Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.3% → 54.0%

Time: 28.4s

Alternatives: 7

Speedup: 151.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 54.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \frac{\mathsf{fma}\left(-0.5, h \cdot M, \frac{\frac{\left(c0 \cdot 2\right) \cdot {d}^{2}}{w}}{{D}^{2}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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 71.9%

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

   3. Simplified 70.4%

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

   6. Applied egg-rr 71.0%

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

      1. associate-/r* 71.0%

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

      2. times-frac 68.9%

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

      3. times-frac 71.1%

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

   8. Simplified 71.1%

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

   9. Taylor expanded in c0 around inf 33.2%

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

       1. *-commutative 33.2%

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

       2. associate-/r* 33.1%

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

   11. Simplified 33.1%

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

   12. Taylor expanded in h around 0 74.9%

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

       1. fma-define 74.9%

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

       2. associate-*r/ 74.9%

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

       3. *-commutative 74.9%

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

       4. associate-/r* 76.0%

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

       5. associate-*r* 76.0%

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

   14. Simplified 76.0%

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

   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 15.8%

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

   5. Taylor expanded in D around 0 1.5%

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

   6. Taylor expanded in d around 0 32.8%

      \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \left(-1 \cdot \frac{M \cdot c0}{h \cdot w} + \frac{M \cdot c0}{h \cdot w}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 32.8%

         \[\leadsto 0.25 \cdot \left(h \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{M \cdot c0}{h \cdot w}\right)}\right) \]

      2. metadata-eval 32.8%

         \[\leadsto 0.25 \cdot \left(h \cdot \left(\color{blue}{0} \cdot \frac{M \cdot c0}{h \cdot w}\right)\right) \]

      3. times-frac 33.5%

         \[\leadsto 0.25 \cdot \left(h \cdot \left(0 \cdot \color{blue}{\left(\frac{M}{h} \cdot \frac{c0}{w}\right)}\right)\right) \]

   8. Simplified 33.5%

      \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{c0}{w}\right)\right)\right)} \]

   9. Taylor expanded in h around 0 43.1%

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

4. Final simplification 53.9%

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

Alternative 2: 53.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \frac{-0.5 \cdot \left(h \cdot M\right) + 2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot {D}^{2}}}{h}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 71.9%

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

   3. Simplified 70.4%

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

   6. Applied egg-rr 71.0%

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

      1. associate-/r* 71.0%

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

      2. times-frac 68.9%

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

      3. times-frac 71.1%

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

   8. Simplified 71.1%

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

   9. Taylor expanded in c0 around inf 33.2%

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

       1. *-commutative 33.2%

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

       2. associate-/r* 33.1%

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

   11. Simplified 33.1%

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

   12. Taylor expanded in h around 0 74.9%

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

   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 15.8%

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

   5. Taylor expanded in D around 0 1.5%

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

   6. Taylor expanded in d around 0 32.8%

      \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \left(-1 \cdot \frac{M \cdot c0}{h \cdot w} + \frac{M \cdot c0}{h \cdot w}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 32.8%

         \[\leadsto 0.25 \cdot \left(h \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{M \cdot c0}{h \cdot w}\right)}\right) \]

      2. metadata-eval 32.8%

         \[\leadsto 0.25 \cdot \left(h \cdot \left(\color{blue}{0} \cdot \frac{M \cdot c0}{h \cdot w}\right)\right) \]

      3. times-frac 33.5%

         \[\leadsto 0.25 \cdot \left(h \cdot \left(0 \cdot \color{blue}{\left(\frac{M}{h} \cdot \frac{c0}{w}\right)}\right)\right) \]

   8. Simplified 33.5%

      \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{c0}{w}\right)\right)\right)} \]

   9. Taylor expanded in h around 0 43.1%

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

4. Final simplification 53.5%

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

Alternative 3: 54.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 71.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 15.8%

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

   5. Taylor expanded in D around 0 1.5%

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

   6. Taylor expanded in d around 0 32.8%

      \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \left(-1 \cdot \frac{M \cdot c0}{h \cdot w} + \frac{M \cdot c0}{h \cdot w}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 32.8%

         \[\leadsto 0.25 \cdot \left(h \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{M \cdot c0}{h \cdot w}\right)}\right) \]

      2. metadata-eval 32.8%

         \[\leadsto 0.25 \cdot \left(h \cdot \left(\color{blue}{0} \cdot \frac{M \cdot c0}{h \cdot w}\right)\right) \]

      3. times-frac 33.5%

         \[\leadsto 0.25 \cdot \left(h \cdot \left(0 \cdot \color{blue}{\left(\frac{M}{h} \cdot \frac{c0}{w}\right)}\right)\right) \]

   8. Simplified 33.5%

      \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{c0}{w}\right)\right)\right)} \]

   9. Taylor expanded in h around 0 43.1%

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

Alternative 4: 31.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\\ \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-247}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ (* d d) (* D D)) (/ c0 (* w h)))))
   (if (<= (* D D) 5e-247)
     0.0
     (* (/ 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 = ((d * d) / (D * D)) * (c0 / (w * h));
	double tmp;
	if ((D * D) <= 5e-247) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((d_1 * d_1) / (d * d)) * (c0 / (w * h))
    if ((d * d) <= 5d-247) then
        tmp = 0.0d0
    else
        tmp = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * d) / (D * D)) * (c0 / (w * h));
	double tmp;
	if ((D * D) <= 5e-247) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) / (D * D)) * (c0 / (w * h))
	tmp = 0
	if (D * D) <= 5e-247:
		tmp = 0.0
	else:
		tmp = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d * d) / Float64(D * D)) * Float64(c0 / Float64(w * h)))
	tmp = 0.0
	if (Float64(D * D) <= 5e-247)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d * d) / (D * D)) * (c0 / (w * h));
	tmp = 0.0;
	if ((D * D) <= 5e-247)
		tmp = 0.0;
	else
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(D * D), $MachinePrecision], 5e-247], 0.0, 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]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 D D) < 4.99999999999999978e-247

   1. Initial program 16.1%

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

   3. Simplified 27.6%

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

   5. Taylor expanded in D around 0 5.9%

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

   6. Taylor expanded in d around 0 35.5%

      \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \left(-1 \cdot \frac{M \cdot c0}{h \cdot w} + \frac{M \cdot c0}{h \cdot w}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 35.5%

         \[\leadsto 0.25 \cdot \left(h \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{M \cdot c0}{h \cdot w}\right)}\right) \]

      2. metadata-eval 35.5%

         \[\leadsto 0.25 \cdot \left(h \cdot \left(\color{blue}{0} \cdot \frac{M \cdot c0}{h \cdot w}\right)\right) \]

      3. times-frac 34.9%

         \[\leadsto 0.25 \cdot \left(h \cdot \left(0 \cdot \color{blue}{\left(\frac{M}{h} \cdot \frac{c0}{w}\right)}\right)\right) \]

   8. Simplified 34.9%

      \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{c0}{w}\right)\right)\right)} \]

   9. Taylor expanded in h around 0 43.1%

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

   if 4.99999999999999978e-247 < (*.f64 D D)

   1. Initial program 31.7%

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

   3. Simplified 35.6%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-247}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h} + \sqrt{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right) - M \cdot M}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot d}{D \cdot D}\\ t_1 := t\_0 \cdot \frac{c0}{w \cdot h}\\ \mathbf{if}\;D \leq 2.05 \cdot 10^{-122}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{w}}{h} \cdot t\_0 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* d d) (* D D))) (t_1 (* t_0 (/ c0 (* w h)))))
   (if (<= D 2.05e-122)
     0.0
     (*
      (/ c0 (* 2.0 w))
      (+ (* (/ (/ c0 w) h) t_0) (sqrt (- (* t_1 t_1) (* M M))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * d) / (D * D);
	double t_1 = t_0 * (c0 / (w * h));
	double tmp;
	if (D <= 2.05e-122) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((((c0 / w) / h) * t_0) + sqrt(((t_1 * t_1) - (M * M))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_1 * d_1) / (d * d)
    t_1 = t_0 * (c0 / (w * h))
    if (d <= 2.05d-122) then
        tmp = 0.0d0
    else
        tmp = (c0 / (2.0d0 * w)) * ((((c0 / w) / h) * t_0) + sqrt(((t_1 * t_1) - (m * m))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * d) / (D * D);
	double t_1 = t_0 * (c0 / (w * h));
	double tmp;
	if (D <= 2.05e-122) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((((c0 / w) / h) * t_0) + Math.sqrt(((t_1 * t_1) - (M * M))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d * d) / (D * D)
	t_1 = t_0 * (c0 / (w * h))
	tmp = 0
	if D <= 2.05e-122:
		tmp = 0.0
	else:
		tmp = (c0 / (2.0 * w)) * ((((c0 / w) / h) * t_0) + math.sqrt(((t_1 * t_1) - (M * M))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d * d) / Float64(D * D))
	t_1 = Float64(t_0 * Float64(c0 / Float64(w * h)))
	tmp = 0.0
	if (D <= 2.05e-122)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(Float64(c0 / w) / h) * t_0) + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d * d) / (D * D);
	t_1 = t_0 * (c0 / (w * h));
	tmp = 0.0;
	if (D <= 2.05e-122)
		tmp = 0.0;
	else
		tmp = (c0 / (2.0 * w)) * ((((c0 / w) / h) * t_0) + sqrt(((t_1 * t_1) - (M * M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 2.05e-122], 0.0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] * t$95$0), $MachinePrecision] + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if D < 2.05e-122

   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 30.6%

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

   5. Taylor expanded in D around 0 6.9%

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

   6. Taylor expanded in d around 0 28.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \left(-1 \cdot \frac{M \cdot c0}{h \cdot w} + \frac{M \cdot c0}{h \cdot w}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 28.7%

         \[\leadsto 0.25 \cdot \left(h \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{M \cdot c0}{h \cdot w}\right)}\right) \]

      2. metadata-eval 28.7%

         \[\leadsto 0.25 \cdot \left(h \cdot \left(\color{blue}{0} \cdot \frac{M \cdot c0}{h \cdot w}\right)\right) \]

      3. times-frac 29.8%

         \[\leadsto 0.25 \cdot \left(h \cdot \left(0 \cdot \color{blue}{\left(\frac{M}{h} \cdot \frac{c0}{w}\right)}\right)\right) \]

   8. Simplified 29.8%

      \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{c0}{w}\right)\right)\right)} \]

   9. Taylor expanded in h around 0 37.1%

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

   if 2.05e-122 < D

   1. Initial program 34.7%

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

   3. Simplified 39.6%

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

   5. Taylor expanded in c0 around 0 39.6%

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

      1. *-commutative 39.6%

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

      2. associate-/r* 39.6%

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

   7. Simplified 39.6%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.05 \cdot 10^{-122}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right) - M \cdot M}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 32.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := \frac{d \cdot d}{D \cdot D} \cdot t\_0\\ \mathbf{if}\;D \leq 1.46 \cdot 10^{-121}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{t\_1 \cdot t\_1 - M \cdot M} + t\_0 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h))) (t_1 (* (/ (* d d) (* D D)) t_0)))
   (if (<= D 1.46e-121)
     0.0
     (*
      (/ c0 (* 2.0 w))
      (+ (sqrt (- (* t_1 t_1) (* M M))) (* t_0 (* (/ d D) (/ d D))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = ((d * d) / (D * D)) * t_0;
	double tmp;
	if (D <= 1.46e-121) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * (sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = ((d_1 * d_1) / (d * d)) * t_0
    if (d <= 1.46d-121) then
        tmp = 0.0d0
    else
        tmp = (c0 / (2.0d0 * w)) * (sqrt(((t_1 * t_1) - (m * m))) + (t_0 * ((d_1 / d) * (d_1 / d))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = ((d * d) / (D * D)) * t_0;
	double tmp;
	if (D <= 1.46e-121) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * (Math.sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w * h)
	t_1 = ((d * d) / (D * D)) * t_0
	tmp = 0
	if D <= 1.46e-121:
		tmp = 0.0
	else:
		tmp = (c0 / (2.0 * w)) * (math.sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(Float64(Float64(d * d) / Float64(D * D)) * t_0)
	tmp = 0.0
	if (D <= 1.46e-121)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))) + Float64(t_0 * Float64(Float64(d / D) * Float64(d / D)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w * h);
	t_1 = ((d * d) / (D * D)) * t_0;
	tmp = 0.0;
	if (D <= 1.46e-121)
		tmp = 0.0;
	else
		tmp = (c0 / (2.0 * w)) * (sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[D, 1.46e-121], 0.0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if D < 1.46000000000000005e-121

   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 30.6%

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

   5. Taylor expanded in D around 0 6.9%

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

   6. Taylor expanded in d around 0 28.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \left(-1 \cdot \frac{M \cdot c0}{h \cdot w} + \frac{M \cdot c0}{h \cdot w}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 28.7%

         \[\leadsto 0.25 \cdot \left(h \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{M \cdot c0}{h \cdot w}\right)}\right) \]

      2. metadata-eval 28.7%

         \[\leadsto 0.25 \cdot \left(h \cdot \left(\color{blue}{0} \cdot \frac{M \cdot c0}{h \cdot w}\right)\right) \]

      3. times-frac 29.8%

         \[\leadsto 0.25 \cdot \left(h \cdot \left(0 \cdot \color{blue}{\left(\frac{M}{h} \cdot \frac{c0}{w}\right)}\right)\right) \]

   8. Simplified 29.8%

      \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{c0}{w}\right)\right)\right)} \]

   9. Taylor expanded in h around 0 37.1%

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

   if 1.46000000000000005e-121 < D

   1. Initial program 34.7%

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

   3. Simplified 39.6%

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

      1. times-frac 38.3%

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

   6. Applied egg-rr 38.3%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.46 \cdot 10^{-121}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right) - M \cdot M} + \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.5% accurate, 151.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.0d0
end function

Java

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

Python

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

Julia

function code(c0, w, h, D, d, M)
	return 0.0
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 23.6%

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

3. Simplified 33.7%

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

5. Taylor expanded in D around 0 9.2%

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

6. Taylor expanded in d around 0 25.6%

   \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \left(-1 \cdot \frac{M \cdot c0}{h \cdot w} + \frac{M \cdot c0}{h \cdot w}\right)\right)} \]
7. Step-by-step derivation

   1. distribute-lft1-in 25.6%

      \[\leadsto 0.25 \cdot \left(h \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{M \cdot c0}{h \cdot w}\right)}\right) \]

   2. metadata-eval 25.6%

      \[\leadsto 0.25 \cdot \left(h \cdot \left(\color{blue}{0} \cdot \frac{M \cdot c0}{h \cdot w}\right)\right) \]

   3. times-frac 25.7%

      \[\leadsto 0.25 \cdot \left(h \cdot \left(0 \cdot \color{blue}{\left(\frac{M}{h} \cdot \frac{c0}{w}\right)}\right)\right) \]

8. Simplified 25.7%

   \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{c0}{w}\right)\right)\right)} \]

9. Taylor expanded in h around 0 32.7%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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))))))