Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.5% → 46.4%

Time: 31.5s

Alternatives: 8

Speedup: 21.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 46.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -9.4 \cdot 10^{-308}:\\ \;\;\;\;\left(c0 \cdot 2\right) \cdot \left(\left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right) \cdot \frac{0.5}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(d \cdot \frac{c0}{\left(D \cdot w\right) \cdot \sqrt{h}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= h -9.4e-308)
   (* (* c0 2.0) (* (* (pow (/ d D) 2.0) (/ (/ c0 w) h)) (/ 0.5 w)))
   (pow (* d (/ c0 (* (* D w) (sqrt h)))) 2.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -9.4e-308) {
		tmp = (c0 * 2.0) * ((pow((d / D), 2.0) * ((c0 / w) / h)) * (0.5 / w));
	} else {
		tmp = pow((d * (c0 / ((D * w) * sqrt(h)))), 2.0);
	}
	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 (h <= (-9.4d-308)) then
        tmp = (c0 * 2.0d0) * ((((d_1 / d) ** 2.0d0) * ((c0 / w) / h)) * (0.5d0 / w))
    else
        tmp = (d_1 * (c0 / ((d * w) * sqrt(h)))) ** 2.0d0
    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 (h <= -9.4e-308) {
		tmp = (c0 * 2.0) * ((Math.pow((d / D), 2.0) * ((c0 / w) / h)) * (0.5 / w));
	} else {
		tmp = Math.pow((d * (c0 / ((D * w) * Math.sqrt(h)))), 2.0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if h <= -9.4e-308:
		tmp = (c0 * 2.0) * ((math.pow((d / D), 2.0) * ((c0 / w) / h)) * (0.5 / w))
	else:
		tmp = math.pow((d * (c0 / ((D * w) * math.sqrt(h)))), 2.0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (h <= -9.4e-308)
		tmp = Float64(Float64(c0 * 2.0) * Float64(Float64((Float64(d / D) ^ 2.0) * Float64(Float64(c0 / w) / h)) * Float64(0.5 / w)));
	else
		tmp = Float64(d * Float64(c0 / Float64(Float64(D * w) * sqrt(h)))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (h <= -9.4e-308)
		tmp = (c0 * 2.0) * ((((d / D) ^ 2.0) * ((c0 / w) / h)) * (0.5 / w));
	else
		tmp = (d * (c0 / ((D * w) * sqrt(h)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[h, -9.4e-308], N[(N[(c0 * 2.0), $MachinePrecision] * N[(N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] * N[(0.5 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(d * N[(c0 / N[(N[(D * w), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -9.4000000000000009e-308

   1. Initial program 22.4%

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

   3. Simplified 37.9%

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

   5. Taylor expanded in w around 0 4.6%

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

   7. Simplified 30.7%

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

   8. Taylor expanded in D around 0 33.1%

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

      1. associate-/l* 33.2%

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

      2. associate-/r* 33.3%

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

      3. unpow2 33.3%

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

      4. unpow2 33.3%

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

      5. times-frac 45.0%

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

      6. unpow2 45.0%

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

   10. Simplified 45.0%

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

       1. associate-*r/ 43.0%

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

       2. associate-*r/ 42.9%

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

       3. *-commutative 42.9%

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

       4. associate-*l/ 43.6%

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

       5. *-commutative 43.6%

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

       6. associate-/r* 44.6%

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

   12. Applied egg-rr 44.6%

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

       1. div-inv 44.6%

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

       2. associate-*r* 44.6%

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

       3. associate-/l/ 43.6%

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

       4. *-commutative 43.6%

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

   14. Applied egg-rr 43.6%

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

       1. associate-*l* 44.4%

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

       2. associate-/r* 44.4%

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

       3. metadata-eval 44.4%

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

       4. associate-/r* 46.2%

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

   16. Simplified 46.2%

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

   if -9.4000000000000009e-308 < h

   1. Initial program 30.2%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in c0 around inf 29.7%

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

      1. div-inv 29.8%

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

      2. pow-prod-down 41.1%

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

   7. Applied egg-rr 41.1%

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

      1. unpow2 41.1%

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

      2. *-commutative 41.1%

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

      3. *-commutative 41.1%

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

   9. Applied egg-rr 41.1%

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

       1. un-div-inv 41.1%

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

       2. pow2 41.1%

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

       3. add-sqr-sqrt 41.1%

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

       4. pow2 41.1%

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

       5. sqrt-prod 41.1%

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

       6. sqrt-pow1 46.7%

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

       7. metadata-eval 46.7%

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

       8. pow1 46.7%

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

       9. *-commutative 46.7%

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

       10. sqrt-prod 47.6%

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

       11. sqrt-pow1 52.5%

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

       12. metadata-eval 52.5%

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

       13. pow1 52.5%

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

   11. Applied egg-rr 52.5%

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

       1. unpow2 52.5%

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

       2. unpow2 52.5%

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

       3. times-frac 61.0%

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

       4. unpow2 61.0%

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

       5. associate-/l* 60.0%

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

       6. associate-*r* 61.1%

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

   13. Simplified 61.1%

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

Alternative 2: 54.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (h * w));
	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 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (h * w))
	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 * (0.0 / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(h * w)))
	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(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (h * w));
	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 * (0.0 / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * w), $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[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 23.9%

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

   5. Taylor expanded in c0 around -inf 1.0%

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

      1. mul-1-neg 1.0%

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

      2. distribute-lft-in 0.3%

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

      3. mul-1-neg 0.3%

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

      4. distribute-rgt-neg-in 0.3%

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

      5. associate-/l* 2.0%

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

      6. mul-1-neg 2.0%

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

      7. associate-/l* 0.8%

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

      8. distribute-lft1-in 0.8%

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

      9. metadata-eval 0.8%

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

      10. mul0-lft 40.2%

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

      11. metadata-eval 40.2%

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

   7. Simplified 40.2%

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

4. Final simplification 53.2%

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

Alternative 3: 44.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -4.4 \cdot 10^{-154} \lor \neg \left(c0 \leq 5.5 \cdot 10^{+49}\right):\\ \;\;\;\;\left(c0 \cdot 2\right) \cdot \left(\left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right) \cdot \frac{0.5}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= c0 -4.4e-154) (not (<= c0 5.5e+49)))
   (* (* c0 2.0) (* (* (pow (/ d D) 2.0) (/ (/ c0 w) h)) (/ 0.5 w)))
   (* c0 (/ 0.0 (* 2.0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((c0 <= -4.4e-154) || !(c0 <= 5.5e+49)) {
		tmp = (c0 * 2.0) * ((pow((d / D), 2.0) * ((c0 / w) / h)) * (0.5 / w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	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 ((c0 <= (-4.4d-154)) .or. (.not. (c0 <= 5.5d+49))) then
        tmp = (c0 * 2.0d0) * ((((d_1 / d) ** 2.0d0) * ((c0 / w) / h)) * (0.5d0 / w))
    else
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    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 ((c0 <= -4.4e-154) || !(c0 <= 5.5e+49)) {
		tmp = (c0 * 2.0) * ((Math.pow((d / D), 2.0) * ((c0 / w) / h)) * (0.5 / w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (c0 <= -4.4e-154) or not (c0 <= 5.5e+49):
		tmp = (c0 * 2.0) * ((math.pow((d / D), 2.0) * ((c0 / w) / h)) * (0.5 / w))
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((c0 <= -4.4e-154) || !(c0 <= 5.5e+49))
		tmp = Float64(Float64(c0 * 2.0) * Float64(Float64((Float64(d / D) ^ 2.0) * Float64(Float64(c0 / w) / h)) * Float64(0.5 / w)));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((c0 <= -4.4e-154) || ~((c0 <= 5.5e+49)))
		tmp = (c0 * 2.0) * ((((d / D) ^ 2.0) * ((c0 / w) / h)) * (0.5 / w));
	else
		tmp = c0 * (0.0 / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[c0, -4.4e-154], N[Not[LessEqual[c0, 5.5e+49]], $MachinePrecision]], N[(N[(c0 * 2.0), $MachinePrecision] * N[(N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] * N[(0.5 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c0 < -4.40000000000000015e-154 or 5.50000000000000042e49 < c0

   1. Initial program 30.9%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in w around 0 7.4%

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

   7. Simplified 38.3%

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

   8. Taylor expanded in D around 0 42.3%

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

      1. associate-/l* 41.8%

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

      2. associate-/r* 41.9%

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

      3. unpow2 41.9%

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

      4. unpow2 41.9%

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

      5. times-frac 55.3%

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

      6. unpow2 55.3%

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

   10. Simplified 55.3%

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

       1. associate-*r/ 54.2%

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

       2. associate-*r/ 53.9%

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

       3. *-commutative 53.9%

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

       4. associate-*l/ 54.2%

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

       5. *-commutative 54.2%

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

       6. associate-/r* 54.9%

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

   12. Applied egg-rr 54.9%

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

       1. div-inv 54.9%

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

       2. associate-*r* 54.9%

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

       3. associate-/l/ 54.2%

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

       4. *-commutative 54.2%

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

   14. Applied egg-rr 54.2%

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

       1. associate-*l* 55.3%

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

       2. associate-/r* 55.3%

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

       3. metadata-eval 55.3%

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

       4. associate-/r* 56.6%

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

   16. Simplified 56.6%

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

   if -4.40000000000000015e-154 < c0 < 5.50000000000000042e49

   1. Initial program 18.5%

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

   3. Simplified 33.6%

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

   5. Taylor expanded in c0 around -inf 2.8%

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

      1. mul-1-neg 2.8%

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

      2. distribute-lft-in 2.8%

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

      3. mul-1-neg 2.8%

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

      4. distribute-rgt-neg-in 2.8%

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

      5. associate-/l* 1.8%

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

      6. mul-1-neg 1.8%

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

      7. associate-/l* 6.2%

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

      8. distribute-lft1-in 6.2%

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

      9. metadata-eval 6.2%

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

      10. mul0-lft 48.2%

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

      11. metadata-eval 48.2%

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

   7. Simplified 48.2%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -4.4 \cdot 10^{-154} \lor \neg \left(c0 \leq 5.5 \cdot 10^{+49}\right):\\ \;\;\;\;\left(c0 \cdot 2\right) \cdot \left(\left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}\right) \cdot \frac{0.5}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -2 \cdot 10^{-159} \lor \neg \left(c0 \leq 2.9 \cdot 10^{+49}\right):\\ \;\;\;\;c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}}{w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= c0 -2e-159) (not (<= c0 2.9e+49)))
   (* c0 (/ (* (pow (/ d D) 2.0) (/ (/ c0 w) h)) w))
   (* c0 (/ 0.0 (* 2.0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((c0 <= -2e-159) || !(c0 <= 2.9e+49)) {
		tmp = c0 * ((pow((d / D), 2.0) * ((c0 / w) / h)) / w);
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	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 ((c0 <= (-2d-159)) .or. (.not. (c0 <= 2.9d+49))) then
        tmp = c0 * ((((d_1 / d) ** 2.0d0) * ((c0 / w) / h)) / w)
    else
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    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 ((c0 <= -2e-159) || !(c0 <= 2.9e+49)) {
		tmp = c0 * ((Math.pow((d / D), 2.0) * ((c0 / w) / h)) / w);
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (c0 <= -2e-159) or not (c0 <= 2.9e+49):
		tmp = c0 * ((math.pow((d / D), 2.0) * ((c0 / w) / h)) / w)
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((c0 <= -2e-159) || !(c0 <= 2.9e+49))
		tmp = Float64(c0 * Float64(Float64((Float64(d / D) ^ 2.0) * Float64(Float64(c0 / w) / h)) / w));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((c0 <= -2e-159) || ~((c0 <= 2.9e+49)))
		tmp = c0 * ((((d / D) ^ 2.0) * ((c0 / w) / h)) / w);
	else
		tmp = c0 * (0.0 / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[c0, -2e-159], N[Not[LessEqual[c0, 2.9e+49]], $MachinePrecision]], N[(c0 * N[(N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c0 < -1.99999999999999998e-159 or 2.9e49 < c0

   1. Initial program 30.9%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in w around 0 7.4%

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

   7. Simplified 38.3%

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

   8. Taylor expanded in D around 0 42.3%

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

      1. associate-/l* 41.8%

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

      2. associate-/r* 41.9%

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

      3. unpow2 41.9%

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

      4. unpow2 41.9%

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

      5. times-frac 55.3%

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

      6. unpow2 55.3%

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

   10. Simplified 55.3%

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

       1. times-frac 55.3%

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

       2. metadata-eval 55.3%

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

       3. associate-*r/ 53.8%

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

       4. *-commutative 53.8%

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

       5. associate-*l/ 55.3%

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

       6. *-commutative 55.3%

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

       7. associate-/r* 56.6%

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

   12. Applied egg-rr 56.6%

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

   if -1.99999999999999998e-159 < c0 < 2.9e49

   1. Initial program 18.5%

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

   3. Simplified 33.6%

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

   5. Taylor expanded in c0 around -inf 2.8%

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

      1. mul-1-neg 2.8%

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

      2. distribute-lft-in 2.8%

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

      3. mul-1-neg 2.8%

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

      4. distribute-rgt-neg-in 2.8%

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

      5. associate-/l* 1.8%

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

      6. mul-1-neg 1.8%

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

      7. associate-/l* 6.2%

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

      8. distribute-lft1-in 6.2%

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

      9. metadata-eval 6.2%

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

      10. mul0-lft 48.2%

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

      11. metadata-eval 48.2%

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

   7. Simplified 48.2%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -2 \cdot 10^{-159} \lor \neg \left(c0 \leq 2.9 \cdot 10^{+49}\right):\\ \;\;\;\;c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{w}}{h}}{w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -8 \cdot 10^{-158} \lor \neg \left(c0 \leq 3.4 \cdot 10^{+49}\right):\\ \;\;\;\;c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{h \cdot w}}{w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= c0 -8e-158) (not (<= c0 3.4e+49)))
   (* c0 (/ (* (pow (/ d D) 2.0) (/ c0 (* h w))) w))
   (* c0 (/ 0.0 (* 2.0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((c0 <= -8e-158) || !(c0 <= 3.4e+49)) {
		tmp = c0 * ((pow((d / D), 2.0) * (c0 / (h * w))) / w);
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	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 ((c0 <= (-8d-158)) .or. (.not. (c0 <= 3.4d+49))) then
        tmp = c0 * ((((d_1 / d) ** 2.0d0) * (c0 / (h * w))) / w)
    else
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    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 ((c0 <= -8e-158) || !(c0 <= 3.4e+49)) {
		tmp = c0 * ((Math.pow((d / D), 2.0) * (c0 / (h * w))) / w);
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (c0 <= -8e-158) or not (c0 <= 3.4e+49):
		tmp = c0 * ((math.pow((d / D), 2.0) * (c0 / (h * w))) / w)
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((c0 <= -8e-158) || !(c0 <= 3.4e+49))
		tmp = Float64(c0 * Float64(Float64((Float64(d / D) ^ 2.0) * Float64(c0 / Float64(h * w))) / w));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((c0 <= -8e-158) || ~((c0 <= 3.4e+49)))
		tmp = c0 * ((((d / D) ^ 2.0) * (c0 / (h * w))) / w);
	else
		tmp = c0 * (0.0 / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[c0, -8e-158], N[Not[LessEqual[c0, 3.4e+49]], $MachinePrecision]], N[(c0 * N[(N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(c0 / N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c0 < -8.00000000000000052e-158 or 3.4000000000000001e49 < c0

   1. Initial program 30.9%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in w around 0 7.4%

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

   7. Simplified 38.3%

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

   8. Taylor expanded in D around 0 42.3%

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

      1. associate-/l* 41.8%

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

      2. associate-/r* 41.9%

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

      3. unpow2 41.9%

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

      4. unpow2 41.9%

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

      5. times-frac 55.3%

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

      6. unpow2 55.3%

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

   10. Simplified 55.3%

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

       1. pow1 55.3%

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

       2. times-frac 55.3%

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

       3. metadata-eval 55.3%

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

       4. associate-*r/ 53.8%

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

       5. *-commutative 53.8%

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

       6. associate-*l/ 55.3%

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

       7. *-commutative 55.3%

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

       8. associate-/r* 56.6%

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

   12. Applied egg-rr 56.6%

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

       1. unpow1 56.6%

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

       2. *-lft-identity 56.6%

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

       3. associate-/l* 55.0%

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

       4. associate-/l/ 54.3%

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

   14. Simplified 54.3%

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

       1. associate-*r/ 55.3%

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

       2. associate-/l/ 56.6%

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

       3. associate-/l/ 55.3%

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

       4. *-commutative 55.3%

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

   16. Applied egg-rr 55.3%

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

   if -8.00000000000000052e-158 < c0 < 3.4000000000000001e49

   1. Initial program 18.5%

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

   3. Simplified 33.6%

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

   5. Taylor expanded in c0 around -inf 2.8%

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

      1. mul-1-neg 2.8%

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

      2. distribute-lft-in 2.8%

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

      3. mul-1-neg 2.8%

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

      4. distribute-rgt-neg-in 2.8%

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

      5. associate-/l* 1.8%

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

      6. mul-1-neg 1.8%

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

      7. associate-/l* 6.2%

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

      8. distribute-lft1-in 6.2%

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

      9. metadata-eval 6.2%

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

      10. mul0-lft 48.2%

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

      11. metadata-eval 48.2%

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

   7. Simplified 48.2%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -8 \cdot 10^{-158} \lor \neg \left(c0 \leq 3.4 \cdot 10^{+49}\right):\\ \;\;\;\;c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{h \cdot w}}{w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -7.2 \cdot 10^{-155} \lor \neg \left(c0 \leq 2.9 \cdot 10^{-21}\right):\\ \;\;\;\;c0 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{h \cdot w}}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= c0 -7.2e-155) (not (<= c0 2.9e-21)))
   (* c0 (* (pow (/ d D) 2.0) (/ (/ c0 (* h w)) w)))
   (* c0 (/ 0.0 (* 2.0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((c0 <= -7.2e-155) || !(c0 <= 2.9e-21)) {
		tmp = c0 * (pow((d / D), 2.0) * ((c0 / (h * w)) / w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	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 ((c0 <= (-7.2d-155)) .or. (.not. (c0 <= 2.9d-21))) then
        tmp = c0 * (((d_1 / d) ** 2.0d0) * ((c0 / (h * w)) / w))
    else
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    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 ((c0 <= -7.2e-155) || !(c0 <= 2.9e-21)) {
		tmp = c0 * (Math.pow((d / D), 2.0) * ((c0 / (h * w)) / w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (c0 <= -7.2e-155) or not (c0 <= 2.9e-21):
		tmp = c0 * (math.pow((d / D), 2.0) * ((c0 / (h * w)) / w))
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((c0 <= -7.2e-155) || !(c0 <= 2.9e-21))
		tmp = Float64(c0 * Float64((Float64(d / D) ^ 2.0) * Float64(Float64(c0 / Float64(h * w)) / w)));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((c0 <= -7.2e-155) || ~((c0 <= 2.9e-21)))
		tmp = c0 * (((d / D) ^ 2.0) * ((c0 / (h * w)) / w));
	else
		tmp = c0 * (0.0 / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[c0, -7.2e-155], N[Not[LessEqual[c0, 2.9e-21]], $MachinePrecision]], N[(c0 * N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(c0 / N[(h * w), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c0 < -7.19999999999999977e-155 or 2.9e-21 < c0

   1. Initial program 29.1%

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

   3. Simplified 44.1%

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

   5. Taylor expanded in w around 0 7.1%

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

   7. Simplified 37.0%

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

   8. Taylor expanded in D around 0 40.6%

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

      1. associate-/l* 40.1%

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

      2. associate-/r* 39.7%

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

      3. unpow2 39.7%

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

      4. unpow2 39.7%

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

      5. times-frac 53.8%

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

      6. unpow2 53.8%

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

   10. Simplified 53.8%

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

       1. pow1 53.8%

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

       2. times-frac 53.8%

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

       3. metadata-eval 53.8%

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

       4. associate-*r/ 52.5%

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

       5. *-commutative 52.5%

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

       6. associate-*l/ 53.8%

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

       7. *-commutative 53.8%

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

       8. associate-/r* 54.9%

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

   12. Applied egg-rr 54.9%

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

       1. unpow1 54.9%

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

       2. *-lft-identity 54.9%

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

       3. associate-/l* 54.0%

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

       4. associate-/l/ 53.4%

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

   14. Simplified 53.4%

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

   if -7.19999999999999977e-155 < c0 < 2.9e-21

   1. Initial program 19.6%

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

   3. Simplified 33.1%

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

   5. Taylor expanded in c0 around -inf 3.5%

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

      1. mul-1-neg 3.5%

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

      2. distribute-lft-in 3.5%

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

      3. mul-1-neg 3.5%

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

      4. distribute-rgt-neg-in 3.5%

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

      5. associate-/l* 2.1%

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

      6. mul-1-neg 2.1%

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

      7. associate-/l* 7.8%

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

      8. distribute-lft1-in 7.8%

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

      9. metadata-eval 7.8%

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

      10. mul0-lft 49.6%

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

      11. metadata-eval 49.6%

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

   7. Simplified 49.6%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -7.2 \cdot 10^{-155} \lor \neg \left(c0 \leq 2.9 \cdot 10^{-21}\right):\\ \;\;\;\;c0 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{h \cdot w}}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -2 \cdot 10^{-158} \lor \neg \left(c0 \leq 2.4 \cdot 10^{-21}\right):\\ \;\;\;\;c0 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot \left(h \cdot w\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= c0 -2e-158) (not (<= c0 2.4e-21)))
   (* c0 (* (pow (/ d D) 2.0) (/ c0 (* w (* h w)))))
   (* c0 (/ 0.0 (* 2.0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((c0 <= -2e-158) || !(c0 <= 2.4e-21)) {
		tmp = c0 * (pow((d / D), 2.0) * (c0 / (w * (h * w))));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	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 ((c0 <= (-2d-158)) .or. (.not. (c0 <= 2.4d-21))) then
        tmp = c0 * (((d_1 / d) ** 2.0d0) * (c0 / (w * (h * w))))
    else
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    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 ((c0 <= -2e-158) || !(c0 <= 2.4e-21)) {
		tmp = c0 * (Math.pow((d / D), 2.0) * (c0 / (w * (h * w))));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (c0 <= -2e-158) or not (c0 <= 2.4e-21):
		tmp = c0 * (math.pow((d / D), 2.0) * (c0 / (w * (h * w))))
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((c0 <= -2e-158) || !(c0 <= 2.4e-21))
		tmp = Float64(c0 * Float64((Float64(d / D) ^ 2.0) * Float64(c0 / Float64(w * Float64(h * w)))));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((c0 <= -2e-158) || ~((c0 <= 2.4e-21)))
		tmp = c0 * (((d / D) ^ 2.0) * (c0 / (w * (h * w))));
	else
		tmp = c0 * (0.0 / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[c0, -2e-158], N[Not[LessEqual[c0, 2.4e-21]], $MachinePrecision]], N[(c0 * N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(c0 / N[(w * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c0 < -2.00000000000000013e-158 or 2.3999999999999999e-21 < c0

   1. Initial program 29.1%

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

   3. Simplified 44.1%

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

   5. Taylor expanded in w around 0 7.1%

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

   7. Simplified 37.0%

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

   8. Taylor expanded in D around 0 40.6%

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

      1. associate-/l* 40.1%

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

      2. associate-/r* 39.7%

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

      3. unpow2 39.7%

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

      4. unpow2 39.7%

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

      5. times-frac 53.8%

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

      6. unpow2 53.8%

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

   10. Simplified 53.8%

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

       1. associate-*r/ 52.8%

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

       2. associate-*r/ 52.5%

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

       3. *-commutative 52.5%

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

       4. associate-*l/ 52.8%

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

       5. *-commutative 52.8%

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

       6. associate-/r* 53.4%

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

   12. Applied egg-rr 53.4%

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

       1. associate-/l* 54.9%

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

       2. times-frac 54.9%

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

       3. metadata-eval 54.9%

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

       4. associate-/l/ 53.8%

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

       5. associate-*r/ 53.4%

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

       6. *-un-lft-identity 53.4%

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

       7. *-commutative 53.4%

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

   14. Applied egg-rr 53.4%

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

       1. associate-/l/ 50.2%

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

       2. *-commutative 50.2%

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

   16. Simplified 50.2%

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

   if -2.00000000000000013e-158 < c0 < 2.3999999999999999e-21

   1. Initial program 19.6%

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

   3. Simplified 33.1%

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

   5. Taylor expanded in c0 around -inf 3.5%

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

      1. mul-1-neg 3.5%

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

      2. distribute-lft-in 3.5%

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

      3. mul-1-neg 3.5%

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

      4. distribute-rgt-neg-in 3.5%

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

      5. associate-/l* 2.1%

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

      6. mul-1-neg 2.1%

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

      7. associate-/l* 7.8%

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

      8. distribute-lft1-in 7.8%

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

      9. metadata-eval 7.8%

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

      10. mul0-lft 49.6%

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

      11. metadata-eval 49.6%

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

   7. Simplified 49.6%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -2 \cdot 10^{-158} \lor \neg \left(c0 \leq 2.4 \cdot 10^{-21}\right):\\ \;\;\;\;c0 \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot \left(h \cdot w\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.3% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ c0 \cdot \frac{0}{2 \cdot w} \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* c0 (/ 0.0 (* 2.0 w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (0.0 / (2.0 * w));
}

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 * (0.0d0 / (2.0d0 * w))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (0.0 / (2.0 * w));
}

Python

def code(c0, w, h, D, d, M):
	return c0 * (0.0 / (2.0 * w))

Julia

function code(c0, w, h, D, d, M)
	return Float64(c0 * Float64(0.0 / Float64(2.0 * w)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = c0 * (0.0 / (2.0 * w));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.6%

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

3. Simplified 41.2%

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

5. Taylor expanded in c0 around -inf 2.5%

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

   1. mul-1-neg 2.5%

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

   2. distribute-lft-in 2.1%

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

   3. mul-1-neg 2.1%

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

   4. distribute-rgt-neg-in 2.1%

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

   5. associate-/l* 3.7%

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

   6. mul-1-neg 3.7%

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

   7. associate-/l* 3.2%

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

   8. distribute-lft1-in 3.2%

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

   9. metadata-eval 3.2%

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

   10. mul0-lft 29.7%

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

   11. metadata-eval 29.7%

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

7. Simplified 29.7%

   \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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))))))