Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.6% → 56.9%

Time: 22.8s

Alternatives: 8

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: 25.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq 10^{-264}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;{\left(\frac{c0 \cdot d}{D \cdot \left(w \cdot \sqrt{h}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 1e-264)
     t_1
     (if (<= t_1 INFINITY) (pow (/ (* c0 d) (* D (* w (sqrt h)))) 2.0) 0.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= 1e-264) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = pow(((c0 * d) / (D * (w * sqrt(h)))), 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= 1e-264) {
		tmp = t_1;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow(((c0 * d) / (D * (w * Math.sqrt(h)))), 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= 1e-264:
		tmp = t_1
	elif t_1 <= math.inf:
		tmp = math.pow(((c0 * d) / (D * (w * math.sqrt(h)))), 2.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= 1e-264)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(c0 * d) / Float64(D * Float64(w * sqrt(h)))) ^ 2.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= 1e-264)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = ((c0 * d) / (D * (w * sqrt(h)))) ^ 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-264], t$95$1, If[LessEqual[t$95$1, Infinity], N[Power[N[(N[(c0 * d), $MachinePrecision] / N[(D * N[(w * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 3 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))))) < 1e-264

   1. Initial program 70.6%

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

   if 1e-264 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in c0 around inf 60.9%

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

      1. add-sqr-sqrt 60.9%

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

      2. pow2 60.9%

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

      3. sqrt-div 60.8%

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

      4. pow-prod-down 74.0%

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

      5. sqrt-pow1 79.6%

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

      6. metadata-eval 79.6%

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

      7. pow1 79.6%

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

      8. sqrt-prod 83.2%

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

      9. sqrt-pow1 83.2%

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

      10. metadata-eval 83.2%

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

      11. pow1 83.2%

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

      12. *-commutative 83.2%

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

      13. sqrt-prod 85.0%

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

      14. sqrt-pow1 92.4%

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

      15. metadata-eval 92.4%

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

      16. pow1 92.4%

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

   7. Applied egg-rr 92.4%

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

   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 22.9%

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

   5. Taylor expanded in c0 around -inf 0.9%

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

      1. associate-*r/ 0.9%

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

      2. neg-mul-1 0.9%

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

      3. distribute-lft-neg-in 0.9%

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

   7. Simplified 0.9%

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

   8. Taylor expanded in h around 0 0.0%

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

      1. associate-*r/ 0.0%

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

      2. distribute-lft1-in 0.0%

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

      3. metadata-eval 0.0%

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

      4. mul0-lft 30.7%

         \[\leadsto c0 \cdot \frac{0.5 \cdot \color{blue}{0}}{h \cdot w} \]

      5. metadata-eval 30.7%

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

      6. mul0-lft 0.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{0 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}}{h \cdot w} \]

      7. metadata-eval 0.0%

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

      8. distribute-lft1-in 0.0%

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

      9. associate-/r* 0.1%

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

   10. Simplified 41.2%

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

   11. Taylor expanded in c0 around 0 41.2%

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

Alternative 2: 56.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq 10^{-264}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;{\left(c0 \cdot \frac{\frac{d}{D}}{w \cdot \sqrt{h}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 1e-264)
     t_1
     (if (<= t_1 INFINITY) (pow (* c0 (/ (/ d D) (* w (sqrt h)))) 2.0) 0.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= 1e-264) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = pow((c0 * ((d / D) / (w * sqrt(h)))), 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= 1e-264) {
		tmp = t_1;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow((c0 * ((d / D) / (w * Math.sqrt(h)))), 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= 1e-264:
		tmp = t_1
	elif t_1 <= math.inf:
		tmp = math.pow((c0 * ((d / D) / (w * math.sqrt(h)))), 2.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_1 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= 1e-264)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(c0 * Float64(Float64(d / D) / Float64(w * sqrt(h)))) ^ 2.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= 1e-264)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = (c0 * ((d / D) / (w * sqrt(h)))) ^ 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-264], t$95$1, If[LessEqual[t$95$1, Infinity], N[Power[N[(c0 * N[(N[(d / D), $MachinePrecision] / N[(w * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 3 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))))) < 1e-264

   1. Initial program 70.6%

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

   if 1e-264 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in c0 around inf 70.8%

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

      1. *-commutative 70.8%

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

   7. Simplified 70.8%

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

   8. Taylor expanded in c0 around 0 60.9%

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

      1. associate-/l* 62.6%

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

      2. associate-*r* 59.0%

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

      3. *-commutative 59.0%

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

      4. associate-/r* 59.1%

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

      5. *-commutative 59.1%

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

      6. associate-/r* 62.7%

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

      7. unpow2 62.7%

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

      8. unpow2 62.7%

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

      9. times-frac 62.7%

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

      10. unpow2 62.7%

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

   10. Simplified 62.7%

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

       1. add-sqr-sqrt 62.6%

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

       2. pow2 62.6%

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

   12. Applied egg-rr 92.4%

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

   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 22.9%

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

   5. Taylor expanded in c0 around -inf 0.9%

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

      1. associate-*r/ 0.9%

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

      2. neg-mul-1 0.9%

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

      3. distribute-lft-neg-in 0.9%

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

   7. Simplified 0.9%

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

   8. Taylor expanded in h around 0 0.0%

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

      1. associate-*r/ 0.0%

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

      2. distribute-lft1-in 0.0%

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

      3. metadata-eval 0.0%

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

      4. mul0-lft 30.7%

         \[\leadsto c0 \cdot \frac{0.5 \cdot \color{blue}{0}}{h \cdot w} \]

      5. metadata-eval 30.7%

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

      6. mul0-lft 0.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{0 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}}{h \cdot w} \]

      7. metadata-eval 0.0%

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

      8. distribute-lft1-in 0.0%

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

      9. associate-/r* 0.1%

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

   10. Simplified 41.2%

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

   11. Taylor expanded in c0 around 0 41.2%

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

Alternative 3: 55.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 71.2%

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

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

   1. Initial program 0.0%

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

   3. Simplified 22.9%

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

   5. Taylor expanded in c0 around -inf 0.9%

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

      1. associate-*r/ 0.9%

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

      2. neg-mul-1 0.9%

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

      3. distribute-lft-neg-in 0.9%

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

   7. Simplified 0.9%

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

   8. Taylor expanded in h around 0 0.0%

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

      1. associate-*r/ 0.0%

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

      2. distribute-lft1-in 0.0%

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

      3. metadata-eval 0.0%

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

      4. mul0-lft 30.7%

         \[\leadsto c0 \cdot \frac{0.5 \cdot \color{blue}{0}}{h \cdot w} \]

      5. metadata-eval 30.7%

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

      6. mul0-lft 0.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{0 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}}{h \cdot w} \]

      7. metadata-eval 0.0%

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

      8. distribute-lft1-in 0.0%

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

      9. associate-/r* 0.1%

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

   10. Simplified 41.2%

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

   11. Taylor expanded in c0 around 0 41.2%

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

Alternative 4: 32.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot d}{D \cdot D}\\ t_1 := t\_0 \cdot \frac{c0}{w \cdot h}\\ \mathbf{if}\;D \cdot D \leq 10^{-243}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{w}}{h} \cdot t\_0 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* d d) (* D D))) (t_1 (* t_0 (/ c0 (* w h)))))
   (if (<= (* D D) 1e-243)
     0.0
     (*
      (/ c0 (* 2.0 w))
      (+ (* (/ (/ c0 w) h) t_0) (sqrt (- (* t_1 t_1) (* M M))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * d) / (D * D);
	double t_1 = t_0 * (c0 / (w * h));
	double tmp;
	if ((D * D) <= 1e-243) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((((c0 / w) / h) * t_0) + sqrt(((t_1 * t_1) - (M * M))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_1 * d_1) / (d * d)
    t_1 = t_0 * (c0 / (w * h))
    if ((d * d) <= 1d-243) then
        tmp = 0.0d0
    else
        tmp = (c0 / (2.0d0 * w)) * ((((c0 / w) / h) * t_0) + sqrt(((t_1 * t_1) - (m * m))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * d) / (D * D);
	double t_1 = t_0 * (c0 / (w * h));
	double tmp;
	if ((D * D) <= 1e-243) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((((c0 / w) / h) * t_0) + Math.sqrt(((t_1 * t_1) - (M * M))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d * d) / (D * D)
	t_1 = t_0 * (c0 / (w * h))
	tmp = 0
	if (D * D) <= 1e-243:
		tmp = 0.0
	else:
		tmp = (c0 / (2.0 * w)) * ((((c0 / w) / h) * t_0) + math.sqrt(((t_1 * t_1) - (M * M))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d * d) / Float64(D * D))
	t_1 = Float64(t_0 * Float64(c0 / Float64(w * h)))
	tmp = 0.0
	if (Float64(D * D) <= 1e-243)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(Float64(c0 / w) / h) * t_0) + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d * d) / (D * D);
	t_1 = t_0 * (c0 / (w * h));
	tmp = 0.0;
	if ((D * D) <= 1e-243)
		tmp = 0.0;
	else
		tmp = (c0 / (2.0 * w)) * ((((c0 / w) / h) * t_0) + sqrt(((t_1 * t_1) - (M * M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(D * D), $MachinePrecision], 1e-243], 0.0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] * t$95$0), $MachinePrecision] + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 D D) < 9.99999999999999995e-244

   1. Initial program 25.8%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in c0 around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

      2. neg-mul-1 0.2%

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

      3. distribute-lft-neg-in 0.2%

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

   7. Simplified 0.2%

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

   8. Taylor expanded in h around 0 1.0%

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

      1. associate-*r/ 1.0%

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

      2. distribute-lft1-in 1.0%

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

      3. metadata-eval 1.0%

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

      4. mul0-lft 29.0%

         \[\leadsto c0 \cdot \frac{0.5 \cdot \color{blue}{0}}{h \cdot w} \]

      5. metadata-eval 29.0%

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

      6. mul0-lft 1.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{0 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}}{h \cdot w} \]

      7. metadata-eval 1.0%

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

      8. distribute-lft1-in 1.0%

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

      9. associate-/r* 1.0%

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

   10. Simplified 42.6%

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

   11. Taylor expanded in c0 around 0 42.6%

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

   if 9.99999999999999995e-244 < (*.f64 D D)

   1. Initial program 28.5%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in c0 around 0 27.1%

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

      1. *-commutative 27.1%

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

      2. associate-/r* 27.8%

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

   7. Simplified 27.8%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 10^{-243}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right) - M \cdot M}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot d}{D \cdot D}\\ t_1 := t\_0 \cdot \frac{c0}{w \cdot h}\\ \mathbf{if}\;D \cdot D \leq 10^{-243}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{\left(\frac{\frac{c0}{w}}{h} \cdot t\_0\right) \cdot t\_1 - M \cdot M}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* d d) (* D D))) (t_1 (* t_0 (/ c0 (* w h)))))
   (if (<= (* D D) 1e-243)
     0.0
     (*
      (/ c0 (* 2.0 w))
      (+ t_1 (sqrt (- (* (* (/ (/ c0 w) h) t_0) t_1) (* M M))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * d) / (D * D);
	double t_1 = t_0 * (c0 / (w * h));
	double tmp;
	if ((D * D) <= 1e-243) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * (t_1 + sqrt((((((c0 / w) / h) * t_0) * t_1) - (M * M))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_1 * d_1) / (d * d)
    t_1 = t_0 * (c0 / (w * h))
    if ((d * d) <= 1d-243) then
        tmp = 0.0d0
    else
        tmp = (c0 / (2.0d0 * w)) * (t_1 + sqrt((((((c0 / w) / h) * t_0) * t_1) - (m * m))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * d) / (D * D);
	double t_1 = t_0 * (c0 / (w * h));
	double tmp;
	if ((D * D) <= 1e-243) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * (t_1 + Math.sqrt((((((c0 / w) / h) * t_0) * t_1) - (M * M))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d * d) / (D * D)
	t_1 = t_0 * (c0 / (w * h))
	tmp = 0
	if (D * D) <= 1e-243:
		tmp = 0.0
	else:
		tmp = (c0 / (2.0 * w)) * (t_1 + math.sqrt((((((c0 / w) / h) * t_0) * t_1) - (M * M))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d * d) / Float64(D * D))
	t_1 = Float64(t_0 * Float64(c0 / Float64(w * h)))
	tmp = 0.0
	if (Float64(D * D) <= 1e-243)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(Float64(Float64(Float64(c0 / w) / h) * t_0) * t_1) - Float64(M * M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d * d) / (D * D);
	t_1 = t_0 * (c0 / (w * h));
	tmp = 0.0;
	if ((D * D) <= 1e-243)
		tmp = 0.0;
	else
		tmp = (c0 / (2.0 * w)) * (t_1 + sqrt((((((c0 / w) / h) * t_0) * t_1) - (M * M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(D * D), $MachinePrecision], 1e-243], 0.0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 D D) < 9.99999999999999995e-244

   1. Initial program 25.8%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in c0 around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

      2. neg-mul-1 0.2%

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

      3. distribute-lft-neg-in 0.2%

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

   7. Simplified 0.2%

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

   8. Taylor expanded in h around 0 1.0%

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

      1. associate-*r/ 1.0%

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

      2. distribute-lft1-in 1.0%

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

      3. metadata-eval 1.0%

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

      4. mul0-lft 29.0%

         \[\leadsto c0 \cdot \frac{0.5 \cdot \color{blue}{0}}{h \cdot w} \]

      5. metadata-eval 29.0%

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

      6. mul0-lft 1.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{0 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}}{h \cdot w} \]

      7. metadata-eval 1.0%

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

      8. distribute-lft1-in 1.0%

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

      9. associate-/r* 1.0%

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

   10. Simplified 42.6%

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

   11. Taylor expanded in c0 around 0 42.6%

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

   if 9.99999999999999995e-244 < (*.f64 D D)

   1. Initial program 28.5%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in c0 around 0 27.1%

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

      1. *-commutative 27.1%

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

      2. associate-/r* 27.8%

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

   7. Simplified 27.1%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 10^{-243}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h} + \sqrt{\left(\frac{\frac{c0}{w}}{h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right) - M \cdot M}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 32.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\\ \mathbf{if}\;D \cdot D \leq 10^{-243}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ (* d d) (* D D)) (/ c0 (* w h)))))
   (if (<= (* D D) 1e-243)
     0.0
     (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * d) / (D * D)) * (c0 / (w * h));
	double tmp;
	if ((D * D) <= 1e-243) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((d_1 * d_1) / (d * d)) * (c0 / (w * h))
    if ((d * d) <= 1d-243) then
        tmp = 0.0d0
    else
        tmp = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * d) / (D * D)) * (c0 / (w * h));
	double tmp;
	if ((D * D) <= 1e-243) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) / (D * D)) * (c0 / (w * h))
	tmp = 0
	if (D * D) <= 1e-243:
		tmp = 0.0
	else:
		tmp = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d * d) / Float64(D * D)) * Float64(c0 / Float64(w * h)))
	tmp = 0.0
	if (Float64(D * D) <= 1e-243)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d * d) / (D * D)) * (c0 / (w * h));
	tmp = 0.0;
	if ((D * D) <= 1e-243)
		tmp = 0.0;
	else
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(D * D), $MachinePrecision], 1e-243], 0.0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 D D) < 9.99999999999999995e-244

   1. Initial program 25.8%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in c0 around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

      2. neg-mul-1 0.2%

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

      3. distribute-lft-neg-in 0.2%

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

   7. Simplified 0.2%

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

   8. Taylor expanded in h around 0 1.0%

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

      1. associate-*r/ 1.0%

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

      2. distribute-lft1-in 1.0%

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

      3. metadata-eval 1.0%

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

      4. mul0-lft 29.0%

         \[\leadsto c0 \cdot \frac{0.5 \cdot \color{blue}{0}}{h \cdot w} \]

      5. metadata-eval 29.0%

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

      6. mul0-lft 1.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{0 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}}{h \cdot w} \]

      7. metadata-eval 1.0%

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

      8. distribute-lft1-in 1.0%

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

      9. associate-/r* 1.0%

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

   10. Simplified 42.6%

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

   11. Taylor expanded in c0 around 0 42.6%

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

   if 9.99999999999999995e-244 < (*.f64 D D)

   1. Initial program 28.5%

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

   3. Simplified 27.1%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 10^{-243}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h} + \sqrt{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right) - M \cdot M}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := \frac{d \cdot d}{D \cdot D} \cdot t\_0\\ \mathbf{if}\;D \leq 7 \cdot 10^{-122}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{t\_1 \cdot t\_1 - M \cdot M} + t\_0 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h))) (t_1 (* (/ (* d d) (* D D)) t_0)))
   (if (<= D 7e-122)
     0.0
     (*
      (/ c0 (* 2.0 w))
      (+ (sqrt (- (* t_1 t_1) (* M M))) (* t_0 (* (/ d D) (/ d D))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = ((d * d) / (D * D)) * t_0;
	double tmp;
	if (D <= 7e-122) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * (sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = ((d_1 * d_1) / (d * d)) * t_0
    if (d <= 7d-122) then
        tmp = 0.0d0
    else
        tmp = (c0 / (2.0d0 * w)) * (sqrt(((t_1 * t_1) - (m * m))) + (t_0 * ((d_1 / d) * (d_1 / d))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = ((d * d) / (D * D)) * t_0;
	double tmp;
	if (D <= 7e-122) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * (Math.sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w * h)
	t_1 = ((d * d) / (D * D)) * t_0
	tmp = 0
	if D <= 7e-122:
		tmp = 0.0
	else:
		tmp = (c0 / (2.0 * w)) * (math.sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(Float64(Float64(d * d) / Float64(D * D)) * t_0)
	tmp = 0.0
	if (D <= 7e-122)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))) + Float64(t_0 * Float64(Float64(d / D) * Float64(d / D)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w * h);
	t_1 = ((d * d) / (D * D)) * t_0;
	tmp = 0.0;
	if (D <= 7e-122)
		tmp = 0.0;
	else
		tmp = (c0 / (2.0 * w)) * (sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[D, 7e-122], 0.0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if D < 7.0000000000000003e-122

   1. Initial program 26.6%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in c0 around -inf 5.1%

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

      1. associate-*r/ 5.1%

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

      2. neg-mul-1 5.1%

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

      3. distribute-lft-neg-in 5.1%

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

   7. Simplified 5.1%

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

   8. Taylor expanded in h around 0 2.4%

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

      1. associate-*r/ 2.4%

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

      2. distribute-lft1-in 2.4%

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

      3. metadata-eval 2.4%

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

      4. mul0-lft 24.5%

         \[\leadsto c0 \cdot \frac{0.5 \cdot \color{blue}{0}}{h \cdot w} \]

      5. metadata-eval 24.5%

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

      6. mul0-lft 2.4%

         \[\leadsto c0 \cdot \frac{\color{blue}{0 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}}{h \cdot w} \]

      7. metadata-eval 2.4%

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

      8. distribute-lft1-in 2.4%

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

      9. associate-/r* 2.5%

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

   10. Simplified 32.8%

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

   11. Taylor expanded in c0 around 0 32.8%

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

   if 7.0000000000000003e-122 < D

   1. Initial program 29.1%

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

   3. Simplified 29.3%

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

      1. times-frac 28.0%

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

   6. Applied egg-rr 28.0%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 7 \cdot 10^{-122}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{\left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right) - M \cdot M} + \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.5% accurate, 151.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.3%

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

3. Simplified 41.8%

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

5. Taylor expanded in c0 around -inf 4.8%

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

   1. associate-*r/ 4.8%

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

   2. neg-mul-1 4.8%

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

   3. distribute-lft-neg-in 4.8%

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

7. Simplified 4.8%

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

8. Taylor expanded in h around 0 3.1%

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

   1. associate-*r/ 3.1%

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

   2. distribute-lft1-in 3.1%

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

   3. metadata-eval 3.1%

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

   4. mul0-lft 22.4%

      \[\leadsto c0 \cdot \frac{0.5 \cdot \color{blue}{0}}{h \cdot w} \]

   5. metadata-eval 22.4%

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

   6. mul0-lft 3.1%

      \[\leadsto c0 \cdot \frac{\color{blue}{0 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}}{h \cdot w} \]

   7. metadata-eval 3.1%

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

   8. distribute-lft1-in 3.1%

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

   9. associate-/r* 3.2%

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

10. Simplified 29.0%

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

11. Taylor expanded in c0 around 0 29.0%

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

Reproduce

herbie shell --seed 2024163 
(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))))))