Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.2% → 44.7%

Time: 33.8s

Alternatives: 8

Speedup: 1.0×

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: 24.2% 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: 44.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in w around 0 78.9%

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

   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 11.1%

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

   5. Taylor expanded in c0 around 0 0.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot \sqrt{-1}}}{2 \cdot w} \]
   6. Step-by-step derivation

      1. add-log-exp 0.0%

         \[\leadsto c0 \cdot \color{blue}{\log \left(e^{\frac{M \cdot \sqrt{-1}}{2 \cdot w}}\right)} \]

      2. associate-/l* 0.0%

         \[\leadsto c0 \cdot \log \left(e^{\color{blue}{M \cdot \frac{\sqrt{-1}}{2 \cdot w}}}\right) \]

      3. exp-prod 35.6%

         \[\leadsto c0 \cdot \log \color{blue}{\left({\left(e^{M}\right)}^{\left(\frac{\sqrt{-1}}{2 \cdot w}\right)}\right)} \]

      4. *-commutative 35.6%

         \[\leadsto c0 \cdot \log \left({\left(e^{M}\right)}^{\left(\frac{\sqrt{-1}}{\color{blue}{w \cdot 2}}\right)}\right) \]

   7. Applied egg-rr 35.6%

      \[\leadsto c0 \cdot \color{blue}{\log \left({\left(e^{M}\right)}^{\left(\frac{\sqrt{-1}}{w \cdot 2}\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.6%

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

Alternative 2: 41.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\\ t_1 := \sqrt{\left(c0 \cdot t\_0 - M\right) \cdot \mathsf{fma}\left(c0, t\_0, M\right)}\\ \mathbf{if}\;c0 \leq -3.1 \cdot 10^{-37}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}, t\_1\right)}{2 \cdot w}\\ \mathbf{elif}\;c0 \leq 2.9 \cdot 10^{-154}:\\ \;\;\;\;\log \left({\left(e^{c0}\right)}^{\left(\frac{M}{2} \cdot \frac{\sqrt{-1}}{w}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{w \cdot \left(D \cdot \left(h \cdot D\right)\right)}, t\_1\right)}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* d (/ d (* D (* w (* h D))))))
        (t_1 (sqrt (* (- (* c0 t_0) M) (fma c0 t_0 M)))))
   (if (<= c0 -3.1e-37)
     (* c0 (/ (fma c0 (* d (/ d (* D (* D (* w h))))) t_1) (* 2.0 w)))
     (if (<= c0 2.9e-154)
       (log (pow (exp c0) (* (/ M 2.0) (/ (sqrt -1.0) w))))
       (* c0 (/ (fma c0 (* d (/ d (* w (* D (* h D))))) t_1) (* 2.0 w)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / (D * (w * (h * D))));
	double t_1 = sqrt((((c0 * t_0) - M) * fma(c0, t_0, M)));
	double tmp;
	if (c0 <= -3.1e-37) {
		tmp = c0 * (fma(c0, (d * (d / (D * (D * (w * h))))), t_1) / (2.0 * w));
	} else if (c0 <= 2.9e-154) {
		tmp = log(pow(exp(c0), ((M / 2.0) * (sqrt(-1.0) / w))));
	} else {
		tmp = c0 * (fma(c0, (d * (d / (w * (D * (h * D))))), t_1) / (2.0 * w));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d * Float64(d / Float64(D * Float64(w * Float64(h * D)))))
	t_1 = sqrt(Float64(Float64(Float64(c0 * t_0) - M) * fma(c0, t_0, M)))
	tmp = 0.0
	if (c0 <= -3.1e-37)
		tmp = Float64(c0 * Float64(fma(c0, Float64(d * Float64(d / Float64(D * Float64(D * Float64(w * h))))), t_1) / Float64(2.0 * w)));
	elseif (c0 <= 2.9e-154)
		tmp = log((exp(c0) ^ Float64(Float64(M / 2.0) * Float64(sqrt(-1.0) / w))));
	else
		tmp = Float64(c0 * Float64(fma(c0, Float64(d * Float64(d / Float64(w * Float64(D * Float64(h * D))))), t_1) / Float64(2.0 * w)));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d * N[(d / N[(D * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(N[(c0 * t$95$0), $MachinePrecision] - M), $MachinePrecision] * N[(c0 * t$95$0 + M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c0, -3.1e-37], N[(c0 * N[(N[(c0 * N[(d * N[(d / N[(D * N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 2.9e-154], N[Log[N[Power[N[Exp[c0], $MachinePrecision], N[(N[(M / 2.0), $MachinePrecision] * N[(N[Sqrt[-1.0], $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(c0 * N[(N[(c0 * N[(d * N[(d / N[(w * N[(D * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -3.09999999999999993e-37

   1. Initial program 23.7%

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

   3. Simplified 38.2%

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

   5. Taylor expanded in w around 0 38.2%

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

   if -3.09999999999999993e-37 < c0 < 2.9e-154

   1. Initial program 13.3%

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

   3. Simplified 19.9%

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

   5. Taylor expanded in c0 around 0 0.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot \sqrt{-1}}}{2 \cdot w} \]
   6. Step-by-step derivation

      1. add-log-exp 0.0%

         \[\leadsto \color{blue}{\log \left(e^{c0 \cdot \frac{M \cdot \sqrt{-1}}{2 \cdot w}}\right)} \]

      2. exp-prod 43.2%

         \[\leadsto \log \color{blue}{\left({\left(e^{c0}\right)}^{\left(\frac{M \cdot \sqrt{-1}}{2 \cdot w}\right)}\right)} \]

      3. times-frac 43.2%

         \[\leadsto \log \left({\left(e^{c0}\right)}^{\color{blue}{\left(\frac{M}{2} \cdot \frac{\sqrt{-1}}{w}\right)}}\right) \]

   7. Applied egg-rr 43.2%

      \[\leadsto \color{blue}{\log \left({\left(e^{c0}\right)}^{\left(\frac{M}{2} \cdot \frac{\sqrt{-1}}{w}\right)}\right)} \]

   if 2.9e-154 < c0

   1. Initial program 26.1%

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

   3. Simplified 41.7%

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

      1. *-un-lft-identity 41.7%

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

      2. *-commutative 41.7%

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

      3. associate-*r* 41.0%

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

      4. associate-*r* 39.1%

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

      5. associate-*l* 40.9%

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

      6. pow2 40.9%

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

   6. Applied egg-rr 40.9%

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

      1. *-lft-identity 40.9%

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

   8. Simplified 40.9%

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

      1. *-commutative 40.9%

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

      2. add-sqr-sqrt 22.1%

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

      3. pow2 22.1%

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

      4. sqrt-prod 21.9%

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

      5. sqrt-pow1 22.7%

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

      6. metadata-eval 22.7%

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

      7. pow1 22.7%

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

   10. Applied egg-rr 22.7%

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

       1. *-commutative 22.7%

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

       2. unpow-prod-down 21.9%

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

       3. pow2 21.9%

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

       4. add-sqr-sqrt 40.9%

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

       5. pow2 40.9%

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

       6. associate-*r* 41.7%

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

       7. *-commutative 41.7%

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

   12. Applied egg-rr 41.7%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -3.1 \cdot 10^{-37}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}, \sqrt{\left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right) \cdot \mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right)}\right)}{2 \cdot w}\\ \mathbf{elif}\;c0 \leq 2.9 \cdot 10^{-154}:\\ \;\;\;\;\log \left({\left(e^{c0}\right)}^{\left(\frac{M}{2} \cdot \frac{\sqrt{-1}}{w}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{w \cdot \left(D \cdot \left(h \cdot D\right)\right)}, \sqrt{\left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right) \cdot \mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right)}\right)}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 35.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\\ t_1 := d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\\ \mathbf{if}\;d \leq 6.4 \cdot 10^{-192}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+161}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}, \sqrt{\left(c0 \cdot t\_1 - M\right) \cdot \mathsf{fma}\left(c0, t\_1, M\right)}\right)}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* (/ d D) (/ d D)) (/ c0 (* w h))))
        (t_1 (* d (/ d (* D (* w (* h D)))))))
   (if (<= d 6.4e-192)
     (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
     (if (<= d 3.6e+161)
       (*
        c0
        (/ (* 2.0 (* c0 (/ (pow d 2.0) (* (* w h) (pow D 2.0))))) (* 2.0 w)))
       (*
        c0
        (/
         (fma
          c0
          (* d (/ d (* D (* D (* w h)))))
          (sqrt (* (- (* c0 t_1) M) (fma c0 t_1 M))))
         (* 2.0 w)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d / D) * (d / D)) * (c0 / (w * h));
	double t_1 = d * (d / (D * (w * (h * D))));
	double tmp;
	if (d <= 6.4e-192) {
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} else if (d <= 3.6e+161) {
		tmp = c0 * ((2.0 * (c0 * (pow(d, 2.0) / ((w * h) * pow(D, 2.0))))) / (2.0 * w));
	} else {
		tmp = c0 * (fma(c0, (d * (d / (D * (D * (w * h))))), sqrt((((c0 * t_1) - M) * fma(c0, t_1, M)))) / (2.0 * w));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(c0 / Float64(w * h)))
	t_1 = Float64(d * Float64(d / Float64(D * Float64(w * Float64(h * D)))))
	tmp = 0.0
	if (d <= 6.4e-192)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	elseif (d <= 3.6e+161)
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(c0 * Float64((d ^ 2.0) / Float64(Float64(w * h) * (D ^ 2.0))))) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(fma(c0, Float64(d * Float64(d / Float64(D * Float64(D * Float64(w * h))))), sqrt(Float64(Float64(Float64(c0 * t_1) - M) * fma(c0, t_1, M)))) / Float64(2.0 * w)));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * N[(d / N[(D * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 6.4e-192], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.6e+161], N[(c0 * N[(N[(2.0 * N[(c0 * N[(N[Power[d, 2.0], $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(c0 * N[(d * N[(d / N[(D * N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(N[(c0 * t$95$1), $MachinePrecision] - M), $MachinePrecision] * N[(c0 * t$95$1 + M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < 6.4000000000000003e-192

   1. Initial program 20.2%

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

   3. Simplified 22.4%

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

      1. times-frac 21.8%

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

   6. Applied egg-rr 21.8%

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

      1. times-frac 21.8%

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

   8. Applied egg-rr 21.8%

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

      1. times-frac 21.8%

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

   10. Applied egg-rr 35.4%

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

   if 6.4000000000000003e-192 < d < 3.59999999999999984e161

   1. Initial program 19.9%

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

   3. Simplified 29.4%

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

   5. Taylor expanded in w around 0 28.3%

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

   6. Taylor expanded in c0 around inf 31.8%

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

      1. associate-*r/ 33.1%

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

   8. Simplified 33.1%

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

   if 3.59999999999999984e161 < d

   1. Initial program 31.5%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in w around 0 48.4%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.4 \cdot 10^{-192}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h} + \sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) - M \cdot M}\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+161}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}, \sqrt{\left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right) \cdot \mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right)}\right)}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 31.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot t\_0\\ t_2 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;d \leq 6.4 \cdot 10^{-192}:\\ \;\;\;\;t\_2 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+161}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(t\_1 + \sqrt{M \cdot \left(t\_0 \cdot {\left(\frac{d}{D}\right)}^{2} - M\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h)))
        (t_1 (* (* (/ d D) (/ d D)) t_0))
        (t_2 (/ c0 (* 2.0 w))))
   (if (<= d 6.4e-192)
     (* t_2 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
     (if (<= d 3.6e+161)
       (*
        c0
        (/ (* 2.0 (* c0 (/ (pow d 2.0) (* (* w h) (pow D 2.0))))) (* 2.0 w)))
       (* t_2 (+ t_1 (sqrt (* M (- (* t_0 (pow (/ d D) 2.0)) M)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = ((d / D) * (d / D)) * t_0;
	double t_2 = c0 / (2.0 * w);
	double tmp;
	if (d <= 6.4e-192) {
		tmp = t_2 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	} else if (d <= 3.6e+161) {
		tmp = c0 * ((2.0 * (c0 * (pow(d, 2.0) / ((w * h) * pow(D, 2.0))))) / (2.0 * w));
	} else {
		tmp = t_2 * (t_1 + sqrt((M * ((t_0 * pow((d / D), 2.0)) - M))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = ((d_1 / d) * (d_1 / d)) * t_0
    t_2 = c0 / (2.0d0 * w)
    if (d_1 <= 6.4d-192) then
        tmp = t_2 * (t_1 + sqrt(((t_1 * t_1) - (m * m))))
    else if (d_1 <= 3.6d+161) then
        tmp = c0 * ((2.0d0 * (c0 * ((d_1 ** 2.0d0) / ((w * h) * (d ** 2.0d0))))) / (2.0d0 * w))
    else
        tmp = t_2 * (t_1 + sqrt((m * ((t_0 * ((d_1 / d) ** 2.0d0)) - m))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = ((d / D) * (d / D)) * t_0;
	double t_2 = c0 / (2.0 * w);
	double tmp;
	if (d <= 6.4e-192) {
		tmp = t_2 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	} else if (d <= 3.6e+161) {
		tmp = c0 * ((2.0 * (c0 * (Math.pow(d, 2.0) / ((w * h) * Math.pow(D, 2.0))))) / (2.0 * w));
	} else {
		tmp = t_2 * (t_1 + Math.sqrt((M * ((t_0 * Math.pow((d / D), 2.0)) - M))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w * h)
	t_1 = ((d / D) * (d / D)) * t_0
	t_2 = c0 / (2.0 * w)
	tmp = 0
	if d <= 6.4e-192:
		tmp = t_2 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	elif d <= 3.6e+161:
		tmp = c0 * ((2.0 * (c0 * (math.pow(d, 2.0) / ((w * h) * math.pow(D, 2.0))))) / (2.0 * w))
	else:
		tmp = t_2 * (t_1 + math.sqrt((M * ((t_0 * math.pow((d / D), 2.0)) - M))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(Float64(Float64(d / D) * Float64(d / D)) * t_0)
	t_2 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (d <= 6.4e-192)
		tmp = Float64(t_2 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))));
	elseif (d <= 3.6e+161)
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(c0 * Float64((d ^ 2.0) / Float64(Float64(w * h) * (D ^ 2.0))))) / Float64(2.0 * w)));
	else
		tmp = Float64(t_2 * Float64(t_1 + sqrt(Float64(M * Float64(Float64(t_0 * (Float64(d / D) ^ 2.0)) - M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w * h);
	t_1 = ((d / D) * (d / D)) * t_0;
	t_2 = c0 / (2.0 * w);
	tmp = 0.0;
	if (d <= 6.4e-192)
		tmp = t_2 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	elseif (d <= 3.6e+161)
		tmp = c0 * ((2.0 * (c0 * ((d ^ 2.0) / ((w * h) * (D ^ 2.0))))) / (2.0 * w));
	else
		tmp = t_2 * (t_1 + sqrt((M * ((t_0 * ((d / D) ^ 2.0)) - M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 6.4e-192], N[(t$95$2 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.6e+161], N[(c0 * N[(N[(2.0 * N[(c0 * N[(N[Power[d, 2.0], $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$1 + N[Sqrt[N[(M * N[(N[(t$95$0 * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < 6.4000000000000003e-192

   1. Initial program 20.2%

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

   3. Simplified 22.4%

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

      1. times-frac 21.8%

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

   6. Applied egg-rr 21.8%

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

      1. times-frac 21.8%

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

   8. Applied egg-rr 21.8%

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

      1. times-frac 21.8%

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

   10. Applied egg-rr 35.4%

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

   if 6.4000000000000003e-192 < d < 3.59999999999999984e161

   1. Initial program 19.9%

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

   3. Simplified 29.4%

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

   5. Taylor expanded in w around 0 28.3%

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

   6. Taylor expanded in c0 around inf 31.8%

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

      1. associate-*r/ 33.1%

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

   8. Simplified 33.1%

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

   if 3.59999999999999984e161 < d

   1. Initial program 31.5%

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

   3. Simplified 29.3%

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

      1. times-frac 29.3%

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

   6. Applied egg-rr 29.3%

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

      1. difference-of-squares 33.5%

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

      2. frac-times 33.5%

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

      3. fma-define 33.5%

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

      4. *-commutative 33.5%

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

      5. pow2 33.5%

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

      6. *-commutative 33.5%

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

      7. frac-times 44.0%

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

      8. pow2 44.0%

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

   8. Applied egg-rr 44.0%

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

   9. Taylor expanded in c0 around 0 18.8%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.4 \cdot 10^{-192}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h} + \sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) - M \cdot M}\right)\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+161}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h} + \sqrt{M \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2} - M\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\\ \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{+270}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\left(w \cdot h\right) \cdot {D}^{2}}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* (/ d D) (/ d D)) (/ c0 (* w h)))))
   (if (<= (* M M) 4e+270)
     (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
     (*
      c0
      (/ (/ (* 2.0 (* c0 (pow d 2.0))) (* (* w h) (pow D 2.0))) (* 2.0 w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d / D) * (d / D)) * (c0 / (w * h));
	double tmp;
	if ((M * M) <= 4e+270) {
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = c0 * (((2.0 * (c0 * pow(d, 2.0))) / ((w * h) * pow(D, 2.0))) / (2.0 * w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((d_1 / d) * (d_1 / d)) * (c0 / (w * h))
    if ((m * m) <= 4d+270) then
        tmp = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
    else
        tmp = c0 * (((2.0d0 * (c0 * (d_1 ** 2.0d0))) / ((w * h) * (d ** 2.0d0))) / (2.0d0 * w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d / D) * (d / D)) * (c0 / (w * h));
	double tmp;
	if ((M * M) <= 4e+270) {
		tmp = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = c0 * (((2.0 * (c0 * Math.pow(d, 2.0))) / ((w * h) * Math.pow(D, 2.0))) / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d / D) * (d / D)) * (c0 / (w * h))
	tmp = 0
	if (M * M) <= 4e+270:
		tmp = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	else:
		tmp = c0 * (((2.0 * (c0 * math.pow(d, 2.0))) / ((w * h) * math.pow(D, 2.0))) / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(c0 / Float64(w * h)))
	tmp = 0.0
	if (Float64(M * M) <= 4e+270)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	else
		tmp = Float64(c0 * Float64(Float64(Float64(2.0 * Float64(c0 * (d ^ 2.0))) / Float64(Float64(w * h) * (D ^ 2.0))) / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d / D) * (d / D)) * (c0 / (w * h));
	tmp = 0.0;
	if ((M * M) <= 4e+270)
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	else
		tmp = c0 * (((2.0 * (c0 * (d ^ 2.0))) / ((w * h) * (D ^ 2.0))) / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * M), $MachinePrecision], 4e+270], 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], N[(c0 * N[(N[(N[(2.0 * N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 4.0000000000000002e270

   1. Initial program 27.2%

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

   3. Simplified 28.6%

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

      1. times-frac 27.9%

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

   6. Applied egg-rr 27.9%

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

      1. times-frac 27.9%

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

   8. Applied egg-rr 27.9%

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

      1. times-frac 27.9%

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

   10. Applied egg-rr 39.1%

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

   if 4.0000000000000002e270 < (*.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 30.2%

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

   5. Taylor expanded in c0 around inf 31.8%

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

      1. associate-*r/ 31.8%

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

   7. Simplified 31.8%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{+270}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h} + \sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\left(w \cdot h\right) \cdot {D}^{2}}}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\\ \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{+270}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* (/ d D) (/ d D)) (/ c0 (* w h)))))
   (if (<= (* M M) 4e+270)
     (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
     (*
      c0
      (/ (* 2.0 (* c0 (/ (pow d 2.0) (* (* w h) (pow D 2.0))))) (* 2.0 w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d / D) * (d / D)) * (c0 / (w * h));
	double tmp;
	if ((M * M) <= 4e+270) {
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = c0 * ((2.0 * (c0 * (pow(d, 2.0) / ((w * h) * pow(D, 2.0))))) / (2.0 * w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((d_1 / d) * (d_1 / d)) * (c0 / (w * h))
    if ((m * m) <= 4d+270) then
        tmp = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
    else
        tmp = c0 * ((2.0d0 * (c0 * ((d_1 ** 2.0d0) / ((w * h) * (d ** 2.0d0))))) / (2.0d0 * w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d / D) * (d / D)) * (c0 / (w * h));
	double tmp;
	if ((M * M) <= 4e+270) {
		tmp = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = c0 * ((2.0 * (c0 * (Math.pow(d, 2.0) / ((w * h) * Math.pow(D, 2.0))))) / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d / D) * (d / D)) * (c0 / (w * h))
	tmp = 0
	if (M * M) <= 4e+270:
		tmp = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	else:
		tmp = c0 * ((2.0 * (c0 * (math.pow(d, 2.0) / ((w * h) * math.pow(D, 2.0))))) / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(c0 / Float64(w * h)))
	tmp = 0.0
	if (Float64(M * M) <= 4e+270)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	else
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(c0 * Float64((d ^ 2.0) / Float64(Float64(w * h) * (D ^ 2.0))))) / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d / D) * (d / D)) * (c0 / (w * h));
	tmp = 0.0;
	if ((M * M) <= 4e+270)
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	else
		tmp = c0 * ((2.0 * (c0 * ((d ^ 2.0) / ((w * h) * (D ^ 2.0))))) / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * M), $MachinePrecision], 4e+270], 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], N[(c0 * N[(N[(2.0 * N[(c0 * N[(N[Power[d, 2.0], $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 4.0000000000000002e270

   1. Initial program 27.2%

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

   3. Simplified 28.6%

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

      1. times-frac 27.9%

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

   6. Applied egg-rr 27.9%

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

      1. times-frac 27.9%

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

   8. Applied egg-rr 27.9%

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

      1. times-frac 27.9%

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

   10. Applied egg-rr 39.1%

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

   if 4.0000000000000002e270 < (*.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 30.2%

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

   5. Taylor expanded in w around 0 28.1%

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

   6. Taylor expanded in c0 around inf 31.8%

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

      1. associate-*r/ 31.7%

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

   8. Simplified 31.7%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{+270}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h} + \sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d / D) * (d / D)) * (c0 / (w * h));
	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 = ((d_1 / d) * (d_1 / d)) * (c0 / (w * h))
    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 = ((d / D) * (d / D)) * (c0 / (w * h));
	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 = ((d / D) * (d / D)) * (c0 / (w * h))
	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(Float64(d / D) * Float64(d / D)) * Float64(c0 / Float64(w * h)))
	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 = ((d / D) * (d / D)) * (c0 / (w * h));
	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[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(w * h), $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]]

Derivation

1. Initial program 22.2%

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

3. Simplified 23.4%

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

   1. times-frac 22.7%

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

6. Applied egg-rr 22.7%

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

   1. times-frac 22.7%

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

8. Applied egg-rr 22.7%

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

   1. times-frac 22.7%

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

10. Applied egg-rr 31.9%

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

11. Final simplification 31.9%

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

Alternative 8: 0.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ c0 \cdot \frac{M \cdot \sqrt{-1}}{2 \cdot w} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* c0 (/ (* M (sqrt -1.0)) (* 2.0 w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * ((M * sqrt(-1.0)) / (2.0 * w));
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = c0 * ((m * sqrt((-1.0d0))) / (2.0d0 * w))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * ((M * Math.sqrt(-1.0)) / (2.0 * w));
}

Python

def code(c0, w, h, D, d, M):
	return c0 * ((M * math.sqrt(-1.0)) / (2.0 * w))

Julia

function code(c0, w, h, D, d, M)
	return Float64(c0 * Float64(Float64(M * sqrt(-1.0)) / Float64(2.0 * w)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = c0 * ((M * sqrt(-1.0)) / (2.0 * w));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(c0 * N[(N[(M * N[Sqrt[-1.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.2%

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

3. Simplified 30.3%

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

5. Taylor expanded in c0 around 0 0.0%

   \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot \sqrt{-1}}}{2 \cdot w} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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))))))