplified 7.3%

      \[\leadsto \color{blue}{\frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{\frac{c0}{w}}{h}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}\right)}}} \]

   8. Taylor expanded in c0 around -inf 2.4%

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

      1. associate-*r* 2.4%

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

      2. mul-1-neg 2.4%

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

      3. distribute-lft1-in 2.4%

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

      4. metadata-eval 2.4%

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

      5. mul0-lft 46.3%

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

      6. distribute-lft-neg-in 46.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{-c0 \cdot 0}}} \]

      7. mul0-rgt 46.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{-\color{blue}{0}}} \]

      8. metadata-eval 46.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{0}}} \]

   10. Simplified 46.3%

       \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

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

Alternative 3: 54.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 INFINITY) t_1 (/ (* c0 0.5) (/ 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 t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (c0 * 0.5) / (w / 0.0);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = (c0 * 0.5) / (w / 0.0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = (c0 * 0.5) / (w / 0.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.0%

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

   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. times-frac 0.0%

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

   3. Simplified 1.3%

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

   5. Applied egg-rr 10.1%

      \[\leadsto \color{blue}{\frac{\left(c0 \cdot 0.5\right) \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)}^{2} - {M}^{2}}\right)}{w}} \]
   6. Step-by-step derivation

      1. associate-/l* 10.6%

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

      2. associate-/r* 10.5%

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

      3. unpow2 10.5%

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

      4. swap-sqr 7.3%

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

      5. unpow2 7.3%

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

      6. pow-sqr 7.3%

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

      7. metadata-eval 7.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0}{w \cdot h}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{\color{blue}{4}} - {M}^{2}}\right)}} \]

      8. associate-/r* 7.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\color{blue}{\left(\frac{\frac{c0}{w}}{h}\right)}}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}\right)}} \]

   7. Simplified 7.3%

      \[\leadsto \color{blue}{\frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{\frac{c0}{w}}{h}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}\right)}}} \]

   8. Taylor expanded in c0 around -inf 2.4%

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

      1. associate-*r* 2.4%

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

      2. mul-1-neg 2.4%

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

      3. distribute-lft1-in 2.4%

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

      4. metadata-eval 2.4%

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

      5. mul0-lft 46.3%

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

      6. distribute-lft-neg-in 46.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{-c0 \cdot 0}}} \]

      7. mul0-rgt 46.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{-\color{blue}{0}}} \]

      8. metadata-eval 46.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{0}}} \]

   10. Simplified 46.3%

       \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.8%

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

Alternative 4: 33.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := t_0 \cdot \frac{d \cdot d}{D \cdot D}\\ \mathbf{if}\;w \leq -5.4 \cdot 10^{-105} \lor \neg \left(w \leq -1.45 \cdot 10^{-240}\right):\\ \;\;\;\;\frac{c0 \cdot 0.5}{\frac{w}{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t_0 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h))) (t_1 (* t_0 (/ (* d d) (* D D)))))
   (if (or (<= w -5.4e-105) (not (<= w -1.45e-240)))
     (/ (* c0 0.5) (/ w 0.0))
     (*
      (/ c0 (* 2.0 w))
      (+ (* t_0 (* (/ d D) (/ d D))) (sqrt (- (* t_1 t_1) (* M M))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d * d) / (D * D));
	double tmp;
	if ((w <= -5.4e-105) || !(w <= -1.45e-240)) {
		tmp = (c0 * 0.5) / (w / 0.0);
	} else {
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d / D) * (d / D))) + sqrt(((t_1 * t_1) - (M * M))));
	}
	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 = t_0 * ((d_1 * d_1) / (d * d))
    if ((w <= (-5.4d-105)) .or. (.not. (w <= (-1.45d-240)))) then
        tmp = (c0 * 0.5d0) / (w / 0.0d0)
    else
        tmp = (c0 / (2.0d0 * w)) * ((t_0 * ((d_1 / d) * (d_1 / d))) + sqrt(((t_1 * t_1) - (m * m))))
    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 = t_0 * ((d * d) / (D * D));
	double tmp;
	if ((w <= -5.4e-105) || !(w <= -1.45e-240)) {
		tmp = (c0 * 0.5) / (w / 0.0);
	} else {
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d / D) * (d / D))) + Math.sqrt(((t_1 * t_1) - (M * M))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w * h)
	t_1 = t_0 * ((d * d) / (D * D))
	tmp = 0
	if (w <= -5.4e-105) or not (w <= -1.45e-240):
		tmp = (c0 * 0.5) / (w / 0.0)
	else:
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d / D) * (d / D))) + math.sqrt(((t_1 * t_1) - (M * M))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(t_0 * Float64(Float64(d * d) / Float64(D * D)))
	tmp = 0.0
	if ((w <= -5.4e-105) || !(w <= -1.45e-240))
		tmp = Float64(Float64(c0 * 0.5) / Float64(w / 0.0));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(t_0 * Float64(Float64(d / D) * Float64(d / D))) + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w * h);
	t_1 = t_0 * ((d * d) / (D * D));
	tmp = 0.0;
	if ((w <= -5.4e-105) || ~((w <= -1.45e-240)))
		tmp = (c0 * 0.5) / (w / 0.0);
	else
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d / D) * (d / D))) + sqrt(((t_1 * t_1) - (M * M))));
	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[(t$95$0 * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[w, -5.4e-105], N[Not[LessEqual[w, -1.45e-240]], $MachinePrecision]], N[(N[(c0 * 0.5), $MachinePrecision] / N[(w / 0.0), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if w < -5.39999999999999985e-105 or -1.4500000000000001e-240 < w

   1. Initial program 17.3%

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

      1. times-frac 16.8%

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

   3. Simplified 17.8%

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

   5. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{\frac{\left(c0 \cdot 0.5\right) \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)}^{2} - {M}^{2}}\right)}{w}} \]
   6. Step-by-step derivation

      1. associate-/l* 25.9%

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

      2. associate-/r* 25.5%

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

      3. unpow2 25.5%

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

      4. swap-sqr 21.2%

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

      5. unpow2 21.2%

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

      6. pow-sqr 21.3%

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

      7. metadata-eval 21.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0}{w \cdot h}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{\color{blue}{4}} - {M}^{2}}\right)}} \]

      8. associate-/r* 21.2%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\color{blue}{\left(\frac{\frac{c0}{w}}{h}\right)}}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}\right)}} \]

   7. Simplified 21.2%

      \[\leadsto \color{blue}{\frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{\frac{c0}{w}}{h}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}\right)}}} \]

   8. Taylor expanded in c0 around -inf 3.7%

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

      1. associate-*r* 3.7%

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

      2. mul-1-neg 3.7%

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

      3. distribute-lft1-in 3.7%

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

      4. metadata-eval 3.7%

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

      5. mul0-lft 38.5%

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

      6. distribute-lft-neg-in 38.5%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{-c0 \cdot 0}}} \]

      7. mul0-rgt 38.5%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{-\color{blue}{0}}} \]

      8. metadata-eval 38.5%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{0}}} \]

   10. Simplified 38.5%

       \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{0}}} \]

   if -5.39999999999999985e-105 < w < -1.4500000000000001e-240

   1. Initial program 49.7%

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

      1. times-frac 49.7%

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

   3. Simplified 49.7%

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

      1. frac-times 49.8%

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

   5. Applied egg-rr 49.8%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -5.4 \cdot 10^{-105} \lor \neg \left(w \leq -1.45 \cdot 10^{-240}\right):\\ \;\;\;\;\frac{c0 \cdot 0.5}{\frac{w}{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) - M \cdot M}\right)\\ \end{array} \]

Alternative 5: 33.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;w \leq -1.8 \cdot 10^{-103} \lor \neg \left(w \leq -2.1 \cdot 10^{-232}\right):\\ \;\;\;\;\frac{c0 \cdot 0.5}{\frac{w}{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{t_0 \cdot t_0 - M \cdot M} + \frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (or (<= w -1.8e-103) (not (<= w -2.1e-232)))
     (/ (* c0 0.5) (/ w 0.0))
     (*
      (/ c0 (* 2.0 w))
      (+
       (sqrt (- (* t_0 t_0) (* M M)))
       (* (/ d D) (* (/ d D) (/ c0 (* w h)))))))))

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 ((w <= -1.8e-103) || !(w <= -2.1e-232)) {
		tmp = (c0 * 0.5) / (w / 0.0);
	} else {
		tmp = (c0 / (2.0 * w)) * (sqrt(((t_0 * t_0) - (M * M))) + ((d / D) * ((d / D) * (c0 / (w * h)))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    if ((w <= (-1.8d-103)) .or. (.not. (w <= (-2.1d-232)))) then
        tmp = (c0 * 0.5d0) / (w / 0.0d0)
    else
        tmp = (c0 / (2.0d0 * w)) * (sqrt(((t_0 * t_0) - (m * m))) + ((d_1 / d) * ((d_1 / d) * (c0 / (w * h)))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((w <= -1.8e-103) || !(w <= -2.1e-232)) {
		tmp = (c0 * 0.5) / (w / 0.0);
	} else {
		tmp = (c0 / (2.0 * w)) * (Math.sqrt(((t_0 * t_0) - (M * M))) + ((d / D) * ((d / D) * (c0 / (w * h)))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (w <= -1.8e-103) or not (w <= -2.1e-232):
		tmp = (c0 * 0.5) / (w / 0.0)
	else:
		tmp = (c0 / (2.0 * w)) * (math.sqrt(((t_0 * t_0) - (M * M))) + ((d / D) * ((d / D) * (c0 / (w * h)))))
	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 ((w <= -1.8e-103) || !(w <= -2.1e-232))
		tmp = Float64(Float64(c0 * 0.5) / Float64(w / 0.0));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))) + Float64(Float64(d / D) * Float64(Float64(d / D) * Float64(c0 / Float64(w * h))))));
	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 ((w <= -1.8e-103) || ~((w <= -2.1e-232)))
		tmp = (c0 * 0.5) / (w / 0.0);
	else
		tmp = (c0 / (2.0 * w)) * (sqrt(((t_0 * t_0) - (M * M))) + ((d / D) * ((d / D) * (c0 / (w * h)))));
	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[Or[LessEqual[w, -1.8e-103], N[Not[LessEqual[w, -2.1e-232]], $MachinePrecision]], N[(N[(c0 * 0.5), $MachinePrecision] / N[(w / 0.0), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(d / D), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < -1.7999999999999999e-103 or -2.1e-232 < w

   1. Initial program 17.3%

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

      1. times-frac 16.8%

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

   3. Simplified 17.8%

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

   5. Applied egg-rr 25.4%

      \[\leadsto \color{blue}{\frac{\left(c0 \cdot 0.5\right) \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)}^{2} - {M}^{2}}\right)}{w}} \]
   6. Step-by-step derivation

      1. associate-/l* 25.9%

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

      2. associate-/r* 25.5%

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

      3. unpow2 25.5%

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

      4. swap-sqr 21.2%

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

      5. unpow2 21.2%

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

      6. pow-sqr 21.3%

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

      7. metadata-eval 21.3%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0}{w \cdot h}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{\color{blue}{4}} - {M}^{2}}\right)}} \]

      8. associate-/r* 21.2%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\color{blue}{\left(\frac{\frac{c0}{w}}{h}\right)}}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}\right)}} \]

   7. Simplified 21.2%

      \[\leadsto \color{blue}{\frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{\frac{c0}{w}}{h}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}\right)}}} \]

   8. Taylor expanded in c0 around -inf 3.7%

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

      1. associate-*r* 3.7%

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

      2. mul-1-neg 3.7%

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

      3. distribute-lft1-in 3.7%

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

      4. metadata-eval 3.7%

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

      5. mul0-lft 38.5%

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

      6. distribute-lft-neg-in 38.5%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{-c0 \cdot 0}}} \]

      7. mul0-rgt 38.5%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{-\color{blue}{0}}} \]

      8. metadata-eval 38.5%

         \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{0}}} \]

   10. Simplified 38.5%

       \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{0}}} \]

   if -1.7999999999999999e-103 < w < -2.1e-232

   1. Initial program 49.7%

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

      1. frac-times 49.7%

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

      2. frac-times 49.7%

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

      3. *-commutative 49.7%

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

      4. associate-*l* 49.8%

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

   3. Applied egg-rr 49.8%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.8 \cdot 10^{-103} \lor \neg \left(w \leq -2.1 \cdot 10^{-232}\right):\\ \;\;\;\;\frac{c0 \cdot 0.5}{\frac{w}{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)\right)\\ \end{array} \]

Alternative 6: 33.3% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \frac{c0 \cdot 0.5}{\frac{w}{0}} \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (/ (* c0 0.5) (/ w 0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 * 0.5) / (w / 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 = (c0 * 0.5d0) / (w / 0.0d0)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 * 0.5) / (w / 0.0);
}

Python

def code(c0, w, h, D, d, M):
	return (c0 * 0.5) / (w / 0.0)

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 * 0.5) / Float64(w / 0.0))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (c0 * 0.5) / (w / 0.0);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 * 0.5), $MachinePrecision] / N[(w / 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.5%

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

   1. times-frac 23.1%

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

3. Simplified 23.9%

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

5. Applied egg-rr 30.8%

   \[\leadsto \color{blue}{\frac{\left(c0 \cdot 0.5\right) \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)}^{2} - {M}^{2}}\right)}{w}} \]
6. Step-by-step derivation

   1. associate-/l* 31.3%

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

   2. associate-/r* 30.9%

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

   3. unpow2 30.9%

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

   4. swap-sqr 25.5%

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

   5. unpow2 25.5%

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

   6. pow-sqr 25.5%

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

   7. metadata-eval 25.5%

      \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{c0}{w \cdot h}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{\color{blue}{4}} - {M}^{2}}\right)}} \]

   8. associate-/r* 25.5%

      \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\color{blue}{\left(\frac{\frac{c0}{w}}{h}\right)}}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}\right)}} \]

7. Simplified 25.5%

   \[\leadsto \color{blue}{\frac{c0 \cdot 0.5}{\frac{w}{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{{\left(\frac{\frac{c0}{w}}{h}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}\right)}}} \]

8. Taylor expanded in c0 around -inf 3.4%

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

   1. associate-*r* 3.4%

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

   2. mul-1-neg 3.4%

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

   3. distribute-lft1-in 3.4%

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

   4. metadata-eval 3.4%

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

   5. mul0-lft 34.9%

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

   6. distribute-lft-neg-in 34.9%

      \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{-c0 \cdot 0}}} \]

   7. mul0-rgt 34.9%

      \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{-\color{blue}{0}}} \]

   8. metadata-eval 34.9%

      \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{0}}} \]

10. Simplified 34.9%

    \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{\color{blue}{0}}} \]

11. Final simplification 34.9%

    \[\leadsto \frac{c0 \cdot 0.5}{\frac{w}{0}} \]

Reproduce

herbie shell --seed 2024024 
(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))))))