Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.8% → 60.6%

Time: 40.6s

Alternatives: 6

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: 24.8% 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: 60.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot {d}^{-2}\right) \cdot {\left(D \cdot M\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 0.25 * ((h * pow(d, -2.0)) * pow((D * M), 2.0));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 0.25 * ((h * Math.pow(d, -2.0)) * Math.pow((D * M), 2.0));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = 0.25 * ((h * math.pow(d, -2.0)) * math.pow((D * M), 2.0))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = 0.25 * ((h * (d ^ -2.0)) * ((D * M) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $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[(0.25 * N[(N[(h * N[Power[d, -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(D * M), $MachinePrecision], 2.0], $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 64.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. 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 0.7%

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

      1. times-frac 1.3%

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

   6. Applied egg-rr 1.3%

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

   7. Taylor expanded in c0 around -inf 0.5%

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

      1. mul-1-neg 0.5%

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

      2. *-commutative 0.5%

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

      3. distribute-rgt-neg-in 0.5%

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

   9. Simplified 28.0%

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

   10. Taylor expanded in c0 around 0 44.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   11. Step-by-step derivation

       1. associate-*r* 44.9%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}} \]

       2. unpow2 44.9%

          \[\leadsto 0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2}} \]

       3. unpow2 44.9%

          \[\leadsto 0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2}} \]

       4. swap-sqr 58.3%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{{d}^{2}} \]

       5. unpow2 58.3%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{{d}^{2}} \]

       6. *-commutative 58.3%

          \[\leadsto 0.25 \cdot \frac{{\color{blue}{\left(M \cdot D\right)}}^{2} \cdot h}{{d}^{2}} \]

   12. Simplified 58.3%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{\left(M \cdot D\right)}^{2} \cdot h}{{d}^{2}}} \]
   13. Step-by-step derivation

       1. div-inv 58.2%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot \frac{1}{{d}^{2}}\right)} \]

       2. pow-flip 58.3%

          \[\leadsto 0.25 \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot \color{blue}{{d}^{\left(-2\right)}}\right) \]

       3. metadata-eval 58.3%

          \[\leadsto 0.25 \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot {d}^{\color{blue}{-2}}\right) \]

   14. Applied egg-rr 58.3%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot {d}^{-2}\right)} \]
   15. Step-by-step derivation

       1. associate-*l* 60.6%

          \[\leadsto 0.25 \cdot \color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot \left(h \cdot {d}^{-2}\right)\right)} \]

       2. *-commutative 60.6%

          \[\leadsto 0.25 \cdot \left({\color{blue}{\left(D \cdot M\right)}}^{2} \cdot \left(h \cdot {d}^{-2}\right)\right) \]

   16. Simplified 60.6%

       \[\leadsto 0.25 \cdot \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot {d}^{-2}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot {d}^{-2}\right) \cdot {\left(D \cdot M\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left({d}^{-2} \cdot \left(h \cdot {\left(D \cdot M\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 0.25 * (pow(d, -2.0) * (h * pow((D * M), 2.0)));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 0.25 * (Math.pow(d, -2.0) * (h * Math.pow((D * M), 2.0)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = 0.25 * (math.pow(d, -2.0) * (h * math.pow((D * M), 2.0)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = 0.25 * ((d ^ -2.0) * (h * ((D * M) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $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[(0.25 * N[(N[Power[d, -2.0], $MachinePrecision] * N[(h * N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision]), $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 64.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. 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 0.7%

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

      1. times-frac 1.3%

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

   6. Applied egg-rr 1.3%

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

   7. Taylor expanded in c0 around -inf 0.5%

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

      1. mul-1-neg 0.5%

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

      2. *-commutative 0.5%

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

      3. distribute-rgt-neg-in 0.5%

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

   9. Simplified 28.0%

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

   10. Taylor expanded in c0 around 0 44.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   11. Step-by-step derivation

       1. associate-*r* 44.9%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}} \]

       2. unpow2 44.9%

          \[\leadsto 0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2}} \]

       3. unpow2 44.9%

          \[\leadsto 0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2}} \]

       4. swap-sqr 58.3%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{{d}^{2}} \]

       5. unpow2 58.3%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{{d}^{2}} \]

       6. *-commutative 58.3%

          \[\leadsto 0.25 \cdot \frac{{\color{blue}{\left(M \cdot D\right)}}^{2} \cdot h}{{d}^{2}} \]

   12. Simplified 58.3%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{\left(M \cdot D\right)}^{2} \cdot h}{{d}^{2}}} \]
   13. Step-by-step derivation

       1. div-inv 58.2%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot \frac{1}{{d}^{2}}\right)} \]

       2. pow-flip 58.3%

          \[\leadsto 0.25 \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot \color{blue}{{d}^{\left(-2\right)}}\right) \]

       3. metadata-eval 58.3%

          \[\leadsto 0.25 \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot {d}^{\color{blue}{-2}}\right) \]

   14. Applied egg-rr 58.3%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot {d}^{-2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left({d}^{-2} \cdot \left(h \cdot {\left(D \cdot M\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{{d}^{2}}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / pow(d, 2.0));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / Math.pow(d, 2.0));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / math.pow(d, 2.0))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / (d ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $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[(0.25 * N[(N[(h * N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $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 64.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. 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 0.7%

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

      1. times-frac 1.3%

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

   6. Applied egg-rr 1.3%

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

   7. Taylor expanded in c0 around -inf 0.5%

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

      1. mul-1-neg 0.5%

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

      2. *-commutative 0.5%

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

      3. distribute-rgt-neg-in 0.5%

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

   9. Simplified 28.0%

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

   10. Taylor expanded in c0 around 0 44.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   11. Step-by-step derivation

       1. associate-*r* 44.9%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}} \]

       2. unpow2 44.9%

          \[\leadsto 0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2}} \]

       3. unpow2 44.9%

          \[\leadsto 0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2}} \]

       4. swap-sqr 58.3%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{{d}^{2}} \]

       5. unpow2 58.3%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{{d}^{2}} \]

       6. *-commutative 58.3%

          \[\leadsto 0.25 \cdot \frac{{\color{blue}{\left(M \cdot D\right)}}^{2} \cdot h}{{d}^{2}} \]

   12. Simplified 58.3%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{\left(M \cdot D\right)}^{2} \cdot h}{{d}^{2}}} \]
   13. Step-by-step derivation

       1. unpow2 58.3%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot h}{{d}^{2}} \]

   14. Applied egg-rr 58.3%

       \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot h}{{d}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{{d}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)\right)\\ \mathbf{if}\;d \leq 7.4 \cdot 10^{-152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{{d}^{2}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+229}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0
         (* (/ c0 (* 2.0 w)) (* 2.0 (* c0 (/ (pow (/ d D) 2.0) (* w h)))))))
   (if (<= d 7.4e-152)
     t_0
     (if (<= d 5.5e+153)
       (* 0.25 (/ (* h (* (* D M) (* D M))) (pow d 2.0)))
       (if (<= d 1.7e+229) t_0 (* 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 / (2.0 * w)) * (2.0 * (c0 * (pow((d / D), 2.0) / (w * h))));
	double tmp;
	if (d <= 7.4e-152) {
		tmp = t_0;
	} else if (d <= 5.5e+153) {
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / pow(d, 2.0));
	} else if (d <= 1.7e+229) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (c0 / (2.0d0 * w)) * (2.0d0 * (c0 * (((d_1 / d) ** 2.0d0) / (w * h))))
    if (d_1 <= 7.4d-152) then
        tmp = t_0
    else if (d_1 <= 5.5d+153) then
        tmp = 0.25d0 * ((h * ((d * m) * (d * m))) / (d_1 ** 2.0d0))
    else if (d_1 <= 1.7d+229) then
        tmp = t_0
    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 t_0 = (c0 / (2.0 * w)) * (2.0 * (c0 * (Math.pow((d / D), 2.0) / (w * h))));
	double tmp;
	if (d <= 7.4e-152) {
		tmp = t_0;
	} else if (d <= 5.5e+153) {
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / Math.pow(d, 2.0));
	} else if (d <= 1.7e+229) {
		tmp = t_0;
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / (2.0 * w)) * (2.0 * (c0 * (math.pow((d / D), 2.0) / (w * h))))
	tmp = 0
	if d <= 7.4e-152:
		tmp = t_0
	elif d <= 5.5e+153:
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / math.pow(d, 2.0))
	elif d <= 1.7e+229:
		tmp = t_0
	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(2.0 * w)) * Float64(2.0 * Float64(c0 * Float64((Float64(d / D) ^ 2.0) / Float64(w * h)))))
	tmp = 0.0
	if (d <= 7.4e-152)
		tmp = t_0;
	elseif (d <= 5.5e+153)
		tmp = Float64(0.25 * Float64(Float64(h * Float64(Float64(D * M) * Float64(D * M))) / (d ^ 2.0)));
	elseif (d <= 1.7e+229)
		tmp = t_0;
	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 / (2.0 * w)) * (2.0 * (c0 * (((d / D) ^ 2.0) / (w * h))));
	tmp = 0.0;
	if (d <= 7.4e-152)
		tmp = t_0;
	elseif (d <= 5.5e+153)
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / (d ^ 2.0));
	elseif (d <= 1.7e+229)
		tmp = t_0;
	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[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(c0 * N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 7.4e-152], t$95$0, If[LessEqual[d, 5.5e+153], N[(0.25 * N[(N[(h * N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e+229], t$95$0, N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < 7.3999999999999997e-152 or 5.5000000000000003e153 < d < 1.7e229

   1. Initial program 18.3%

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

   3. Simplified 18.5%

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

      1. times-frac 17.9%

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

   6. Applied egg-rr 17.9%

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

   8. Applied egg-rr 21.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(1 \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, \sqrt{\mathsf{fma}\left({\left(\frac{c0}{w \cdot h}\right)}^{2}, {\left(\frac{d}{D}\right)}^{4}, -{M}^{2}\right)}\right)\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 21.9%

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

      2. *-commutative 21.9%

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

      3. fma-undefine 21.9%

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

      4. unsub-neg 21.9%

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

      5. *-commutative 21.9%

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

   10. Simplified 21.9%

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

   11. Taylor expanded in c0 around inf 25.6%

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

       1. *-commutative 25.6%

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

       2. times-frac 26.2%

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

       3. unpow2 26.2%

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

       4. unpow2 26.2%

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

       5. times-frac 39.9%

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

       6. unpow2 39.9%

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

       7. associate-*l/ 39.7%

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

       8. associate-*r/ 39.3%

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

   13. Simplified 39.3%

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

   if 7.3999999999999997e-152 < d < 5.5000000000000003e153

   1. Initial program 25.2%

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

   3. Simplified 22.1%

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

      1. times-frac 23.6%

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

   6. Applied egg-rr 23.6%

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

   7. Taylor expanded in c0 around -inf 4.9%

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

      1. mul-1-neg 4.9%

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

      2. *-commutative 4.9%

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

      3. distribute-rgt-neg-in 4.9%

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

   9. Simplified 16.4%

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

   10. Taylor expanded in c0 around 0 51.3%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   11. Step-by-step derivation

       1. associate-*r* 49.8%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}} \]

       2. unpow2 49.8%

          \[\leadsto 0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2}} \]

       3. unpow2 49.8%

          \[\leadsto 0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2}} \]

       4. swap-sqr 54.8%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{{d}^{2}} \]

       5. unpow2 54.8%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{{d}^{2}} \]

       6. *-commutative 54.8%

          \[\leadsto 0.25 \cdot \frac{{\color{blue}{\left(M \cdot D\right)}}^{2} \cdot h}{{d}^{2}} \]

   12. Simplified 54.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{\left(M \cdot D\right)}^{2} \cdot h}{{d}^{2}}} \]
   13. Step-by-step derivation

       1. unpow2 54.8%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot h}{{d}^{2}} \]

   14. Applied egg-rr 54.8%

       \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot h}{{d}^{2}} \]

   if 1.7e229 < d

   1. Initial program 3.1%

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

   3. Simplified 25.0%

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

   5. Taylor expanded in c0 around -inf 0.0%

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

      1. distribute-lft-in 0.0%

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

      2. mul-1-neg 0.0%

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

      3. distribute-rgt-neg-in 0.0%

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

      4. associate-/l* 0.0%

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

      5. mul-1-neg 0.0%

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

      6. associate-/l* 0.0%

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

      7. distribute-lft1-in 0.0%

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

      8. metadata-eval 0.0%

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

      9. mul0-lft 60.7%

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

      10. metadata-eval 60.7%

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

   7. Simplified 60.7%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 7.4 \cdot 10^{-152}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)\right)\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{{d}^{2}}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+229}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-160}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{{d}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 2e-160)
   (* c0 (/ 0.0 (* 2.0 w)))
   (* 0.25 (/ (* h (* (* D M) (* D M))) (pow d 2.0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 2e-160) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / pow(d, 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 (m <= 2d-160) then
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    else
        tmp = 0.25d0 * ((h * ((d * m) * (d * m))) / (d_1 ** 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 (M <= 2e-160) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / Math.pow(d, 2.0));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 2e-160:
		tmp = c0 * (0.0 / (2.0 * w))
	else:
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / math.pow(d, 2.0))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 2e-160)
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	else
		tmp = Float64(0.25 * Float64(Float64(h * Float64(Float64(D * M) * Float64(D * M))) / (d ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 2e-160)
		tmp = c0 * (0.0 / (2.0 * w));
	else
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / (d ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 2e-160], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(h * N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2e-160

   1. Initial program 20.1%

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

   3. Simplified 29.4%

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

   5. Taylor expanded in c0 around -inf 10.6%

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

      1. distribute-lft-in 9.4%

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

      2. mul-1-neg 9.4%

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

      3. distribute-rgt-neg-in 9.4%

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

      4. associate-/l* 7.2%

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

      5. mul-1-neg 7.2%

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

      6. associate-/l* 10.0%

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

      7. distribute-lft1-in 10.0%

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

      8. metadata-eval 10.0%

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

      9. mul0-lft 49.7%

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

      10. metadata-eval 49.7%

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

   7. Simplified 49.7%

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

   if 2e-160 < M

   1. Initial program 13.2%

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

   3. Simplified 13.2%

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

      1. times-frac 13.1%

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

   6. Applied egg-rr 13.1%

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

   7. Taylor expanded in c0 around -inf 0.2%

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

      1. mul-1-neg 0.2%

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

      2. *-commutative 0.2%

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

      3. distribute-rgt-neg-in 0.2%

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

   9. Simplified 19.7%

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

   10. Taylor expanded in c0 around 0 32.8%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   11. Step-by-step derivation

       1. associate-*r* 30.6%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2}} \]

       2. unpow2 30.6%

          \[\leadsto 0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2}} \]

       3. unpow2 30.6%

          \[\leadsto 0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2}} \]

       4. swap-sqr 44.9%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{{d}^{2}} \]

       5. unpow2 44.9%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{{d}^{2}} \]

       6. *-commutative 44.9%

          \[\leadsto 0.25 \cdot \frac{{\color{blue}{\left(M \cdot D\right)}}^{2} \cdot h}{{d}^{2}} \]

   12. Simplified 44.9%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{\left(M \cdot D\right)}^{2} \cdot h}{{d}^{2}}} \]
   13. Step-by-step derivation

       1. unpow2 44.9%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot h}{{d}^{2}} \]

   14. Applied egg-rr 44.9%

       \[\leadsto 0.25 \cdot \frac{\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot h}{{d}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-160}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{{d}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.1% 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 17.8%

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

3. Simplified 33.8%

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

5. Taylor expanded in c0 around -inf 7.2%

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

   1. distribute-lft-in 6.4%

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

   2. mul-1-neg 6.4%

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

   3. distribute-rgt-neg-in 6.4%

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

   4. associate-/l* 5.6%

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

   5. mul-1-neg 5.6%

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

   6. associate-/l* 6.8%

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

   7. distribute-lft1-in 6.8%

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

   8. metadata-eval 6.8%

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

   9. mul0-lft 42.1%

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

   10. metadata-eval 42.1%

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

7. Simplified 42.1%

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

Reproduce

herbie shell --seed 2024093 
(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))))))