Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.9% → 61.9%

Time: 29.4s

Alternatives: 5

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = 8.674929312406072e+304
  2. w = -2.4006177732026406e-262
  3. h = 7.740481612397995e-138
  4. D = -7.938889642396208e-212
  5. d = 3.0193600182023346e+297
  6. M = -1.169716018518994e+110

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: 24.9% 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: 61.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{d}{D}}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := t_1 \cdot \left(2 \cdot \frac{\frac{\frac{c0}{w}}{h}}{t_0 \cdot t_0}\right)\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_4 := t_1 \cdot \left(t_3 + \sqrt{t_3 \cdot t_3 - M \cdot M}\right)\\ \mathbf{if}\;t_4 \leq -5 \cdot 10^{-95}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;-0.5 \cdot \left(-0.5 \cdot \frac{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2}}\right)\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{{M}^{2}}{\frac{{\left(\frac{d}{D}\right)}^{2}}{h} \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ d D)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (* t_1 (* 2.0 (/ (/ (/ c0 w) h) (* t_0 t_0)))))
        (t_3 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_4 (* t_1 (+ t_3 (sqrt (- (* t_3 t_3) (* M M)))))))
   (if (<= t_4 -5e-95)
     t_2
     (if (<= t_4 0.0)
       (* -0.5 (* -0.5 (/ (* h (* (pow D 2.0) (pow M 2.0))) (pow d 2.0))))
       (if (<= t_4 INFINITY)
         t_2
         (* -0.5 (/ (pow M 2.0) (* (/ (pow (/ d D) 2.0) h) -2.0))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = 1.0 / (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_1 * (2.0 * (((c0 / w) / h) / (t_0 * t_0)));
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_4 = t_1 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	double tmp;
	if (t_4 <= -5e-95) {
		tmp = t_2;
	} else if (t_4 <= 0.0) {
		tmp = -0.5 * (-0.5 * ((h * (pow(D, 2.0) * pow(M, 2.0))) / pow(d, 2.0)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = -0.5 * (pow(M, 2.0) / ((pow((d / D), 2.0) / h) * -2.0));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = 1.0 / (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_1 * (2.0 * (((c0 / w) / h) / (t_0 * t_0)));
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_4 = t_1 * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))));
	double tmp;
	if (t_4 <= -5e-95) {
		tmp = t_2;
	} else if (t_4 <= 0.0) {
		tmp = -0.5 * (-0.5 * ((h * (Math.pow(D, 2.0) * Math.pow(M, 2.0))) / Math.pow(d, 2.0)));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = -0.5 * (Math.pow(M, 2.0) / ((Math.pow((d / D), 2.0) / h) * -2.0));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = 1.0 / (d / D)
	t_1 = c0 / (2.0 * w)
	t_2 = t_1 * (2.0 * (((c0 / w) / h) / (t_0 * t_0)))
	t_3 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_4 = t_1 * (t_3 + math.sqrt(((t_3 * t_3) - (M * M))))
	tmp = 0
	if t_4 <= -5e-95:
		tmp = t_2
	elif t_4 <= 0.0:
		tmp = -0.5 * (-0.5 * ((h * (math.pow(D, 2.0) * math.pow(M, 2.0))) / math.pow(d, 2.0)))
	elif t_4 <= math.inf:
		tmp = t_2
	else:
		tmp = -0.5 * (math.pow(M, 2.0) / ((math.pow((d / D), 2.0) / h) * -2.0))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(1.0 / Float64(d / D))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(t_1 * Float64(2.0 * Float64(Float64(Float64(c0 / w) / h) / Float64(t_0 * t_0))))
	t_3 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_4 = Float64(t_1 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M)))))
	tmp = 0.0
	if (t_4 <= -5e-95)
		tmp = t_2;
	elseif (t_4 <= 0.0)
		tmp = Float64(-0.5 * Float64(-0.5 * Float64(Float64(h * Float64((D ^ 2.0) * (M ^ 2.0))) / (d ^ 2.0))));
	elseif (t_4 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(-0.5 * Float64((M ^ 2.0) / Float64(Float64((Float64(d / D) ^ 2.0) / h) * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = 1.0 / (d / D);
	t_1 = c0 / (2.0 * w);
	t_2 = t_1 * (2.0 * (((c0 / w) / h) / (t_0 * t_0)));
	t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_4 = t_1 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	tmp = 0.0;
	if (t_4 <= -5e-95)
		tmp = t_2;
	elseif (t_4 <= 0.0)
		tmp = -0.5 * (-0.5 * ((h * ((D ^ 2.0) * (M ^ 2.0))) / (d ^ 2.0)));
	elseif (t_4 <= Inf)
		tmp = t_2;
	else
		tmp = -0.5 * ((M ^ 2.0) / ((((d / D) ^ 2.0) / h) * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(1.0 / N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(2.0 * N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e-95], t$95$2, If[LessEqual[t$95$4, 0.0], N[(-0.5 * N[(-0.5 * N[(N[(h * N[(N[Power[D, 2.0], $MachinePrecision] * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$2, N[(-0.5 * N[(N[Power[M, 2.0], $MachinePrecision] / N[(N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] / h), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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))))) < -4.9999999999999998e-95 or -0.0 < (*.f64 (/.f64 c0 (*.f64 2 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 77.5%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in c0 around inf 78.7%

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

      1. *-commutative 78.7%

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

      2. associate-/l/ 77.4%

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

      3. associate-*l/ 75.9%

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

      4. associate-/l* 77.3%

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

      5. *-commutative 77.3%

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

   6. Simplified 77.3%

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

      1. expm1-log1p-u 77.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2}}{{d}^{2}}\right)\right)}}\right) \]

      2. expm1-udef 73.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{D}^{2}}{{d}^{2}}\right)} - 1}}\right) \]

      3. clear-num 73.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{{d}^{2}}{{D}^{2}}}}\right)} - 1}\right) \]

      4. unpow2 73.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}}\right)} - 1}\right) \]

      5. unpow2 73.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\frac{d \cdot d}{\color{blue}{D \cdot D}}}\right)} - 1}\right) \]

      6. frac-times 73.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}\right)} - 1}\right) \]

      7. pow2 73.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{\left(\frac{d}{D}\right)}^{2}}}\right)} - 1}\right) \]

      8. pow-flip 73.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{d}{D}\right)}^{\left(-2\right)}}\right)} - 1}\right) \]

      9. metadata-eval 73.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left({\left(\frac{d}{D}\right)}^{\color{blue}{-2}}\right)} - 1}\right) \]

   8. Applied egg-rr 73.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{e^{\mathsf{log1p}\left({\left(\frac{d}{D}\right)}^{-2}\right)} - 1}}\right) \]
   9. Step-by-step derivation

      1. expm1-def 81.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{D}\right)}^{-2}\right)\right)}}\right) \]

      2. expm1-log1p 82.2%

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

   10. Simplified 82.2%

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

       1. associate-/l/ 84.7%

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

       2. div-inv 84.6%

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

   12. Applied egg-rr 84.6%

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

       1. associate-*r/ 84.7%

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

       2. *-rgt-identity 84.7%

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

   14. Simplified 84.7%

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

       1. metadata-eval 84.7%

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

       2. pow-prod-up 84.7%

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

       3. unpow-1 84.7%

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

       4. unpow-1 84.7%

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

   16. Applied egg-rr 84.7%

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

   if -4.9999999999999998e-95 < (*.f64 (/.f64 c0 (*.f64 2 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))))) < -0.0

   1. Initial program 59.6%

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

   3. Simplified 59.4%

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

   5. Applied egg-rr 31.9%

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

      1. associate--r- 37.7%

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

      2. +-inverses 37.7%

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

      3. *-commutative 37.7%

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

   7. Simplified 37.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{0 + {M}^{2}}{\frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2} - \sqrt{{\left(\frac{c0}{h \cdot w}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}}} \]

   8. Taylor expanded in c0 around -inf 54.3%

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

      1. associate-/r* 54.3%

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

      2. associate-*r/ 54.3%

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

      3. neg-mul-1 54.3%

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

      4. associate-/r* 54.3%

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

   10. Simplified 54.3%

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

   11. Taylor expanded in d around 0 68.5%

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

       1. *-commutative 68.5%

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

       2. *-commutative 68.5%

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

       3. associate-*l* 68.6%

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

       4. *-commutative 68.6%

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

   13. Simplified 68.6%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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 2.5%

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

   5. Applied egg-rr 3.5%

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

      1. associate--r- 5.9%

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

      2. +-inverses 27.6%

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

      3. *-commutative 27.6%

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

   7. Simplified 23.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{0 + {M}^{2}}{\frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2} - \sqrt{{\left(\frac{c0}{h \cdot w}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}}} \]

   8. Taylor expanded in c0 around -inf 31.1%

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

      1. associate-/r* 27.5%

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

      2. associate-*r/ 27.5%

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

      3. neg-mul-1 27.5%

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

      4. associate-/r* 28.2%

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

   10. Simplified 28.2%

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

   11. Taylor expanded in w around 0 42.9%

       \[\leadsto -0.5 \cdot \color{blue}{\frac{{M}^{2}}{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot h} - \frac{{d}^{2}}{{D}^{2} \cdot h}}} \]
   12. Step-by-step derivation

       1. sub-neg 42.9%

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

       2. mul-1-neg 42.9%

          \[\leadsto -0.5 \cdot \frac{{M}^{2}}{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot h} + \color{blue}{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot h}}} \]

       3. distribute-rgt-out 42.9%

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

       4. associate-/r* 43.3%

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

       5. unpow2 43.3%

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

       6. unpow2 43.3%

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

       7. times-frac 51.0%

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

       8. unpow2 51.0%

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

       9. metadata-eval 51.0%

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

   13. Simplified 51.0%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -5 \cdot 10^{-95}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{\frac{c0}{w}}{h}}{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;-0.5 \cdot \left(-0.5 \cdot \frac{h \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{\frac{c0}{w}}{h}}{\frac{1}{\frac{d}{D}} \cdot \frac{1}{\frac{d}{D}}}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{{M}^{2}}{\frac{{\left(\frac{d}{D}\right)}^{2}}{h} \cdot -2}\\ \end{array} \]

Alternative 2: 60.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{d}{D}}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t_1 \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\frac{\frac{c0}{w}}{h}}{t_0 \cdot t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{{M}^{2}}{\frac{{\left(\frac{d}{D}\right)}^{2}}{h} \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ d D)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (* 2.0 (/ (/ (/ c0 w) h) (* t_0 t_0))))
     (* -0.5 (/ (pow M 2.0) (* (/ (pow (/ d D) 2.0) h) -2.0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = 1.0 / (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * (2.0 * (((c0 / w) / h) / (t_0 * t_0)));
	} else {
		tmp = -0.5 * (pow(M, 2.0) / ((pow((d / D), 2.0) / h) * -2.0));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = 1.0 / (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * (2.0 * (((c0 / w) / h) / (t_0 * t_0)));
	} else {
		tmp = -0.5 * (Math.pow(M, 2.0) / ((Math.pow((d / D), 2.0) / h) * -2.0));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = 1.0 / (d / D)
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * (2.0 * (((c0 / w) / h) / (t_0 * t_0)))
	else:
		tmp = -0.5 * (math.pow(M, 2.0) / ((math.pow((d / D), 2.0) / h) * -2.0))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(1.0 / Float64(d / D))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(Float64(c0 / w) / h) / Float64(t_0 * t_0))));
	else
		tmp = Float64(-0.5 * Float64((M ^ 2.0) / Float64(Float64((Float64(d / D) ^ 2.0) / h) * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = 1.0 / (d / D);
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_1 * (2.0 * (((c0 / w) / h) / (t_0 * t_0)));
	else
		tmp = -0.5 * ((M ^ 2.0) / ((((d / D) ^ 2.0) / h) * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(1.0 / N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(2.0 * N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[Power[M, 2.0], $MachinePrecision] / N[(N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] / h), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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 74.4%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in c0 around inf 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-/l/ 70.8%

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

      3. associate-*l/ 69.6%

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

      4. associate-/l* 71.0%

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

      5. *-commutative 71.0%

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

   6. Simplified 71.0%

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

      1. expm1-log1p-u 70.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2}}{{d}^{2}}\right)\right)}}\right) \]

      2. expm1-udef 64.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{D}^{2}}{{d}^{2}}\right)} - 1}}\right) \]

      3. clear-num 64.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{{d}^{2}}{{D}^{2}}}}\right)} - 1}\right) \]

      4. unpow2 64.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}}\right)} - 1}\right) \]

      5. unpow2 64.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\frac{d \cdot d}{\color{blue}{D \cdot D}}}\right)} - 1}\right) \]

      6. frac-times 65.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}\right)} - 1}\right) \]

      7. pow2 65.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{\left(\frac{d}{D}\right)}^{2}}}\right)} - 1}\right) \]

      8. pow-flip 65.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{d}{D}\right)}^{\left(-2\right)}}\right)} - 1}\right) \]

      9. metadata-eval 65.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left({\left(\frac{d}{D}\right)}^{\color{blue}{-2}}\right)} - 1}\right) \]

   8. Applied egg-rr 65.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{e^{\mathsf{log1p}\left({\left(\frac{d}{D}\right)}^{-2}\right)} - 1}}\right) \]
   9. Step-by-step derivation

      1. expm1-def 75.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{D}\right)}^{-2}\right)\right)}}\right) \]

      2. expm1-log1p 76.0%

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

   10. Simplified 76.0%

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

       1. associate-/l/ 77.0%

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

       2. div-inv 77.0%

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

   12. Applied egg-rr 77.0%

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

       1. associate-*r/ 77.0%

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

       2. *-rgt-identity 77.0%

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

   14. Simplified 77.0%

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

       1. metadata-eval 77.0%

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

       2. pow-prod-up 77.0%

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

       3. unpow-1 77.0%

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

       4. unpow-1 77.0%

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

   16. Applied egg-rr 77.0%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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 2.5%

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

   5. Applied egg-rr 3.5%

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

      1. associate--r- 5.9%

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

      2. +-inverses 27.6%

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

      3. *-commutative 27.6%

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

   7. Simplified 23.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{0 + {M}^{2}}{\frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2} - \sqrt{{\left(\frac{c0}{h \cdot w}\right)}^{2} \cdot {\left(\frac{d}{D}\right)}^{4} - {M}^{2}}}} \]

   8. Taylor expanded in c0 around -inf 31.1%

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

      1. associate-/r* 27.5%

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

      2. associate-*r/ 27.5%

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

      3. neg-mul-1 27.5%

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

      4. associate-/r* 28.2%

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

   10. Simplified 28.2%

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

   11. Taylor expanded in w around 0 42.9%

       \[\leadsto -0.5 \cdot \color{blue}{\frac{{M}^{2}}{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot h} - \frac{{d}^{2}}{{D}^{2} \cdot h}}} \]
   12. Step-by-step derivation

       1. sub-neg 42.9%

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

       2. mul-1-neg 42.9%

          \[\leadsto -0.5 \cdot \frac{{M}^{2}}{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot h} + \color{blue}{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot h}}} \]

       3. distribute-rgt-out 42.9%

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

       4. associate-/r* 43.3%

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

       5. unpow2 43.3%

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

       6. unpow2 43.3%

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

       7. times-frac 51.0%

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

       8. unpow2 51.0%

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

       9. metadata-eval 51.0%

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

   13. Simplified 51.0%

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

4. Final simplification 60.4%

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

Alternative 3: 54.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{d}{D}}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t_1 \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\frac{\frac{c0}{w}}{h}}{t_0 \cdot t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ d D)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (* 2.0 (/ (/ (/ c0 w) h) (* t_0 t_0))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = 1.0 / (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * (2.0 * (((c0 / w) / h) / (t_0 * t_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 = 1.0 / (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * (2.0 * (((c0 / w) / h) / (t_0 * t_0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = 1.0 / (d / D)
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * (2.0 * (((c0 / w) / h) / (t_0 * t_0)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(1.0 / Float64(d / D))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(Float64(c0 / w) / h) / Float64(t_0 * t_0))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = 1.0 / (d / D);
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_1 * (2.0 * (((c0 / w) / h) / (t_0 * t_0)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(1.0 / N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(2.0 * N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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 74.4%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in c0 around inf 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-/l/ 70.8%

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

      3. associate-*l/ 69.6%

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

      4. associate-/l* 71.0%

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

      5. *-commutative 71.0%

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

   6. Simplified 71.0%

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

      1. expm1-log1p-u 70.8%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2}}{{d}^{2}}\right)\right)}}\right) \]

      2. expm1-udef 64.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{D}^{2}}{{d}^{2}}\right)} - 1}}\right) \]

      3. clear-num 64.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{{d}^{2}}{{D}^{2}}}}\right)} - 1}\right) \]

      4. unpow2 64.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}}\right)} - 1}\right) \]

      5. unpow2 64.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\frac{d \cdot d}{\color{blue}{D \cdot D}}}\right)} - 1}\right) \]

      6. frac-times 65.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}\right)} - 1}\right) \]

      7. pow2 65.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{\left(\frac{d}{D}\right)}^{2}}}\right)} - 1}\right) \]

      8. pow-flip 65.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{d}{D}\right)}^{\left(-2\right)}}\right)} - 1}\right) \]

      9. metadata-eval 65.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left({\left(\frac{d}{D}\right)}^{\color{blue}{-2}}\right)} - 1}\right) \]

   8. Applied egg-rr 65.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{e^{\mathsf{log1p}\left({\left(\frac{d}{D}\right)}^{-2}\right)} - 1}}\right) \]
   9. Step-by-step derivation

      1. expm1-def 75.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{D}\right)}^{-2}\right)\right)}}\right) \]

      2. expm1-log1p 76.0%

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

   10. Simplified 76.0%

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

       1. associate-/l/ 77.0%

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

       2. div-inv 77.0%

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

   12. Applied egg-rr 77.0%

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

       1. associate-*r/ 77.0%

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

       2. *-rgt-identity 77.0%

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

   14. Simplified 77.0%

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

       1. metadata-eval 77.0%

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

       2. pow-prod-up 77.0%

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

       3. unpow-1 77.0%

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

       4. unpow-1 77.0%

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

   16. Applied egg-rr 77.0%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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 2.5%

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

   4. Taylor expanded in c0 around -inf 1.9%

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

      1. mul-1-neg 1.9%

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

      2. distribute-lft-in 1.2%

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

   6. Simplified 32.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   7. Taylor expanded in c0 around 0 40.5%

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

4. Final simplification 53.6%

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

Alternative 4: 43.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 8.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{w \cdot h}}{\frac{1}{\frac{d}{D}} \cdot \frac{D}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= w 8.5e+98)
   (* (/ c0 (* 2.0 w)) (* 2.0 (/ (/ c0 (* w h)) (* (/ 1.0 (/ d D)) (/ D d)))))
   0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= 8.5e+98) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) / ((1.0 / (d / D)) * (D / d))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (w <= 8.5d+98) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * ((c0 / (w * h)) / ((1.0d0 / (d_1 / d)) * (d / d_1))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= 8.5e+98) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) / ((1.0 / (d / D)) * (D / d))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if w <= 8.5e+98:
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) / ((1.0 / (d / D)) * (D / d))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (w <= 8.5e+98)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(c0 / Float64(w * h)) / Float64(Float64(1.0 / Float64(d / D)) * Float64(D / d)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (w <= 8.5e+98)
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (w * h)) / ((1.0 / (d / D)) * (D / d))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[w, 8.5e+98], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 8.4999999999999996e98

   1. Initial program 28.8%

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

   3. Simplified 29.2%

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

   4. Taylor expanded in c0 around inf 37.5%

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

      1. *-commutative 37.5%

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

      2. associate-/l/ 37.0%

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

      3. associate-*l/ 37.0%

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

      4. associate-/l* 37.5%

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

      5. *-commutative 37.5%

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

   6. Simplified 37.5%

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

      1. expm1-log1p-u 37.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2}}{{d}^{2}}\right)\right)}}\right) \]

      2. expm1-udef 34.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{D}^{2}}{{d}^{2}}\right)} - 1}}\right) \]

      3. clear-num 34.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{{d}^{2}}{{D}^{2}}}}\right)} - 1}\right) \]

      4. unpow2 34.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\frac{\color{blue}{d \cdot d}}{{D}^{2}}}\right)} - 1}\right) \]

      5. unpow2 34.1%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\frac{d \cdot d}{\color{blue}{D \cdot D}}}\right)} - 1}\right) \]

      6. frac-times 42.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}\right)} - 1}\right) \]

      7. pow2 42.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{\left(\frac{d}{D}\right)}^{2}}}\right)} - 1}\right) \]

      8. pow-flip 42.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{d}{D}\right)}^{\left(-2\right)}}\right)} - 1}\right) \]

      9. metadata-eval 42.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{e^{\mathsf{log1p}\left({\left(\frac{d}{D}\right)}^{\color{blue}{-2}}\right)} - 1}\right) \]

   8. Applied egg-rr 42.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{e^{\mathsf{log1p}\left({\left(\frac{d}{D}\right)}^{-2}\right)} - 1}}\right) \]
   9. Step-by-step derivation

      1. expm1-def 46.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{h \cdot w}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{D}\right)}^{-2}\right)\right)}}\right) \]

      2. expm1-log1p 46.9%

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

   10. Simplified 46.9%

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

       1. metadata-eval 49.3%

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

       2. pow-prod-up 49.3%

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

       3. unpow-1 49.3%

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

       4. unpow-1 49.3%

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

   12. Applied egg-rr 46.9%

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

   13. Taylor expanded in d around 0 46.9%

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

   if 8.4999999999999996e98 < w

   1. Initial program 16.9%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in c0 around -inf 11.9%

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

      1. mul-1-neg 11.9%

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

      2. distribute-lft-in 9.6%

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

   6. Simplified 45.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   7. Taylor expanded in c0 around 0 45.5%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 8.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{w \cdot h}}{\frac{1}{\frac{d}{D}} \cdot \frac{D}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 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 26.7%

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

3. Simplified 27.9%

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

4. Taylor expanded in c0 around -inf 5.1%

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

   1. mul-1-neg 5.1%

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

   2. distribute-lft-in 4.6%

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

6. Simplified 25.7%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

7. Taylor expanded in c0 around 0 30.6%

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

8. Final simplification 30.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023314 
(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))))))