Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.9% → 63.0%

Time: 34.4s

Alternatives: 9

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = -2.4437268891229528e+196
  2. w = -3.523487723397485e-264
  3. h = -6.322967645771543e-223
  4. D = 1.8735767596864516e-245
  5. d = 4.990786902775376e+301
  6. M = 10605607578.986292

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: 63.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := t_1 \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-48} \lor \neg \left(t_2 \leq 0\right) \land t_2 \leq \infty:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\left(D \cdot M\right)}^{-2}}\\ \end{array} \end{array} \]

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_2 (* t_1 (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (or (<= t_2 -4e-48) (and (not (<= t_2 0.0)) (<= t_2 INFINITY)))
     (* t_1 (* 2.0 (/ (* c0 (pow d 2.0)) (* (* w h) (pow D 2.0)))))
     (* h (/ (/ 0.25 (pow d 2.0)) (pow (* D M) -2.0))))))

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);
	double t_2 = t_1 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if ((t_2 <= -4e-48) || (!(t_2 <= 0.0) && (t_2 <= ((double) INFINITY)))) {
		tmp = t_1 * (2.0 * ((c0 * pow(d, 2.0)) / ((w * h) * pow(D, 2.0))));
	} else {
		tmp = h * ((0.25 / pow(d, 2.0)) / pow((D * M), -2.0));
	}
	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);
	double t_2 = t_1 * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if ((t_2 <= -4e-48) || (!(t_2 <= 0.0) && (t_2 <= Double.POSITIVE_INFINITY))) {
		tmp = t_1 * (2.0 * ((c0 * Math.pow(d, 2.0)) / ((w * h) * Math.pow(D, 2.0))));
	} else {
		tmp = h * ((0.25 / Math.pow(d, 2.0)) / Math.pow((D * M), -2.0));
	}
	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_2 = t_1 * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if (t_2 <= -4e-48) or (not (t_2 <= 0.0) and (t_2 <= math.inf)):
		tmp = t_1 * (2.0 * ((c0 * math.pow(d, 2.0)) / ((w * h) * math.pow(D, 2.0))))
	else:
		tmp = h * ((0.25 / math.pow(d, 2.0)) / math.pow((D * M), -2.0))
	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(c0 / Float64(2.0 * w))
	t_2 = Float64(t_1 * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if ((t_2 <= -4e-48) || (!(t_2 <= 0.0) && (t_2 <= Inf)))
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(c0 * (d ^ 2.0)) / Float64(Float64(w * h) * (D ^ 2.0)))));
	else
		tmp = Float64(h * Float64(Float64(0.25 / (d ^ 2.0)) / (Float64(D * M) ^ -2.0)));
	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_2 = t_1 * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if ((t_2 <= -4e-48) || (~((t_2 <= 0.0)) && (t_2 <= Inf)))
		tmp = t_1 * (2.0 * ((c0 * (d ^ 2.0)) / ((w * h) * (D ^ 2.0))));
	else
		tmp = h * ((0.25 / (d ^ 2.0)) / ((D * M) ^ -2.0));
	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[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -4e-48], And[N[Not[LessEqual[t$95$2, 0.0]], $MachinePrecision], LessEqual[t$95$2, Infinity]]], N[(t$95$1 * N[(2.0 * N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(h * N[(N[(0.25 / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(D * M), $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))))) < -3.9999999999999999e-48 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 76.5%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in c0 around inf 79.4%

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

   if -3.9999999999999999e-48 < (*.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 or +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 2.4%

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

   3. Simplified 3.6%

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

   5. Applied egg-rr 3.0%

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

      1. *-commutative 3.0%

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

      2. *-commutative 3.0%

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

      3. times-frac 2.9%

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

   7. Simplified 31.5%

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

   8. Taylor expanded in c0 around -inf 22.2%

      \[\leadsto \frac{0 + {M}^{2}}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h} - \color{blue}{-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}} \cdot \frac{c0}{2 \cdot w} \]
   9. Step-by-step derivation

      1. mul-1-neg 22.2%

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

      2. associate-*r* 25.1%

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

      3. *-commutative 25.1%

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

      4. *-commutative 25.1%

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

      5. times-frac 27.8%

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

      6. associate-/r* 26.4%

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

   10. Simplified 26.4%

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

   11. Taylor expanded in M around 0 39.4%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 39.4%

          \[\leadsto \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

       2. associate-*r* 40.9%

          \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2}} \]

       3. unpow2 40.9%

          \[\leadsto \frac{0.25 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{2}} \]

       4. unpow2 40.9%

          \[\leadsto \frac{0.25 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2}} \]

       5. swap-sqr 52.1%

          \[\leadsto \frac{0.25 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right)}{{d}^{2}} \]

       6. unpow2 52.1%

          \[\leadsto \frac{0.25 \cdot \left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right)}{{d}^{2}} \]

       7. associate-/l* 52.0%

          \[\leadsto \color{blue}{\frac{0.25}{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2} \cdot h}}} \]

       8. associate-/r* 55.7%

          \[\leadsto \frac{0.25}{\color{blue}{\frac{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2}}}{h}}} \]

       9. *-commutative 55.7%

          \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{{\color{blue}{\left(M \cdot D\right)}}^{2}}}{h}} \]

   13. Simplified 55.7%

       \[\leadsto \color{blue}{\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}} \]
   14. Step-by-step derivation

       1. expm1-log1p-u 51.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}\right)\right)} \]

       2. expm1-udef 43.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}\right)} - 1} \]

       3. associate-/r/ 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.25}{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}} \cdot h}\right)} - 1 \]

       4. div-inv 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{{d}^{2} \cdot \frac{1}{{\left(M \cdot D\right)}^{2}}}} \cdot h\right)} - 1 \]

       5. *-commutative 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot \frac{1}{{\color{blue}{\left(D \cdot M\right)}}^{2}}} \cdot h\right)} - 1 \]

       6. pow-flip 43.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot \color{blue}{{\left(D \cdot M\right)}^{\left(-2\right)}}} \cdot h\right)} - 1 \]

       7. *-commutative 43.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\color{blue}{\left(M \cdot D\right)}}^{\left(-2\right)}} \cdot h\right)} - 1 \]

       8. metadata-eval 43.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{\color{blue}{-2}}} \cdot h\right)} - 1 \]

   15. Applied egg-rr 43.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}} \cdot h\right)} - 1} \]
   16. Step-by-step derivation

       1. expm1-def 51.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}} \cdot h\right)\right)} \]

       2. expm1-log1p 55.7%

          \[\leadsto \color{blue}{\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}} \cdot h} \]

       3. *-commutative 55.7%

          \[\leadsto \color{blue}{h \cdot \frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}}} \]

       4. associate-/r* 55.9%

          \[\leadsto h \cdot \color{blue}{\frac{\frac{0.25}{{d}^{2}}}{{\left(M \cdot D\right)}^{-2}}} \]

       5. *-commutative 55.9%

          \[\leadsto h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\color{blue}{\left(D \cdot M\right)}}^{-2}} \]

   17. Simplified 55.9%

       \[\leadsto \color{blue}{h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\left(D \cdot M\right)}^{-2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -4 \cdot 10^{-48} \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0\right) \land \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\left(D \cdot M\right)}^{-2}}\\ \end{array} \]

Alternative 2: 62.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0 \lor \neg \left(t_2 \leq \infty\right):\\ \;\;\;\;h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\left(D \cdot M\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot {D}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -4e-48)
     t_2
     (if (or (<= t_2 0.0) (not (<= t_2 INFINITY)))
       (* h (/ (/ 0.25 (pow d 2.0)) (pow (* D M) -2.0)))
       (* t_0 (* 2.0 (* (/ c0 w) (/ (pow d 2.0) (* h (pow D 2.0))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -4e-48) {
		tmp = t_2;
	} else if ((t_2 <= 0.0) || !(t_2 <= ((double) INFINITY))) {
		tmp = h * ((0.25 / pow(d, 2.0)) / pow((D * M), -2.0));
	} else {
		tmp = t_0 * (2.0 * ((c0 / w) * (pow(d, 2.0) / (h * pow(D, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -4e-48) {
		tmp = t_2;
	} else if ((t_2 <= 0.0) || !(t_2 <= Double.POSITIVE_INFINITY)) {
		tmp = h * ((0.25 / Math.pow(d, 2.0)) / Math.pow((D * M), -2.0));
	} else {
		tmp = t_0 * (2.0 * ((c0 / w) * (Math.pow(d, 2.0) / (h * Math.pow(D, 2.0)))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -4e-48:
		tmp = t_2
	elif (t_2 <= 0.0) or not (t_2 <= math.inf):
		tmp = h * ((0.25 / math.pow(d, 2.0)) / math.pow((D * M), -2.0))
	else:
		tmp = t_0 * (2.0 * ((c0 / w) * (math.pow(d, 2.0) / (h * math.pow(D, 2.0)))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -4e-48)
		tmp = t_2;
	elseif ((t_2 <= 0.0) || !(t_2 <= Inf))
		tmp = Float64(h * Float64(Float64(0.25 / (d ^ 2.0)) / (Float64(D * M) ^ -2.0)));
	else
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(c0 / w) * Float64((d ^ 2.0) / Float64(h * (D ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -4e-48)
		tmp = t_2;
	elseif ((t_2 <= 0.0) || ~((t_2 <= Inf)))
		tmp = h * ((0.25 / (d ^ 2.0)) / ((D * M) ^ -2.0));
	else
		tmp = t_0 * (2.0 * ((c0 / w) * ((d ^ 2.0) / (h * (D ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-48], t$95$2, If[Or[LessEqual[t$95$2, 0.0], N[Not[LessEqual[t$95$2, Infinity]], $MachinePrecision]], N[(h * N[(N[(0.25 / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(D * M), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[(N[(c0 / w), $MachinePrecision] * N[(N[Power[d, 2.0], $MachinePrecision] / N[(h * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))))) < -3.9999999999999999e-48

   1. Initial program 79.8%

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

   if -3.9999999999999999e-48 < (*.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 or +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 2.4%

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

   3. Simplified 3.6%

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

   5. Applied egg-rr 3.0%

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

      1. *-commutative 3.0%

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

      2. *-commutative 3.0%

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

      3. times-frac 2.9%

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

   7. Simplified 31.5%

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

   8. Taylor expanded in c0 around -inf 22.2%

      \[\leadsto \frac{0 + {M}^{2}}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h} - \color{blue}{-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}} \cdot \frac{c0}{2 \cdot w} \]
   9. Step-by-step derivation

      1. mul-1-neg 22.2%

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

      2. associate-*r* 25.1%

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

      3. *-commutative 25.1%

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

      4. *-commutative 25.1%

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

      5. times-frac 27.8%

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

      6. associate-/r* 26.4%

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

   10. Simplified 26.4%

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

   11. Taylor expanded in M around 0 39.4%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 39.4%

          \[\leadsto \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

       2. associate-*r* 40.9%

          \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2}} \]

       3. unpow2 40.9%

          \[\leadsto \frac{0.25 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{2}} \]

       4. unpow2 40.9%

          \[\leadsto \frac{0.25 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2}} \]

       5. swap-sqr 52.1%

          \[\leadsto \frac{0.25 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right)}{{d}^{2}} \]

       6. unpow2 52.1%

          \[\leadsto \frac{0.25 \cdot \left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right)}{{d}^{2}} \]

       7. associate-/l* 52.0%

          \[\leadsto \color{blue}{\frac{0.25}{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2} \cdot h}}} \]

       8. associate-/r* 55.7%

          \[\leadsto \frac{0.25}{\color{blue}{\frac{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2}}}{h}}} \]

       9. *-commutative 55.7%

          \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{{\color{blue}{\left(M \cdot D\right)}}^{2}}}{h}} \]

   13. Simplified 55.7%

       \[\leadsto \color{blue}{\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}} \]
   14. Step-by-step derivation

       1. expm1-log1p-u 51.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}\right)\right)} \]

       2. expm1-udef 43.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}\right)} - 1} \]

       3. associate-/r/ 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.25}{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}} \cdot h}\right)} - 1 \]

       4. div-inv 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{{d}^{2} \cdot \frac{1}{{\left(M \cdot D\right)}^{2}}}} \cdot h\right)} - 1 \]

       5. *-commutative 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot \frac{1}{{\color{blue}{\left(D \cdot M\right)}}^{2}}} \cdot h\right)} - 1 \]

       6. pow-flip 43.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot \color{blue}{{\left(D \cdot M\right)}^{\left(-2\right)}}} \cdot h\right)} - 1 \]

       7. *-commutative 43.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\color{blue}{\left(M \cdot D\right)}}^{\left(-2\right)}} \cdot h\right)} - 1 \]

       8. metadata-eval 43.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{\color{blue}{-2}}} \cdot h\right)} - 1 \]

   15. Applied egg-rr 43.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}} \cdot h\right)} - 1} \]
   16. Step-by-step derivation

       1. expm1-def 51.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}} \cdot h\right)\right)} \]

       2. expm1-log1p 55.7%

          \[\leadsto \color{blue}{\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}} \cdot h} \]

       3. *-commutative 55.7%

          \[\leadsto \color{blue}{h \cdot \frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}}} \]

       4. associate-/r* 55.9%

          \[\leadsto h \cdot \color{blue}{\frac{\frac{0.25}{{d}^{2}}}{{\left(M \cdot D\right)}^{-2}}} \]

       5. *-commutative 55.9%

          \[\leadsto h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\color{blue}{\left(D \cdot M\right)}}^{-2}} \]

   17. Simplified 55.9%

       \[\leadsto \color{blue}{h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\left(D \cdot M\right)}^{-2}}} \]

   if 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 72.5%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in c0 around inf 75.9%

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

      1. *-commutative 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-*r* 73.0%

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

      4. times-frac 72.9%

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

      5. *-commutative 72.9%

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

   6. Simplified 72.9%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -4 \cdot 10^{-48}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0 \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty\right):\\ \;\;\;\;h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\left(D \cdot M\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{{d}^{2}}{h \cdot {D}^{2}}\right)\right)\\ \end{array} \]

Alternative 3: 62.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0 \lor \neg \left(t_2 \leq \infty\right):\\ \;\;\;\;h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\left(D \cdot M\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot {D}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -4e-48)
     t_2
     (if (or (<= t_2 0.0) (not (<= t_2 INFINITY)))
       (* h (/ (/ 0.25 (pow d 2.0)) (pow (* D M) -2.0)))
       (* t_0 (* 2.0 (/ (* c0 (pow d 2.0)) (* w (* h (pow D 2.0))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -4e-48) {
		tmp = t_2;
	} else if ((t_2 <= 0.0) || !(t_2 <= ((double) INFINITY))) {
		tmp = h * ((0.25 / pow(d, 2.0)) / pow((D * M), -2.0));
	} else {
		tmp = t_0 * (2.0 * ((c0 * pow(d, 2.0)) / (w * (h * pow(D, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -4e-48) {
		tmp = t_2;
	} else if ((t_2 <= 0.0) || !(t_2 <= Double.POSITIVE_INFINITY)) {
		tmp = h * ((0.25 / Math.pow(d, 2.0)) / Math.pow((D * M), -2.0));
	} else {
		tmp = t_0 * (2.0 * ((c0 * Math.pow(d, 2.0)) / (w * (h * Math.pow(D, 2.0)))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -4e-48:
		tmp = t_2
	elif (t_2 <= 0.0) or not (t_2 <= math.inf):
		tmp = h * ((0.25 / math.pow(d, 2.0)) / math.pow((D * M), -2.0))
	else:
		tmp = t_0 * (2.0 * ((c0 * math.pow(d, 2.0)) / (w * (h * math.pow(D, 2.0)))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -4e-48)
		tmp = t_2;
	elseif ((t_2 <= 0.0) || !(t_2 <= Inf))
		tmp = Float64(h * Float64(Float64(0.25 / (d ^ 2.0)) / (Float64(D * M) ^ -2.0)));
	else
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(c0 * (d ^ 2.0)) / Float64(w * Float64(h * (D ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -4e-48)
		tmp = t_2;
	elseif ((t_2 <= 0.0) || ~((t_2 <= Inf)))
		tmp = h * ((0.25 / (d ^ 2.0)) / ((D * M) ^ -2.0));
	else
		tmp = t_0 * (2.0 * ((c0 * (d ^ 2.0)) / (w * (h * (D ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-48], t$95$2, If[Or[LessEqual[t$95$2, 0.0], N[Not[LessEqual[t$95$2, Infinity]], $MachinePrecision]], N[(h * N[(N[(0.25 / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(D * M), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[(w * N[(h * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))))) < -3.9999999999999999e-48

   1. Initial program 79.8%

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

   if -3.9999999999999999e-48 < (*.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 or +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 2.4%

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

   3. Simplified 3.6%

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

   5. Applied egg-rr 3.0%

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

      1. *-commutative 3.0%

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

      2. *-commutative 3.0%

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

      3. times-frac 2.9%

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

   7. Simplified 31.5%

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

   8. Taylor expanded in c0 around -inf 22.2%

      \[\leadsto \frac{0 + {M}^{2}}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h} - \color{blue}{-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}} \cdot \frac{c0}{2 \cdot w} \]
   9. Step-by-step derivation

      1. mul-1-neg 22.2%

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

      2. associate-*r* 25.1%

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

      3. *-commutative 25.1%

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

      4. *-commutative 25.1%

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

      5. times-frac 27.8%

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

      6. associate-/r* 26.4%

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

   10. Simplified 26.4%

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

   11. Taylor expanded in M around 0 39.4%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 39.4%

          \[\leadsto \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

       2. associate-*r* 40.9%

          \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2}} \]

       3. unpow2 40.9%

          \[\leadsto \frac{0.25 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{2}} \]

       4. unpow2 40.9%

          \[\leadsto \frac{0.25 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2}} \]

       5. swap-sqr 52.1%

          \[\leadsto \frac{0.25 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right)}{{d}^{2}} \]

       6. unpow2 52.1%

          \[\leadsto \frac{0.25 \cdot \left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right)}{{d}^{2}} \]

       7. associate-/l* 52.0%

          \[\leadsto \color{blue}{\frac{0.25}{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2} \cdot h}}} \]

       8. associate-/r* 55.7%

          \[\leadsto \frac{0.25}{\color{blue}{\frac{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2}}}{h}}} \]

       9. *-commutative 55.7%

          \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{{\color{blue}{\left(M \cdot D\right)}}^{2}}}{h}} \]

   13. Simplified 55.7%

       \[\leadsto \color{blue}{\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}} \]
   14. Step-by-step derivation

       1. expm1-log1p-u 51.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}\right)\right)} \]

       2. expm1-udef 43.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}\right)} - 1} \]

       3. associate-/r/ 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.25}{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}} \cdot h}\right)} - 1 \]

       4. div-inv 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{{d}^{2} \cdot \frac{1}{{\left(M \cdot D\right)}^{2}}}} \cdot h\right)} - 1 \]

       5. *-commutative 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot \frac{1}{{\color{blue}{\left(D \cdot M\right)}}^{2}}} \cdot h\right)} - 1 \]

       6. pow-flip 43.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot \color{blue}{{\left(D \cdot M\right)}^{\left(-2\right)}}} \cdot h\right)} - 1 \]

       7. *-commutative 43.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\color{blue}{\left(M \cdot D\right)}}^{\left(-2\right)}} \cdot h\right)} - 1 \]

       8. metadata-eval 43.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{\color{blue}{-2}}} \cdot h\right)} - 1 \]

   15. Applied egg-rr 43.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}} \cdot h\right)} - 1} \]
   16. Step-by-step derivation

       1. expm1-def 51.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}} \cdot h\right)\right)} \]

       2. expm1-log1p 55.7%

          \[\leadsto \color{blue}{\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}} \cdot h} \]

       3. *-commutative 55.7%

          \[\leadsto \color{blue}{h \cdot \frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}}} \]

       4. associate-/r* 55.9%

          \[\leadsto h \cdot \color{blue}{\frac{\frac{0.25}{{d}^{2}}}{{\left(M \cdot D\right)}^{-2}}} \]

       5. *-commutative 55.9%

          \[\leadsto h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\color{blue}{\left(D \cdot M\right)}}^{-2}} \]

   17. Simplified 55.9%

       \[\leadsto \color{blue}{h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\left(D \cdot M\right)}^{-2}}} \]

   if 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 72.5%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in c0 around inf 75.9%

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

      1. *-commutative 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-*r* 73.0%

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

      4. times-frac 72.9%

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

      5. *-commutative 72.9%

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

   6. Simplified 72.9%

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

      1. frac-times 73.0%

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

      2. *-commutative 73.0%

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

   8. Applied egg-rr 73.0%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -4 \cdot 10^{-48}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0 \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty\right):\\ \;\;\;\;h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\left(D \cdot M\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0 \cdot {d}^{2}}{w \cdot \left(h \cdot {D}^{2}\right)}\right)\\ \end{array} \]

Alternative 4: 62.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right)\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{-48} \lor \neg \left(t_1 \leq 0\right) \land t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\left(D \cdot M\right)}^{-2}}\\ \end{array} \end{array} \]

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 (or (<= t_1 -4e-48) (and (not (<= t_1 0.0)) (<= t_1 INFINITY)))
     t_1
     (* h (/ (/ 0.25 (pow d 2.0)) (pow (* D M) -2.0))))))

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 <= -4e-48) || (!(t_1 <= 0.0) && (t_1 <= ((double) INFINITY)))) {
		tmp = t_1;
	} else {
		tmp = h * ((0.25 / pow(d, 2.0)) / pow((D * M), -2.0));
	}
	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 <= -4e-48) || (!(t_1 <= 0.0) && (t_1 <= Double.POSITIVE_INFINITY))) {
		tmp = t_1;
	} else {
		tmp = h * ((0.25 / Math.pow(d, 2.0)) / Math.pow((D * M), -2.0));
	}
	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 <= -4e-48) or (not (t_1 <= 0.0) and (t_1 <= math.inf)):
		tmp = t_1
	else:
		tmp = h * ((0.25 / math.pow(d, 2.0)) / math.pow((D * M), -2.0))
	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 <= -4e-48) || (!(t_1 <= 0.0) && (t_1 <= Inf)))
		tmp = t_1;
	else
		tmp = Float64(h * Float64(Float64(0.25 / (d ^ 2.0)) / (Float64(D * M) ^ -2.0)));
	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 <= -4e-48) || (~((t_1 <= 0.0)) && (t_1 <= Inf)))
		tmp = t_1;
	else
		tmp = h * ((0.25 / (d ^ 2.0)) / ((D * M) ^ -2.0));
	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[Or[LessEqual[t$95$1, -4e-48], And[N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision], LessEqual[t$95$1, Infinity]]], t$95$1, N[(h * N[(N[(0.25 / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(D * M), $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))))) < -3.9999999999999999e-48 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 76.5%

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

   if -3.9999999999999999e-48 < (*.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 or +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 2.4%

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

   3. Simplified 3.6%

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

   5. Applied egg-rr 3.0%

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

      1. *-commutative 3.0%

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

      2. *-commutative 3.0%

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

      3. times-frac 2.9%

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

   7. Simplified 31.5%

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

   8. Taylor expanded in c0 around -inf 22.2%

      \[\leadsto \frac{0 + {M}^{2}}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h} - \color{blue}{-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}} \cdot \frac{c0}{2 \cdot w} \]
   9. Step-by-step derivation

      1. mul-1-neg 22.2%

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

      2. associate-*r* 25.1%

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

      3. *-commutative 25.1%

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

      4. *-commutative 25.1%

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

      5. times-frac 27.8%

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

      6. associate-/r* 26.4%

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

   10. Simplified 26.4%

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

   11. Taylor expanded in M around 0 39.4%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 39.4%

          \[\leadsto \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

       2. associate-*r* 40.9%

          \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2}} \]

       3. unpow2 40.9%

          \[\leadsto \frac{0.25 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{2}} \]

       4. unpow2 40.9%

          \[\leadsto \frac{0.25 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2}} \]

       5. swap-sqr 52.1%

          \[\leadsto \frac{0.25 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right)}{{d}^{2}} \]

       6. unpow2 52.1%

          \[\leadsto \frac{0.25 \cdot \left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right)}{{d}^{2}} \]

       7. associate-/l* 52.0%

          \[\leadsto \color{blue}{\frac{0.25}{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2} \cdot h}}} \]

       8. associate-/r* 55.7%

          \[\leadsto \frac{0.25}{\color{blue}{\frac{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2}}}{h}}} \]

       9. *-commutative 55.7%

          \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{{\color{blue}{\left(M \cdot D\right)}}^{2}}}{h}} \]

   13. Simplified 55.7%

       \[\leadsto \color{blue}{\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}} \]
   14. Step-by-step derivation

       1. expm1-log1p-u 51.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}\right)\right)} \]

       2. expm1-udef 43.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}\right)} - 1} \]

       3. associate-/r/ 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.25}{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}} \cdot h}\right)} - 1 \]

       4. div-inv 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{{d}^{2} \cdot \frac{1}{{\left(M \cdot D\right)}^{2}}}} \cdot h\right)} - 1 \]

       5. *-commutative 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot \frac{1}{{\color{blue}{\left(D \cdot M\right)}}^{2}}} \cdot h\right)} - 1 \]

       6. pow-flip 43.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot \color{blue}{{\left(D \cdot M\right)}^{\left(-2\right)}}} \cdot h\right)} - 1 \]

       7. *-commutative 43.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\color{blue}{\left(M \cdot D\right)}}^{\left(-2\right)}} \cdot h\right)} - 1 \]

       8. metadata-eval 43.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{\color{blue}{-2}}} \cdot h\right)} - 1 \]

   15. Applied egg-rr 43.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}} \cdot h\right)} - 1} \]
   16. Step-by-step derivation

       1. expm1-def 51.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}} \cdot h\right)\right)} \]

       2. expm1-log1p 55.7%

          \[\leadsto \color{blue}{\frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}} \cdot h} \]

       3. *-commutative 55.7%

          \[\leadsto \color{blue}{h \cdot \frac{0.25}{{d}^{2} \cdot {\left(M \cdot D\right)}^{-2}}} \]

       4. associate-/r* 55.9%

          \[\leadsto h \cdot \color{blue}{\frac{\frac{0.25}{{d}^{2}}}{{\left(M \cdot D\right)}^{-2}}} \]

       5. *-commutative 55.9%

          \[\leadsto h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\color{blue}{\left(D \cdot M\right)}}^{-2}} \]

   17. Simplified 55.9%

       \[\leadsto \color{blue}{h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\left(D \cdot M\right)}^{-2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -4 \cdot 10^{-48} \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0\right) \land \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;h \cdot \frac{\frac{0.25}{{d}^{2}}}{{\left(D \cdot M\right)}^{-2}}\\ \end{array} \]

Alternative 5: 62.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right)\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{-48} \lor \neg \left(t_1 \leq 0\right) \land t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\frac{\frac{{d}^{2}}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{h}}\\ \end{array} \end{array} \]

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 (or (<= t_1 -4e-48) (and (not (<= t_1 0.0)) (<= t_1 INFINITY)))
     t_1
     (/ 0.25 (/ (/ (pow d 2.0) (* (* D M) (* D M))) h)))))

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 <= -4e-48) || (!(t_1 <= 0.0) && (t_1 <= ((double) INFINITY)))) {
		tmp = t_1;
	} else {
		tmp = 0.25 / ((pow(d, 2.0) / ((D * M) * (D * M))) / h);
	}
	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 <= -4e-48) || (!(t_1 <= 0.0) && (t_1 <= Double.POSITIVE_INFINITY))) {
		tmp = t_1;
	} else {
		tmp = 0.25 / ((Math.pow(d, 2.0) / ((D * M) * (D * M))) / h);
	}
	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 <= -4e-48) or (not (t_1 <= 0.0) and (t_1 <= math.inf)):
		tmp = t_1
	else:
		tmp = 0.25 / ((math.pow(d, 2.0) / ((D * M) * (D * M))) / h)
	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 <= -4e-48) || (!(t_1 <= 0.0) && (t_1 <= Inf)))
		tmp = t_1;
	else
		tmp = Float64(0.25 / Float64(Float64((d ^ 2.0) / Float64(Float64(D * M) * Float64(D * M))) / h));
	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 <= -4e-48) || (~((t_1 <= 0.0)) && (t_1 <= Inf)))
		tmp = t_1;
	else
		tmp = 0.25 / (((d ^ 2.0) / ((D * M) * (D * M))) / h);
	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[Or[LessEqual[t$95$1, -4e-48], And[N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision], LessEqual[t$95$1, Infinity]]], t$95$1, N[(0.25 / N[(N[(N[Power[d, 2.0], $MachinePrecision] / N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $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))))) < -3.9999999999999999e-48 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 76.5%

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

   if -3.9999999999999999e-48 < (*.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 or +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 2.4%

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

   3. Simplified 3.6%

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

   5. Applied egg-rr 3.0%

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

      1. *-commutative 3.0%

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

      2. *-commutative 3.0%

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

      3. times-frac 2.9%

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

   7. Simplified 31.5%

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

   8. Taylor expanded in c0 around -inf 22.2%

      \[\leadsto \frac{0 + {M}^{2}}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h} - \color{blue}{-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}} \cdot \frac{c0}{2 \cdot w} \]
   9. Step-by-step derivation

      1. mul-1-neg 22.2%

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

      2. associate-*r* 25.1%

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

      3. *-commutative 25.1%

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

      4. *-commutative 25.1%

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

      5. times-frac 27.8%

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

      6. associate-/r* 26.4%

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

   10. Simplified 26.4%

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

   11. Taylor expanded in M around 0 39.4%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 39.4%

          \[\leadsto \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

       2. associate-*r* 40.9%

          \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2}} \]

       3. unpow2 40.9%

          \[\leadsto \frac{0.25 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{2}} \]

       4. unpow2 40.9%

          \[\leadsto \frac{0.25 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2}} \]

       5. swap-sqr 52.1%

          \[\leadsto \frac{0.25 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right)}{{d}^{2}} \]

       6. unpow2 52.1%

          \[\leadsto \frac{0.25 \cdot \left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right)}{{d}^{2}} \]

       7. associate-/l* 52.0%

          \[\leadsto \color{blue}{\frac{0.25}{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2} \cdot h}}} \]

       8. associate-/r* 55.7%

          \[\leadsto \frac{0.25}{\color{blue}{\frac{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2}}}{h}}} \]

       9. *-commutative 55.7%

          \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{{\color{blue}{\left(M \cdot D\right)}}^{2}}}{h}} \]

   13. Simplified 55.7%

       \[\leadsto \color{blue}{\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}} \]
   14. Step-by-step derivation

       1. unpow2 55.7%

          \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}}{h}} \]

   15. Applied egg-rr 55.7%

       \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}}{h}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -4 \cdot 10^{-48} \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0\right) \land \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\frac{\frac{{d}^{2}}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{h}}\\ \end{array} \]

Alternative 6: 62.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right)\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot {d}^{-2}\right)\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\frac{\frac{{d}^{2}}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{h}}\\ \end{array} \end{array} \]

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 -4e-48)
     t_1
     (if (<= t_1 0.0)
       (* 0.25 (* (pow (* D M) 2.0) (* h (pow d -2.0))))
       (if (<= t_1 INFINITY)
         t_1
         (/ 0.25 (/ (/ (pow d 2.0) (* (* D M) (* D M))) h)))))))

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 <= -4e-48) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = 0.25 * (pow((D * M), 2.0) * (h * pow(d, -2.0)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 0.25 / ((pow(d, 2.0) / ((D * M) * (D * M))) / h);
	}
	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 <= -4e-48) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = 0.25 * (Math.pow((D * M), 2.0) * (h * Math.pow(d, -2.0)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 0.25 / ((Math.pow(d, 2.0) / ((D * M) * (D * M))) / h);
	}
	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 <= -4e-48:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = 0.25 * (math.pow((D * M), 2.0) * (h * math.pow(d, -2.0)))
	elif t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = 0.25 / ((math.pow(d, 2.0) / ((D * M) * (D * M))) / h)
	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 <= -4e-48)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(0.25 * Float64((Float64(D * M) ^ 2.0) * Float64(h * (d ^ -2.0))));
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(0.25 / Float64(Float64((d ^ 2.0) / Float64(Float64(D * M) * Float64(D * M))) / h));
	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 <= -4e-48)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = 0.25 * (((D * M) ^ 2.0) * (h * (d ^ -2.0)));
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = 0.25 / (((d ^ 2.0) / ((D * M) * (D * M))) / h);
	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, -4e-48], t$95$1, If[LessEqual[t$95$1, 0.0], N[(0.25 * N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * N[Power[d, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[(0.25 / N[(N[(N[Power[d, 2.0], $MachinePrecision] / N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $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))))) < -3.9999999999999999e-48 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 76.5%

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

   if -3.9999999999999999e-48 < (*.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 34.7%

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

   3. Simplified 33.9%

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

   4. Taylor expanded in c0 around -inf 45.9%

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

      1. +-commutative 45.9%

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

      2. fma-def 45.9%

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

      3. times-frac 45.8%

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

      4. associate-*r* 45.8%

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

      5. neg-mul-1 45.8%

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

      6. distribute-lft1-in 45.8%

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

      7. metadata-eval 45.8%

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

      8. mul0-lft 45.8%

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

      9. distribute-lft-neg-in 45.8%

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

      10. distribute-rgt-neg-in 45.8%

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

      11. metadata-eval 45.8%

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

   6. Simplified 45.8%

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

   7. Taylor expanded in c0 around 0 40.4%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 40.4%

         \[\leadsto 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)\right)} \]

      2. expm1-udef 26.1%

         \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)} - 1\right)} \]

      3. div-inv 26.1%

         \[\leadsto 0.25 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{{d}^{2}}}\right)} - 1\right) \]

      4. pow2 26.1%

         \[\leadsto 0.25 \cdot \left(e^{\mathsf{log1p}\left(\left({D}^{2} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \cdot \frac{1}{{d}^{2}}\right)} - 1\right) \]

      5. associate-*r* 32.5%

         \[\leadsto 0.25 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\left({D}^{2} \cdot \left(M \cdot M\right)\right) \cdot h\right)} \cdot \frac{1}{{d}^{2}}\right)} - 1\right) \]

      6. pow2 32.5%

         \[\leadsto 0.25 \cdot \left(e^{\mathsf{log1p}\left(\left(\left({D}^{2} \cdot \color{blue}{{M}^{2}}\right) \cdot h\right) \cdot \frac{1}{{d}^{2}}\right)} - 1\right) \]

      7. pow-prod-down 41.4%

         \[\leadsto 0.25 \cdot \left(e^{\mathsf{log1p}\left(\left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right) \cdot \frac{1}{{d}^{2}}\right)} - 1\right) \]

      8. pow-flip 41.4%

         \[\leadsto 0.25 \cdot \left(e^{\mathsf{log1p}\left(\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot \color{blue}{{d}^{\left(-2\right)}}\right)} - 1\right) \]

      9. metadata-eval 41.4%

         \[\leadsto 0.25 \cdot \left(e^{\mathsf{log1p}\left(\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot {d}^{\color{blue}{-2}}\right)} - 1\right) \]

   9. Applied egg-rr 41.4%

      \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot {d}^{-2}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 76.8%

          \[\leadsto 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot {d}^{-2}\right)\right)} \]

       2. expm1-log1p 77.7%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\left({\left(D \cdot M\right)}^{2} \cdot h\right) \cdot {d}^{-2}\right)} \]

       3. associate-*l* 78.1%

          \[\leadsto 0.25 \cdot \color{blue}{\left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot {d}^{-2}\right)\right)} \]

       4. *-commutative 78.1%

          \[\leadsto 0.25 \cdot \left({\color{blue}{\left(M \cdot D\right)}}^{2} \cdot \left(h \cdot {d}^{-2}\right)\right) \]

   11. Simplified 78.1%

       \[\leadsto 0.25 \cdot \color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot \left(h \cdot {d}^{-2}\right)\right)} \]

   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 1.3%

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

   5. Applied egg-rr 1.9%

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

      1. *-commutative 1.9%

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

      2. *-commutative 1.9%

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

      3. times-frac 1.9%

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

   7. Simplified 29.8%

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

   8. Taylor expanded in c0 around -inf 19.8%

      \[\leadsto \frac{0 + {M}^{2}}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h} - \color{blue}{-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}} \cdot \frac{c0}{2 \cdot w} \]
   9. Step-by-step derivation

      1. mul-1-neg 19.8%

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

      2. associate-*r* 22.8%

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

      3. *-commutative 22.8%

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

      4. *-commutative 22.8%

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

      5. times-frac 25.8%

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

      6. associate-/r* 24.7%

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

   10. Simplified 24.7%

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

   11. Taylor expanded in M around 0 39.3%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 39.3%

          \[\leadsto \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

       2. associate-*r* 39.9%

          \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2}} \]

       3. unpow2 39.9%

          \[\leadsto \frac{0.25 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{2}} \]

       4. unpow2 39.9%

          \[\leadsto \frac{0.25 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2}} \]

       5. swap-sqr 50.1%

          \[\leadsto \frac{0.25 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right)}{{d}^{2}} \]

       6. unpow2 50.1%

          \[\leadsto \frac{0.25 \cdot \left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right)}{{d}^{2}} \]

       7. associate-/l* 50.1%

          \[\leadsto \color{blue}{\frac{0.25}{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2} \cdot h}}} \]

       8. associate-/r* 54.0%

          \[\leadsto \frac{0.25}{\color{blue}{\frac{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2}}}{h}}} \]

       9. *-commutative 54.0%

          \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{{\color{blue}{\left(M \cdot D\right)}}^{2}}}{h}} \]

   13. Simplified 54.0%

       \[\leadsto \color{blue}{\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}} \]
   14. Step-by-step derivation

       1. unpow2 54.0%

          \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}}{h}} \]

   15. Applied egg-rr 54.0%

       \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}}{h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.4%

   \[\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 -4 \cdot 10^{-48}:\\ \;\;\;\;\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)\\ \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:\\ \;\;\;\;0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot {d}^{-2}\right)\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(\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)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\frac{\frac{{d}^{2}}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{h}}\\ \end{array} \]

Alternative 7: 41.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\\ \mathbf{if}\;h \leq -2.4 \cdot 10^{-279} \lor \neg \left(h \leq 7.4 \cdot 10^{-295}\right):\\ \;\;\;\;\frac{0.25}{\frac{\frac{{d}^{2}}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* w h)) (/ (* d d) (* D D)))))
   (if (or (<= h -2.4e-279) (not (<= h 7.4e-295)))
     (/ 0.25 (/ (/ (pow d 2.0) (* (* D M) (* D M))) h))
     (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double tmp;
	if ((h <= -2.4e-279) || !(h <= 7.4e-295)) {
		tmp = 0.25 / ((pow(d, 2.0) / ((D * M) * (D * M))) / h);
	} 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 = (c0 / (w * h)) * ((d_1 * d_1) / (d * d))
    if ((h <= (-2.4d-279)) .or. (.not. (h <= 7.4d-295))) then
        tmp = 0.25d0 / (((d_1 ** 2.0d0) / ((d * m) * (d * m))) / h)
    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 = (c0 / (w * h)) * ((d * d) / (D * D));
	double tmp;
	if ((h <= -2.4e-279) || !(h <= 7.4e-295)) {
		tmp = 0.25 / ((Math.pow(d, 2.0) / ((D * M) * (D * M))) / h);
	} 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 = (c0 / (w * h)) * ((d * d) / (D * D))
	tmp = 0
	if (h <= -2.4e-279) or not (h <= 7.4e-295):
		tmp = 0.25 / ((math.pow(d, 2.0) / ((D * M) * (D * M))) / h)
	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(c0 / Float64(w * h)) * Float64(Float64(d * d) / Float64(D * D)))
	tmp = 0.0
	if ((h <= -2.4e-279) || !(h <= 7.4e-295))
		tmp = Float64(0.25 / Float64(Float64((d ^ 2.0) / Float64(Float64(D * M) * Float64(D * M))) / h));
	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 = (c0 / (w * h)) * ((d * d) / (D * D));
	tmp = 0.0;
	if ((h <= -2.4e-279) || ~((h <= 7.4e-295)))
		tmp = 0.25 / (((d ^ 2.0) / ((D * M) * (D * M))) / h);
	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[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[h, -2.4e-279], N[Not[LessEqual[h, 7.4e-295]], $MachinePrecision]], N[(0.25 / N[(N[(N[Power[d, 2.0], $MachinePrecision] / N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -2.3999999999999999e-279 or 7.3999999999999999e-295 < h

   1. Initial program 19.9%

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

   3. Simplified 20.3%

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

   5. Applied egg-rr 2.3%

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

      1. *-commutative 2.3%

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

      2. *-commutative 2.3%

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

      3. times-frac 2.3%

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

   7. Simplified 24.8%

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

   8. Taylor expanded in c0 around -inf 18.4%

      \[\leadsto \frac{0 + {M}^{2}}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h} - \color{blue}{-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}} \cdot \frac{c0}{2 \cdot w} \]
   9. Step-by-step derivation

      1. mul-1-neg 18.4%

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

      2. associate-*r* 20.2%

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

      3. *-commutative 20.2%

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

      4. *-commutative 20.2%

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

      5. times-frac 22.3%

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

      6. associate-/r* 21.6%

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

   10. Simplified 21.6%

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

   11. Taylor expanded in M around 0 32.9%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 32.9%

          \[\leadsto \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

       2. associate-*r* 34.1%

          \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2}} \]

       3. unpow2 34.1%

          \[\leadsto \frac{0.25 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{2}} \]

       4. unpow2 34.1%

          \[\leadsto \frac{0.25 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2}} \]

       5. swap-sqr 42.7%

          \[\leadsto \frac{0.25 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right)}{{d}^{2}} \]

       6. unpow2 42.7%

          \[\leadsto \frac{0.25 \cdot \left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right)}{{d}^{2}} \]

       7. associate-/l* 42.6%

          \[\leadsto \color{blue}{\frac{0.25}{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2} \cdot h}}} \]

       8. associate-/r* 44.7%

          \[\leadsto \frac{0.25}{\color{blue}{\frac{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2}}}{h}}} \]

       9. *-commutative 44.7%

          \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{{\color{blue}{\left(M \cdot D\right)}}^{2}}}{h}} \]

   13. Simplified 44.7%

       \[\leadsto \color{blue}{\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}} \]
   14. Step-by-step derivation

       1. unpow2 44.7%

          \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}}{h}} \]

   15. Applied egg-rr 44.7%

       \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}}{h}} \]

   if -2.3999999999999999e-279 < h < 7.3999999999999999e-295

   1. Initial program 66.7%

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

   3. Simplified 66.7%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.4 \cdot 10^{-279} \lor \neg \left(h \leq 7.4 \cdot 10^{-295}\right):\\ \;\;\;\;\frac{0.25}{\frac{\frac{{d}^{2}}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) - M \cdot M}\right)\\ \end{array} \]

Alternative 8: 42.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\frac{\frac{{d}^{2}}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{h}} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (/ 0.25 (/ (/ (pow d 2.0) (* (* D M) (* D M))) h)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 / ((pow(d, 2.0) / ((D * M) * (D * M))) / h);
}

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 = 0.25d0 / (((d_1 ** 2.0d0) / ((d * m) * (d * m))) / h)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 / ((Math.pow(d, 2.0) / ((D * M) * (D * M))) / h);
}

Python

def code(c0, w, h, D, d, M):
	return 0.25 / ((math.pow(d, 2.0) / ((D * M) * (D * M))) / h)

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.25 / Float64(Float64((d ^ 2.0) / Float64(Float64(D * M) * Float64(D * M))) / h))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.25 / (((d ^ 2.0) / ((D * M) * (D * M))) / h);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.25 / N[(N[(N[Power[d, 2.0], $MachinePrecision] / N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.7%

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

3. Simplified 23.1%

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

5. Applied egg-rr 2.2%

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

   1. *-commutative 2.2%

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

   2. *-commutative 2.2%

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

   3. times-frac 2.2%

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

7. Simplified 23.4%

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

8. Taylor expanded in c0 around -inf 17.3%

   \[\leadsto \frac{0 + {M}^{2}}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h} - \color{blue}{-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}} \cdot \frac{c0}{2 \cdot w} \]
9. Step-by-step derivation

   1. mul-1-neg 17.3%

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

   2. associate-*r* 19.0%

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

   3. *-commutative 19.0%

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

   4. *-commutative 19.0%

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

   5. times-frac 21.0%

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

   6. associate-/r* 20.4%

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

10. Simplified 20.4%

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

11. Taylor expanded in M around 0 31.5%

    \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
12. Step-by-step derivation

    1. associate-*r/ 31.5%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

    2. associate-*r* 32.6%

       \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}}{{d}^{2}} \]

    3. unpow2 32.6%

       \[\leadsto \frac{0.25 \cdot \left(\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{2}} \]

    4. unpow2 32.6%

       \[\leadsto \frac{0.25 \cdot \left(\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h\right)}{{d}^{2}} \]

    5. swap-sqr 40.7%

       \[\leadsto \frac{0.25 \cdot \left(\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h\right)}{{d}^{2}} \]

    6. unpow2 40.7%

       \[\leadsto \frac{0.25 \cdot \left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right)}{{d}^{2}} \]

    7. associate-/l* 40.6%

       \[\leadsto \color{blue}{\frac{0.25}{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2} \cdot h}}} \]

    8. associate-/r* 43.3%

       \[\leadsto \frac{0.25}{\color{blue}{\frac{\frac{{d}^{2}}{{\left(D \cdot M\right)}^{2}}}{h}}} \]

    9. *-commutative 43.3%

       \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{{\color{blue}{\left(M \cdot D\right)}}^{2}}}{h}} \]

13. Simplified 43.3%

    \[\leadsto \color{blue}{\frac{0.25}{\frac{\frac{{d}^{2}}{{\left(M \cdot D\right)}^{2}}}{h}}} \]
14. Step-by-step derivation

    1. unpow2 43.3%

       \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}}{h}} \]

15. Applied egg-rr 43.3%

    \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{\color{blue}{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}}}{h}} \]

16. Final simplification 43.3%

    \[\leadsto \frac{0.25}{\frac{\frac{{d}^{2}}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{h}} \]

Alternative 9: 34.1% 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 22.7%

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

3. Simplified 23.1%

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

4. Taylor expanded in c0 around -inf 2.3%

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

   1. associate-*r* 2.3%

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

   2. neg-mul-1 2.3%

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

   3. distribute-lft1-in 2.3%

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

   4. metadata-eval 2.3%

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

   5. mul0-lft 26.6%

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

   6. distribute-lft-neg-in 26.6%

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

   7. distribute-rgt-neg-in 26.6%

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

   8. metadata-eval 26.6%

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

6. Simplified 26.6%

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

7. Taylor expanded in c0 around 0 31.1%

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

8. Final simplification 31.1%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023336 
(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))))))