Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.5% → 52.8%

Time: 25.1s

Alternatives: 9

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: 24.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 52.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{\frac{c0}{w}}{h}\\ t_1 := {\left(\frac{d}{D}\right)}^{2}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_2 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(t\_0, t\_1, M\_m\right)}, \sqrt{t\_0 \cdot t\_1 - M\_m}, \frac{\frac{c0}{w} \cdot t\_1}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{M\_m \cdot \left(-c0\right)}\right)}{w} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (/ c0 w) h))
        (t_1 (pow (/ d D) 2.0))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M_m M_m))))) INFINITY)
     (*
      t_2
      (fma
       (sqrt (fma t_0 t_1 M_m))
       (sqrt (- (* t_0 t_1) M_m))
       (/ (* (/ c0 w) t_1) h)))
     (* (/ (log (exp (* M_m (- c0)))) w) -0.5))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = (c0 / w) / h;
	double t_1 = pow((d / D), 2.0);
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_2 * fma(sqrt(fma(t_0, t_1, M_m)), sqrt(((t_0 * t_1) - M_m)), (((c0 / w) * t_1) / h));
	} else {
		tmp = (log(exp((M_m * -c0))) / w) * -0.5;
	}
	return tmp;
}

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(c0 / w) / h)
	t_1 = Float64(d / D) ^ 2.0
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_2 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(t_2 * fma(sqrt(fma(t_0, t_1, M_m)), sqrt(Float64(Float64(t_0 * t_1) - M_m)), Float64(Float64(Float64(c0 / w) * t_1) / h)));
	else
		tmp = Float64(Float64(log(exp(Float64(M_m * Float64(-c0)))) / w) * -0.5);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$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[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$2 * N[(N[Sqrt[N[(t$95$0 * t$95$1 + M$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(t$95$0 * t$95$1), $MachinePrecision] - M$95$m), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(c0 / w), $MachinePrecision] * t$95$1), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[Exp[N[(M$95$m * (-c0)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / w), $MachinePrecision] * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.4%

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

   3. Simplified 71.6%

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

   6. Applied egg-rr 75.1%

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

      1. associate-/r* 73.9%

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

      2. times-frac 69.0%

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

      3. times-frac 72.5%

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

   8. Simplified 72.5%

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

      1. *-un-lft-identity 72.5%

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

      2. frac-times 70.3%

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

      3. associate-/l* 72.6%

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

   10. Applied egg-rr 72.6%

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

       1. *-lft-identity 72.6%

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

       2. associate-*r/ 70.3%

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

       3. associate-*l/ 72.6%

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

       4. associate-/r* 72.4%

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

   12. Simplified 72.4%

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

       1. associate-*r/ 74.9%

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

   14. Applied egg-rr 74.9%

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

   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 14.1%

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

   5. Taylor expanded in M around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. exp-prod 27.3%

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

      3. exp-prod 30.8%

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

   9. Applied egg-rr 30.8%

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

   11. Applied egg-rr 36.8%

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

       1. *-commutative 36.8%

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

       2. associate-*l* 36.8%

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

       3. neg-mul-1 36.8%

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

   13. Simplified 36.8%

       \[\leadsto \frac{\log \color{blue}{\left(e^{c0 \cdot \left(-M\right)}\right)}}{w} \cdot -0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.7%

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

Alternative 2: 52.8% accurate, 0.4× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_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 M_m))))) INFINITY)
     (* t_0 (* 2.0 (/ (* c0 (/ (pow d 2.0) (pow D 2.0))) (* w h))))
     (* (/ (log (exp (* M_m (- c0)))) w) -0.5))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_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 * M_m))))) <= ((double) INFINITY)) {
		tmp = t_0 * (2.0 * ((c0 * (pow(d, 2.0) / pow(D, 2.0))) / (w * h)));
	} else {
		tmp = (log(exp((M_m * -c0))) / w) * -0.5;
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_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 * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (2.0 * ((c0 * (Math.pow(d, 2.0) / Math.pow(D, 2.0))) / (w * h)));
	} else {
		tmp = (Math.log(Math.exp((M_m * -c0))) / w) * -0.5;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_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 * M_m))))) <= math.inf:
		tmp = t_0 * (2.0 * ((c0 * (math.pow(d, 2.0) / math.pow(D, 2.0))) / (w * h)))
	else:
		tmp = (math.log(math.exp((M_m * -c0))) / w) * -0.5
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_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 * M_m))))) <= Inf)
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(c0 * Float64((d ^ 2.0) / (D ^ 2.0))) / Float64(w * h))));
	else
		tmp = Float64(Float64(log(exp(Float64(M_m * Float64(-c0)))) / w) * -0.5);
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_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 * M_m))))) <= Inf)
		tmp = t_0 * (2.0 * ((c0 * ((d ^ 2.0) / (D ^ 2.0))) / (w * h)));
	else
		tmp = (log(exp((M_m * -c0))) / w) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$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$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(2.0 * N[(N[(c0 * N[(N[Power[d, 2.0], $MachinePrecision] / N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[Exp[N[(M$95$m * (-c0)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / w), $MachinePrecision] * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.4%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in D around 0 59.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{-0.5 \cdot \frac{{D}^{4} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w}}{{D}^{2}}} \]
   6. Step-by-step derivation

      1. fma-define 59.0%

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

      2. associate-*r* 57.7%

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

      3. *-commutative 57.7%

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

      4. *-commutative 57.7%

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

      5. *-commutative 57.7%

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

      6. associate-*l/ 57.6%

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

      7. *-commutative 57.6%

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

      8. associate-/r* 57.7%

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

   7. Simplified 57.7%

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

   8. Taylor expanded in D around 0 69.3%

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

      1. associate-/r* 71.8%

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

      2. associate-/l* 74.1%

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

   10. Simplified 74.1%

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

   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 14.1%

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

   5. Taylor expanded in M around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. exp-prod 27.3%

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

      3. exp-prod 30.8%

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

   9. Applied egg-rr 30.8%

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

   11. Applied egg-rr 36.8%

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

       1. *-commutative 36.8%

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

       2. associate-*l* 36.8%

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

       3. neg-mul-1 36.8%

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

   13. Simplified 36.8%

       \[\leadsto \frac{\log \color{blue}{\left(e^{c0 \cdot \left(-M\right)}\right)}}{w} \cdot -0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.5%

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

Alternative 3: 52.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;\left(c0 \cdot \frac{1}{2 \cdot w}\right) \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M\_m \cdot M\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{M\_m \cdot \left(-c0\right)}\right)}{w} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* w h)) (/ (* d d) (* D D))))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m)))))
        INFINITY)
     (* (* c0 (/ 1.0 (* 2.0 w))) (+ t_0 (sqrt (- (* t_0 t_0) (* M_m M_m)))))
     (* (/ (log (exp (* M_m (- c0)))) w) -0.5))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = (c0 * (1.0 / (2.0 * w))) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))));
	} else {
		tmp = (log(exp((M_m * -c0))) / w) * -0.5;
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * (1.0 / (2.0 * w))) * (t_0 + Math.sqrt(((t_0 * t_0) - (M_m * M_m))));
	} else {
		tmp = (Math.log(Math.exp((M_m * -c0))) / w) * -0.5;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D))
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= math.inf:
		tmp = (c0 * (1.0 / (2.0 * w))) * (t_0 + math.sqrt(((t_0 * t_0) - (M_m * M_m))))
	else:
		tmp = (math.log(math.exp((M_m * -c0))) / w) * -0.5
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d * d) / Float64(D * D)))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(Float64(c0 * Float64(1.0 / Float64(2.0 * w))) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M_m * M_m)))));
	else
		tmp = Float64(Float64(log(exp(Float64(M_m * Float64(-c0)))) / w) * -0.5);
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Inf)
		tmp = (c0 * (1.0 / (2.0 * w))) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))));
	else
		tmp = (log(exp((M_m * -c0))) / w) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $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[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * N[(1.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[Exp[N[(M$95$m * (-c0)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / w), $MachinePrecision] * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.4%

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

   3. Simplified 71.6%

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

      1. div-inv 71.6%

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

      2. *-commutative 71.6%

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

   6. Applied egg-rr 71.6%

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

   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 14.1%

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

   5. Taylor expanded in M around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. exp-prod 27.3%

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

      3. exp-prod 30.8%

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

   9. Applied egg-rr 30.8%

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

   11. Applied egg-rr 36.8%

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

       1. *-commutative 36.8%

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

       2. associate-*l* 36.8%

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

       3. neg-mul-1 36.8%

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

   13. Simplified 36.8%

       \[\leadsto \frac{\log \color{blue}{\left(e^{c0 \cdot \left(-M\right)}\right)}}{w} \cdot -0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(c0 \cdot \frac{1}{2 \cdot w}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{M \cdot \left(-c0\right)}\right)}{w} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.0% accurate, 0.5× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* w h)) (/ (* d d) (* D D))))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m)))))
        INFINITY)
     (* (* c0 (/ 1.0 (* 2.0 w))) (+ t_0 (sqrt (- (* t_0 t_0) (* M_m M_m)))))
     0.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $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[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * N[(1.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $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 69.4%

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

   3. Simplified 71.6%

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

      1. div-inv 71.6%

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

      2. *-commutative 71.6%

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

   6. Applied egg-rr 71.6%

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

   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 23.3%

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

      1. fma-undefine 24.3%

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

      2. associate-*r/ 14.6%

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

      3. *-commutative 14.6%

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

      4. associate-*r* 12.7%

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

      5. associate-*r* 11.2%

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

      6. associate-/l* 9.9%

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

      7. frac-times 10.6%

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

      8. associate-*l/ 11.2%

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

      9. times-frac 19.3%

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

      10. pow2 19.3%

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

   6. Applied egg-rr 15.2%

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

   7. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. distribute-lft1-in 0.0%

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

      3. metadata-eval 0.0%

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

      4. mul0-lft 32.3%

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

   9. Simplified 32.3%

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

   10. Taylor expanded in c0 around 0 41.5%

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

4. Final simplification 50.9%

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

Alternative 5: 52.8% accurate, 0.5× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h)))
        (t_1 (* t_0 (/ (* d d) (* D D))))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M_m M_m))))) INFINITY)
     (* t_2 (+ t_1 (sqrt (- (* t_1 (* t_0 (* (/ d D) (/ d D)))) (* M_m M_m)))))
     0.0)))

C

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

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d * d) / (D * D));
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_2 * (t_3 + Math.sqrt(((t_3 * t_3) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_2 * (t_1 + Math.sqrt(((t_1 * (t_0 * ((d / D) * (d / D)))) - (M_m * M_m))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 / (w * h)
	t_1 = t_0 * ((d * d) / (D * D))
	t_2 = c0 / (2.0 * w)
	t_3 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_2 * (t_3 + math.sqrt(((t_3 * t_3) - (M_m * M_m))))) <= math.inf:
		tmp = t_2 * (t_1 + math.sqrt(((t_1 * (t_0 * ((d / D) * (d / D)))) - (M_m * M_m))))
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = c0 / (w * h);
	t_1 = t_0 * ((d * d) / (D * D));
	t_2 = c0 / (2.0 * w);
	t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M_m * M_m))))) <= Inf)
		tmp = t_2 * (t_1 + sqrt(((t_1 * (t_0 * ((d / D) * (d / D)))) - (M_m * M_m))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$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]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$2 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $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 69.4%

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

   3. Simplified 71.6%

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

      1. times-frac 71.6%

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

   6. Applied egg-rr 71.6%

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

   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 23.3%

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

      1. fma-undefine 24.3%

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

      2. associate-*r/ 14.6%

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

      3. *-commutative 14.6%

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

      4. associate-*r* 12.7%

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

      5. associate-*r* 11.2%

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

      6. associate-/l* 9.9%

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

      7. frac-times 10.6%

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

      8. associate-*l/ 11.2%

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

      9. times-frac 19.3%

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

      10. pow2 19.3%

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

   6. Applied egg-rr 15.2%

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

   7. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. distribute-lft1-in 0.0%

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

      3. metadata-eval 0.0%

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

      4. mul0-lft 32.3%

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

   9. Simplified 32.3%

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

   10. Taylor expanded in c0 around 0 41.5%

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

Alternative 6: 36.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;c0 \leq -3 \cdot 10^{+213}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -7.5 \cdot 10^{+59}:\\ \;\;\;\;c0 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(h \cdot {w}^{2}\right)}\\ \mathbf{elif}\;c0 \leq 4.9 \cdot 10^{-6}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;{\left(w \cdot 0\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= c0 -3e+213)
   0.0
   (if (<= c0 -7.5e+59)
     (* c0 (/ (* c0 (* d d)) (* (* D D) (* h (pow w 2.0)))))
     (if (<= c0 4.9e-6) 0.0 (pow (* w 0.0) -1.0)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (c0 <= -3e+213) {
		tmp = 0.0;
	} else if (c0 <= -7.5e+59) {
		tmp = c0 * ((c0 * (d * d)) / ((D * D) * (h * pow(w, 2.0))));
	} else if (c0 <= 4.9e-6) {
		tmp = 0.0;
	} else {
		tmp = pow((w * 0.0), -1.0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (c0 <= (-3d+213)) then
        tmp = 0.0d0
    else if (c0 <= (-7.5d+59)) then
        tmp = c0 * ((c0 * (d_1 * d_1)) / ((d * d) * (h * (w ** 2.0d0))))
    else if (c0 <= 4.9d-6) then
        tmp = 0.0d0
    else
        tmp = (w * 0.0d0) ** (-1.0d0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (c0 <= -3e+213) {
		tmp = 0.0;
	} else if (c0 <= -7.5e+59) {
		tmp = c0 * ((c0 * (d * d)) / ((D * D) * (h * Math.pow(w, 2.0))));
	} else if (c0 <= 4.9e-6) {
		tmp = 0.0;
	} else {
		tmp = Math.pow((w * 0.0), -1.0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if c0 <= -3e+213:
		tmp = 0.0
	elif c0 <= -7.5e+59:
		tmp = c0 * ((c0 * (d * d)) / ((D * D) * (h * math.pow(w, 2.0))))
	elif c0 <= 4.9e-6:
		tmp = 0.0
	else:
		tmp = math.pow((w * 0.0), -1.0)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (c0 <= -3e+213)
		tmp = 0.0;
	elseif (c0 <= -7.5e+59)
		tmp = Float64(c0 * Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(h * (w ^ 2.0)))));
	elseif (c0 <= 4.9e-6)
		tmp = 0.0;
	else
		tmp = Float64(w * 0.0) ^ -1.0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (c0 <= -3e+213)
		tmp = 0.0;
	elseif (c0 <= -7.5e+59)
		tmp = c0 * ((c0 * (d * d)) / ((D * D) * (h * (w ^ 2.0))));
	elseif (c0 <= 4.9e-6)
		tmp = 0.0;
	else
		tmp = (w * 0.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[c0, -3e+213], 0.0, If[LessEqual[c0, -7.5e+59], N[(c0 * N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * N[Power[w, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 4.9e-6], 0.0, N[Power[N[(w * 0.0), $MachinePrecision], -1.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -3.0000000000000001e213 or -7.4999999999999996e59 < c0 < 4.89999999999999967e-6

   1. Initial program 16.4%

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

   3. Simplified 29.7%

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

      1. fma-undefine 31.7%

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

      2. associate-*r/ 24.0%

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

      3. *-commutative 24.0%

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

      4. associate-*r* 22.5%

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

      5. associate-*r* 22.3%

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

      6. associate-/l* 22.2%

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

      7. frac-times 22.2%

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

      8. associate-*l/ 22.3%

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

      9. times-frac 27.7%

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

      10. pow2 27.7%

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

   6. Applied egg-rr 27.1%

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

   7. Taylor expanded in c0 around -inf 4.5%

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

      1. associate-*r/ 4.5%

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

      2. distribute-lft1-in 4.5%

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

      3. metadata-eval 4.5%

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

      4. mul0-lft 36.3%

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

   9. Simplified 36.3%

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

   10. Taylor expanded in c0 around 0 41.5%

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

   if -3.0000000000000001e213 < c0 < -7.4999999999999996e59

   1. Initial program 37.5%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in c0 around inf 48.4%

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

      1. *-commutative 48.4%

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

   7. Simplified 48.4%

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

      1. unpow2 48.4%

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

   9. Applied egg-rr 48.4%

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

   11. Applied egg-rr 48.4%

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

   if 4.89999999999999967e-6 < c0

   1. Initial program 24.2%

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

   3. Simplified 41.6%

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

      1. fma-undefine 41.6%

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

      2. associate-*r/ 37.3%

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

      3. *-commutative 37.3%

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

      4. associate-*r* 34.5%

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

      5. associate-*r* 28.9%

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

      6. associate-/l* 26.0%

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

      7. frac-times 27.4%

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

      8. associate-*l/ 28.9%

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

      9. times-frac 35.0%

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

      10. pow2 35.0%

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

   6. Applied egg-rr 35.7%

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

   7. Taylor expanded in c0 around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

      2. distribute-lft1-in 0.2%

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

      3. metadata-eval 0.2%

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

      4. mul0-lft 12.1%

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

   9. Simplified 12.1%

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

   11. Applied egg-rr 38.3%

       \[\leadsto \color{blue}{{\left(0 \cdot w\right)}^{-1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -3 \cdot 10^{+213}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -7.5 \cdot 10^{+59}:\\ \;\;\;\;c0 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(h \cdot {w}^{2}\right)}\\ \mathbf{elif}\;c0 \leq 4.9 \cdot 10^{-6}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;{\left(w \cdot 0\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;c0 \leq 0.00022:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;{\left(w \cdot 0\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= c0 0.00022) 0.0 (pow (* w 0.0) -1.0)))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (c0 <= 0.00022) {
		tmp = 0.0;
	} else {
		tmp = pow((w * 0.0), -1.0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (c0 <= 0.00022d0) then
        tmp = 0.0d0
    else
        tmp = (w * 0.0d0) ** (-1.0d0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (c0 <= 0.00022) {
		tmp = 0.0;
	} else {
		tmp = Math.pow((w * 0.0), -1.0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if c0 <= 0.00022:
		tmp = 0.0
	else:
		tmp = math.pow((w * 0.0), -1.0)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (c0 <= 0.00022)
		tmp = 0.0;
	else
		tmp = Float64(w * 0.0) ^ -1.0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (c0 <= 0.00022)
		tmp = 0.0;
	else
		tmp = (w * 0.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[c0, 0.00022], 0.0, N[Power[N[(w * 0.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c0 < 2.20000000000000008e-4

   1. Initial program 20.7%

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

   3. Simplified 35.6%

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

      1. fma-undefine 37.7%

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

      2. associate-*r/ 29.6%

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

      3. *-commutative 29.6%

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

      4. associate-*r* 28.3%

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

      5. associate-*r* 27.5%

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

      6. associate-/l* 26.9%

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

      7. frac-times 27.4%

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

      8. associate-*l/ 27.5%

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

      9. times-frac 33.4%

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

      10. pow2 33.4%

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

   6. Applied egg-rr 31.9%

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

   7. Taylor expanded in c0 around -inf 3.7%

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

      1. associate-*r/ 3.7%

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

      2. distribute-lft1-in 3.7%

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

      3. metadata-eval 3.7%

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

      4. mul0-lft 30.7%

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

   9. Simplified 30.7%

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

   10. Taylor expanded in c0 around 0 36.0%

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

   if 2.20000000000000008e-4 < c0

   1. Initial program 24.2%

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

   3. Simplified 41.6%

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

      1. fma-undefine 41.6%

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

      2. associate-*r/ 37.3%

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

      3. *-commutative 37.3%

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

      4. associate-*r* 34.5%

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

      5. associate-*r* 28.9%

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

      6. associate-/l* 26.0%

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

      7. frac-times 27.4%

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

      8. associate-*l/ 28.9%

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

      9. times-frac 35.0%

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

      10. pow2 35.0%

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

   6. Applied egg-rr 35.7%

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

   7. Taylor expanded in c0 around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

      2. distribute-lft1-in 0.2%

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

      3. metadata-eval 0.2%

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

      4. mul0-lft 12.1%

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

   9. Simplified 12.1%

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

   11. Applied egg-rr 38.3%

       \[\leadsto \color{blue}{{\left(0 \cdot w\right)}^{-1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq 0.00022:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;{\left(w \cdot 0\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 9 \cdot 10^{+136}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{M\_m \cdot \left(-c0\right)}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 9e+136) 0.0 (* -0.5 (/ (* M_m (- c0)) w))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 9e+136) {
		tmp = 0.0;
	} else {
		tmp = -0.5 * ((M_m * -c0) / w);
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (m_m <= 9d+136) then
        tmp = 0.0d0
    else
        tmp = (-0.5d0) * ((m_m * -c0) / w)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 9e+136) {
		tmp = 0.0;
	} else {
		tmp = -0.5 * ((M_m * -c0) / w);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if M_m <= 9e+136:
		tmp = 0.0
	else:
		tmp = -0.5 * ((M_m * -c0) / w)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 9e+136)
		tmp = 0.0;
	else
		tmp = Float64(-0.5 * Float64(Float64(M_m * Float64(-c0)) / w));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (M_m <= 9e+136)
		tmp = 0.0;
	else
		tmp = -0.5 * ((M_m * -c0) / w);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 9e+136], 0.0, N[(-0.5 * N[(N[(M$95$m * (-c0)), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 8.9999999999999999e136

   1. Initial program 24.0%

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

   3. Simplified 36.1%

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

      1. fma-undefine 37.9%

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

      2. associate-*r/ 31.2%

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

      3. *-commutative 31.2%

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

      4. associate-*r* 29.4%

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

      5. associate-*r* 27.4%

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

      6. associate-/l* 26.0%

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

      7. frac-times 26.9%

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

      8. associate-*l/ 27.4%

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

      9. times-frac 32.4%

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

      10. pow2 32.4%

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

   6. Applied egg-rr 36.3%

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

   7. Taylor expanded in c0 around -inf 3.0%

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

      1. associate-*r/ 3.0%

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

      2. distribute-lft1-in 3.0%

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

      3. metadata-eval 3.0%

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

      4. mul0-lft 28.3%

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

   9. Simplified 28.3%

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

   10. Taylor expanded in c0 around 0 34.5%

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

   if 8.9999999999999999e136 < M

   1. Initial program 3.4%

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

   3. Simplified 35.5%

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

   5. Taylor expanded in M around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. exp-prod 0.1%

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

      3. exp-prod 0.0%

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

   9. Applied egg-rr 0.0%

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

   11. Applied egg-rr 36.1%

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

       1. *-commutative 36.1%

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

       2. mul-1-neg 36.1%

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

       3. distribute-rgt-neg-in 36.1%

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

   13. Simplified 36.1%

       \[\leadsto \frac{\color{blue}{c0 \cdot \left(-M\right)}}{w} \cdot -0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{+136}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{M \cdot \left(-c0\right)}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.1% accurate, 151.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ 0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m) :precision binary64 0.0)

C

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

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    code = 0.0d0
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	return 0.0;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	return 0.0

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := 0.0

Derivation

1. Initial program 21.7%

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

3. Simplified 37.2%

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

   1. fma-undefine 38.8%

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

   2. associate-*r/ 31.7%

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

   3. *-commutative 31.7%

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

   4. associate-*r* 30.0%

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

   5. associate-*r* 27.9%

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

   6. associate-/l* 26.7%

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

   7. frac-times 27.4%

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

   8. associate-*l/ 27.9%

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

   9. times-frac 33.9%

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

   10. pow2 33.9%

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

6. Applied egg-rr 33.0%

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

7. Taylor expanded in c0 around -inf 2.7%

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

   1. associate-*r/ 2.7%

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

   2. distribute-lft1-in 2.7%

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

   3. metadata-eval 2.7%

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

   4. mul0-lft 25.6%

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

9. Simplified 25.6%

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

10. Taylor expanded in c0 around 0 32.0%

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

Reproduce

herbie shell --seed 2024180 
(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))))))