Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.6% → 53.9%

Time: 47.9s

Alternatives: 6

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: 24.6% 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: 53.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{c0}{w}}{h}\\ t_1 := {\left(\frac{d}{D}\right)}^{2}\\ t_2 := t\_0 \cdot t\_1\\ t_3 := \frac{c0}{2 \cdot w}\\ t_4 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_3 \cdot \left(t\_4 + \sqrt{t\_4 \cdot t\_4 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_3 \cdot \sqrt{\mathsf{fma}\left(t\_0, t\_1, M\right) \cdot \left(t\_2 - M\right)} + t\_3 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ c0 w) h))
        (t_1 (pow (/ d D) 2.0))
        (t_2 (* t_0 t_1))
        (t_3 (/ c0 (* 2.0 w)))
        (t_4 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_3 (+ t_4 (sqrt (- (* t_4 t_4) (* M M))))) INFINITY)
     (+ (* t_3 (sqrt (* (fma t_0 t_1 M) (- t_2 M)))) (* t_3 t_2))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / w) / h;
	double t_1 = pow((d / D), 2.0);
	double t_2 = t_0 * t_1;
	double t_3 = c0 / (2.0 * w);
	double t_4 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_3 * (t_4 + sqrt(((t_4 * t_4) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (t_3 * sqrt((fma(t_0, t_1, M) * (t_2 - M)))) + (t_3 * t_2);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / w) / h)
	t_1 = Float64(d / D) ^ 2.0
	t_2 = Float64(t_0 * t_1)
	t_3 = Float64(c0 / Float64(2.0 * w))
	t_4 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_3 * Float64(t_4 + sqrt(Float64(Float64(t_4 * t_4) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(t_3 * sqrt(Float64(fma(t_0, t_1, M) * Float64(t_2 - M)))) + Float64(t_3 * t_2));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$3 * N[(t$95$4 + N[Sqrt[N[(N[(t$95$4 * t$95$4), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$3 * N[Sqrt[N[(N[(t$95$0 * t$95$1 + M), $MachinePrecision] * N[(t$95$2 - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * t$95$2), $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 73.0%

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

   3. Simplified 70.3%

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

   6. Applied egg-rr 70.3%

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

      1. fma-undefine 70.3%

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

      2. fma-define 70.3%

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

      3. pow1/2 70.3%

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

      4. pow1/2 70.3%

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

      5. pow-prod-down 73.0%

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

      6. fma-define 73.0%

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

      7. fma-neg 73.0%

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

   8. Applied egg-rr 73.0%

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

      1. clear-num 73.0%

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

      2. inv-pow 73.0%

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

      3. *-commutative 73.0%

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

   10. Applied egg-rr 73.0%

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

       1. unpow-1 73.0%

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

       2. associate-/l* 73.0%

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

   12. Simplified 73.0%

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

       1. distribute-lft-in 72.9%

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

   14. Applied egg-rr 73.6%

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

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

   1. Initial program 0.0%

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

   3. Simplified 23.7%

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

   5. Taylor expanded in M around -inf 0.0%

      \[\leadsto c0 \cdot \color{blue}{\left(-0.5 \cdot \frac{M \cdot \sqrt{-1}}{w}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto c0 \cdot \color{blue}{\frac{-0.5 \cdot \left(M \cdot \sqrt{-1}\right)}{w}} \]

      2. associate-*r* 0.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}}{w} \]

   7. Simplified 0.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}{w}} \]
   8. Step-by-step derivation

      1. add-log-exp 0.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{\log \left(e^{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}\right)}}{w} \]

      2. exp-prod 27.7%

         \[\leadsto c0 \cdot \frac{\log \color{blue}{\left({\left(e^{-0.5 \cdot M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \]

      3. exp-prod 27.7%

         \[\leadsto c0 \cdot \frac{\log \left({\color{blue}{\left({\left(e^{-0.5}\right)}^{M}\right)}}^{\left(\sqrt{-1}\right)}\right)}{w} \]

   9. Applied egg-rr 27.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{\log \left({\left({\left(e^{-0.5}\right)}^{M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \]

   10. Taylor expanded in M around 0 38.2%

       \[\leadsto c0 \cdot \frac{\log \color{blue}{1}}{w} \]

   11. Taylor expanded in c0 around 0 38.2%

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

4. Final simplification 50.1%

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

Alternative 2: 54.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot t\_1 \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(c0 \cdot \frac{1}{2 \cdot w}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_1 (+ t_0 (sqrt (- (* t_0 t_0) (* M M))))))
   (if (<= (* (/ c0 (* 2.0 w)) t_1) INFINITY)
     (* t_1 (* c0 (/ 1.0 (* 2.0 w))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_1 = t_0 + sqrt(((t_0 * t_0) - (M * M)));
	double tmp;
	if (((c0 / (2.0 * w)) * t_1) <= ((double) INFINITY)) {
		tmp = t_1 * (c0 * (1.0 / (2.0 * w)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_1 = t_0 + Math.sqrt(((t_0 * t_0) - (M * M)));
	double tmp;
	if (((c0 / (2.0 * w)) * t_1) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * (c0 * (1.0 / (2.0 * w)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_1 = t_0 + math.sqrt(((t_0 * t_0) - (M * M)))
	tmp = 0
	if ((c0 / (2.0 * w)) * t_1) <= math.inf:
		tmp = t_1 * (c0 * (1.0 / (2.0 * w)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_1 = Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * t_1) <= Inf)
		tmp = Float64(t_1 * Float64(c0 * Float64(1.0 / Float64(2.0 * w))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_1 = t_0 + sqrt(((t_0 * t_0) - (M * M)));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * t_1) <= Inf)
		tmp = t_1 * (c0 * (1.0 / (2.0 * w)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], Infinity], N[(t$95$1 * N[(c0 * N[(1.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $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 73.0%

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

      1. div-inv 73.1%

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

      2. *-commutative 73.1%

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

   4. Applied egg-rr 73.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 23.7%

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

   5. Taylor expanded in M around -inf 0.0%

      \[\leadsto c0 \cdot \color{blue}{\left(-0.5 \cdot \frac{M \cdot \sqrt{-1}}{w}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto c0 \cdot \color{blue}{\frac{-0.5 \cdot \left(M \cdot \sqrt{-1}\right)}{w}} \]

      2. associate-*r* 0.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}}{w} \]

   7. Simplified 0.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}{w}} \]
   8. Step-by-step derivation

      1. add-log-exp 0.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{\log \left(e^{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}\right)}}{w} \]

      2. exp-prod 27.7%

         \[\leadsto c0 \cdot \frac{\log \color{blue}{\left({\left(e^{-0.5 \cdot M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \]

      3. exp-prod 27.7%

         \[\leadsto c0 \cdot \frac{\log \left({\color{blue}{\left({\left(e^{-0.5}\right)}^{M}\right)}}^{\left(\sqrt{-1}\right)}\right)}{w} \]

   9. Applied egg-rr 27.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{\log \left({\left({\left(e^{-0.5}\right)}^{M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \]

   10. Taylor expanded in M around 0 38.2%

       \[\leadsto c0 \cdot \frac{\log \color{blue}{1}}{w} \]

   11. Taylor expanded in c0 around 0 38.2%

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

4. Final simplification 49.9%

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

Alternative 3: 54.3% 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 73.0%

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

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

   1. Initial program 0.0%

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

   3. Simplified 23.7%

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

   5. Taylor expanded in M around -inf 0.0%

      \[\leadsto c0 \cdot \color{blue}{\left(-0.5 \cdot \frac{M \cdot \sqrt{-1}}{w}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto c0 \cdot \color{blue}{\frac{-0.5 \cdot \left(M \cdot \sqrt{-1}\right)}{w}} \]

      2. associate-*r* 0.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}}{w} \]

   7. Simplified 0.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}{w}} \]
   8. Step-by-step derivation

      1. add-log-exp 0.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{\log \left(e^{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}\right)}}{w} \]

      2. exp-prod 27.7%

         \[\leadsto c0 \cdot \frac{\log \color{blue}{\left({\left(e^{-0.5 \cdot M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \]

      3. exp-prod 27.7%

         \[\leadsto c0 \cdot \frac{\log \left({\color{blue}{\left({\left(e^{-0.5}\right)}^{M}\right)}}^{\left(\sqrt{-1}\right)}\right)}{w} \]

   9. Applied egg-rr 27.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{\log \left({\left({\left(e^{-0.5}\right)}^{M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \]

   10. Taylor expanded in M around 0 38.2%

       \[\leadsto c0 \cdot \frac{\log \color{blue}{1}}{w} \]

   11. Taylor expanded in c0 around 0 38.2%

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

Alternative 4: 34.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := t\_0 \cdot \frac{d \cdot d}{D \cdot D}\\ \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-275}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 10^{-230} \lor \neg \left(M \cdot M \leq 2 \cdot 10^{-6}\right) \land M \cdot M \leq 10^{+296}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{t\_1 \cdot t\_1 - M \cdot M} + t\_0 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h))) (t_1 (* t_0 (/ (* d d) (* D D)))))
   (if (<= (* M M) 2e-275)
     0.0
     (if (or (<= (* M M) 1e-230)
             (and (not (<= (* M M) 2e-6)) (<= (* M M) 1e+296)))
       (*
        (/ c0 (* 2.0 w))
        (+ (sqrt (- (* t_1 t_1) (* M M))) (* t_0 (* (/ d D) (/ d D)))))
       0.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d * d) / (D * D));
	double tmp;
	if ((M * M) <= 2e-275) {
		tmp = 0.0;
	} else if (((M * M) <= 1e-230) || (!((M * M) <= 2e-6) && ((M * M) <= 1e+296))) {
		tmp = (c0 / (2.0 * w)) * (sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = t_0 * ((d_1 * d_1) / (d * d))
    if ((m * m) <= 2d-275) then
        tmp = 0.0d0
    else if (((m * m) <= 1d-230) .or. (.not. ((m * m) <= 2d-6)) .and. ((m * m) <= 1d+296)) then
        tmp = (c0 / (2.0d0 * w)) * (sqrt(((t_1 * t_1) - (m * m))) + (t_0 * ((d_1 / d) * (d_1 / d))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d * d) / (D * D));
	double tmp;
	if ((M * M) <= 2e-275) {
		tmp = 0.0;
	} else if (((M * M) <= 1e-230) || (!((M * M) <= 2e-6) && ((M * M) <= 1e+296))) {
		tmp = (c0 / (2.0 * w)) * (Math.sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w * h)
	t_1 = t_0 * ((d * d) / (D * D))
	tmp = 0
	if (M * M) <= 2e-275:
		tmp = 0.0
	elif ((M * M) <= 1e-230) or (not ((M * M) <= 2e-6) and ((M * M) <= 1e+296)):
		tmp = (c0 / (2.0 * w)) * (math.sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(t_0 * Float64(Float64(d * d) / Float64(D * D)))
	tmp = 0.0
	if (Float64(M * M) <= 2e-275)
		tmp = 0.0;
	elseif ((Float64(M * M) <= 1e-230) || (!(Float64(M * M) <= 2e-6) && (Float64(M * M) <= 1e+296)))
		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)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w * h);
	t_1 = t_0 * ((d * d) / (D * D));
	tmp = 0.0;
	if ((M * M) <= 2e-275)
		tmp = 0.0;
	elseif (((M * M) <= 1e-230) || (~(((M * M) <= 2e-6)) && ((M * M) <= 1e+296)))
		tmp = (c0 / (2.0 * w)) * (sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * M), $MachinePrecision], 2e-275], 0.0, If[Or[LessEqual[N[(M * M), $MachinePrecision], 1e-230], And[N[Not[LessEqual[N[(M * M), $MachinePrecision], 2e-6]], $MachinePrecision], LessEqual[N[(M * M), $MachinePrecision], 1e+296]]], 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], 0.0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 1.99999999999999987e-275 or 1.00000000000000005e-230 < (*.f64 M M) < 1.99999999999999991e-6 or 9.99999999999999981e295 < (*.f64 M M)

   1. Initial program 18.1%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in M around -inf 0.0%

      \[\leadsto c0 \cdot \color{blue}{\left(-0.5 \cdot \frac{M \cdot \sqrt{-1}}{w}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto c0 \cdot \color{blue}{\frac{-0.5 \cdot \left(M \cdot \sqrt{-1}\right)}{w}} \]

      2. associate-*r* 0.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}}{w} \]

   7. Simplified 0.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}{w}} \]
   8. Step-by-step derivation

      1. add-log-exp 0.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{\log \left(e^{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}\right)}}{w} \]

      2. exp-prod 30.4%

         \[\leadsto c0 \cdot \frac{\log \color{blue}{\left({\left(e^{-0.5 \cdot M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \]

      3. exp-prod 30.4%

         \[\leadsto c0 \cdot \frac{\log \left({\color{blue}{\left({\left(e^{-0.5}\right)}^{M}\right)}}^{\left(\sqrt{-1}\right)}\right)}{w} \]

   9. Applied egg-rr 30.4%

      \[\leadsto c0 \cdot \frac{\color{blue}{\log \left({\left({\left(e^{-0.5}\right)}^{M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \]

   10. Taylor expanded in M around 0 34.9%

       \[\leadsto c0 \cdot \frac{\log \color{blue}{1}}{w} \]

   11. Taylor expanded in c0 around 0 34.9%

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

   if 1.99999999999999987e-275 < (*.f64 M M) < 1.00000000000000005e-230 or 1.99999999999999991e-6 < (*.f64 M M) < 9.99999999999999981e295

   1. Initial program 39.8%

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

   3. Simplified 39.5%

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

      1. times-frac 39.6%

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

   6. Applied egg-rr 39.6%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-275}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 10^{-230} \lor \neg \left(M \cdot M \leq 2 \cdot 10^{-6}\right) \land M \cdot M \leq 10^{+296}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) - M \cdot M} + \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 33.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\\ \mathbf{if}\;D \cdot D \leq 5 \cdot 10^{-254}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{+126}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* w h)) (/ (* d d) (* D D)))))
   (if (<= (* D D) 5e-254)
     0.0
     (if (<= (* D D) 5e+126)
       (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
       0.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double tmp;
	if ((D * D) <= 5e-254) {
		tmp = 0.0;
	} else if ((D * D) <= 5e+126) {
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c0 / (w * h)) * ((d_1 * d_1) / (d * d))
    if ((d * d) <= 5d-254) then
        tmp = 0.0d0
    else if ((d * d) <= 5d+126) then
        tmp = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double tmp;
	if ((D * D) <= 5e-254) {
		tmp = 0.0;
	} else if ((D * D) <= 5e+126) {
		tmp = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D))
	tmp = 0
	if (D * D) <= 5e-254:
		tmp = 0.0
	elif (D * D) <= 5e+126:
		tmp = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d * d) / Float64(D * D)))
	tmp = 0.0
	if (Float64(D * D) <= 5e-254)
		tmp = 0.0;
	elseif (Float64(D * D) <= 5e+126)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	tmp = 0.0;
	if ((D * D) <= 5e-254)
		tmp = 0.0;
	elseif ((D * D) <= 5e+126)
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	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[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(D * D), $MachinePrecision], 5e-254], 0.0, If[LessEqual[N[(D * D), $MachinePrecision], 5e+126], 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], 0.0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 D D) < 5.0000000000000003e-254 or 4.99999999999999977e126 < (*.f64 D D)

   1. Initial program 17.5%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in M around -inf 0.0%

      \[\leadsto c0 \cdot \color{blue}{\left(-0.5 \cdot \frac{M \cdot \sqrt{-1}}{w}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto c0 \cdot \color{blue}{\frac{-0.5 \cdot \left(M \cdot \sqrt{-1}\right)}{w}} \]

      2. associate-*r* 0.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}}{w} \]

   7. Simplified 0.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}{w}} \]
   8. Step-by-step derivation

      1. add-log-exp 0.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{\log \left(e^{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}\right)}}{w} \]

      2. exp-prod 22.0%

         \[\leadsto c0 \cdot \frac{\log \color{blue}{\left({\left(e^{-0.5 \cdot M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \]

      3. exp-prod 22.0%

         \[\leadsto c0 \cdot \frac{\log \left({\color{blue}{\left({\left(e^{-0.5}\right)}^{M}\right)}}^{\left(\sqrt{-1}\right)}\right)}{w} \]

   9. Applied egg-rr 22.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{\log \left({\left({\left(e^{-0.5}\right)}^{M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \]

   10. Taylor expanded in M around 0 29.9%

       \[\leadsto c0 \cdot \frac{\log \color{blue}{1}}{w} \]

   11. Taylor expanded in c0 around 0 29.9%

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

   if 5.0000000000000003e-254 < (*.f64 D D) < 4.99999999999999977e126

   1. Initial program 39.8%

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

   3. Simplified 40.8%

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

Alternative 6: 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 24.5%

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

3. Simplified 38.5%

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

5. Taylor expanded in M around -inf 0.0%

   \[\leadsto c0 \cdot \color{blue}{\left(-0.5 \cdot \frac{M \cdot \sqrt{-1}}{w}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 0.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{-0.5 \cdot \left(M \cdot \sqrt{-1}\right)}{w}} \]

   2. associate-*r* 0.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}}{w} \]

7. Simplified 0.0%

   \[\leadsto c0 \cdot \color{blue}{\frac{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}{w}} \]
8. Step-by-step derivation

   1. add-log-exp 0.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{\log \left(e^{\left(-0.5 \cdot M\right) \cdot \sqrt{-1}}\right)}}{w} \]

   2. exp-prod 21.8%

      \[\leadsto c0 \cdot \frac{\log \color{blue}{\left({\left(e^{-0.5 \cdot M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \]

   3. exp-prod 21.8%

      \[\leadsto c0 \cdot \frac{\log \left({\color{blue}{\left({\left(e^{-0.5}\right)}^{M}\right)}}^{\left(\sqrt{-1}\right)}\right)}{w} \]

9. Applied egg-rr 21.8%

   \[\leadsto c0 \cdot \frac{\color{blue}{\log \left({\left({\left(e^{-0.5}\right)}^{M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \]

10. Taylor expanded in M around 0 29.1%

    \[\leadsto c0 \cdot \frac{\log \color{blue}{1}}{w} \]

11. Taylor expanded in c0 around 0 29.1%

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

Reproduce

herbie shell --seed 2024110 
(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))))))