Henrywood and Agarwal, Equation (13)

Percentage Accurate: 23.8% → 62.8%

Time: 26.8s

Alternatives: 6

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = 9.755812787869991e+229
  2. w = 4.313441016808423e-281
  3. h = -2.7109880571819004e-99
  4. D = 3.0219812894258817e-290
  5. d = 2.6903260272662014e+305
  6. M = -7.232445161097534e-153

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: 23.8% 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: 62.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{w}{\frac{d}{D}}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := t_1 \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right)\\ \mathbf{if}\;t_3 \leq -4 \cdot 10^{-42}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{c0}{\frac{h}{\frac{d}{D}} \cdot t_0}\right)\\ \mathbf{elif}\;t_3 \leq 0 \lor \neg \left(t_3 \leq \infty\right):\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{{d}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{c0}{t_0 \cdot \frac{h \cdot D}{d}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ w (/ d D)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_3 (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))))))
   (if (<= t_3 -4e-42)
     (* t_1 (* 2.0 (/ c0 (* (/ h (/ d D)) t_0))))
     (if (or (<= t_3 0.0) (not (<= t_3 INFINITY)))
       (* 0.25 (/ (* h (* (* D M) (* D M))) (pow d 2.0)))
       (* t_1 (* 2.0 (/ c0 (* t_0 (/ (* h D) d)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = w / (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -4e-42) {
		tmp = t_1 * (2.0 * (c0 / ((h / (d / D)) * t_0)));
	} else if ((t_3 <= 0.0) || !(t_3 <= ((double) INFINITY))) {
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / pow(d, 2.0));
	} else {
		tmp = t_1 * (2.0 * (c0 / (t_0 * ((h * D) / d))));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = w / (d / D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -4e-42) {
		tmp = t_1 * (2.0 * (c0 / ((h / (d / D)) * t_0)));
	} else if ((t_3 <= 0.0) || !(t_3 <= Double.POSITIVE_INFINITY)) {
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / Math.pow(d, 2.0));
	} else {
		tmp = t_1 * (2.0 * (c0 / (t_0 * ((h * D) / d))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = w / (d / D)
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_3 = t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))
	tmp = 0
	if t_3 <= -4e-42:
		tmp = t_1 * (2.0 * (c0 / ((h / (d / D)) * t_0)))
	elif (t_3 <= 0.0) or not (t_3 <= math.inf):
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / math.pow(d, 2.0))
	else:
		tmp = t_1 * (2.0 * (c0 / (t_0 * ((h * D) / d))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(w / Float64(d / D))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_3 = Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))))
	tmp = 0.0
	if (t_3 <= -4e-42)
		tmp = Float64(t_1 * Float64(2.0 * Float64(c0 / Float64(Float64(h / Float64(d / D)) * t_0))));
	elseif ((t_3 <= 0.0) || !(t_3 <= Inf))
		tmp = Float64(0.25 * Float64(Float64(h * Float64(Float64(D * M) * Float64(D * M))) / (d ^ 2.0)));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(c0 / Float64(t_0 * Float64(Float64(h * D) / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = w / (d / D);
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	tmp = 0.0;
	if (t_3 <= -4e-42)
		tmp = t_1 * (2.0 * (c0 / ((h / (d / D)) * t_0)));
	elseif ((t_3 <= 0.0) || ~((t_3 <= Inf)))
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / (d ^ 2.0));
	else
		tmp = t_1 * (2.0 * (c0 / (t_0 * ((h * D) / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(w / N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -4e-42], N[(t$95$1 * N[(2.0 * N[(c0 / N[(N[(h / N[(d / D), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$3, 0.0], N[Not[LessEqual[t$95$3, Infinity]], $MachinePrecision]], N[(0.25 * N[(N[(h * N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(c0 / N[(t$95$0 * N[(N[(h * D), $MachinePrecision] / d), $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))))) < -4.00000000000000015e-42

   1. Initial program 82.5%

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

      1. times-frac 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in c0 around inf 82.5%

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

      1. *-commutative 82.5%

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

      2. *-commutative 82.5%

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

      3. associate-*r* 81.5%

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

      4. associate-/r* 81.5%

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

      5. associate-*l/ 81.5%

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

      6. times-frac 84.8%

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

      7. unpow2 84.8%

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

      8. associate-*r/ 84.8%

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

      9. unpow2 84.8%

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

      10. associate-/l/ 86.3%

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

      11. associate-*r/ 86.3%

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

      12. associate-/r* 84.0%

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

      13. associate-*l/ 84.1%

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

      14. unpow2 84.1%

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

      15. associate-*l/ 84.0%

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

      16. associate-/l* 84.1%

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

      17. *-commutative 84.1%

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

   6. Simplified 84.1%

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

      1. unpow2 84.1%

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

   8. Applied egg-rr 84.1%

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

      1. times-frac 88.6%

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

   10. Applied egg-rr 88.6%

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

   if -4.00000000000000015e-42 < (*.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%

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

      1. times-frac 0.3%

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

   3. Simplified 2.0%

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

   4. Taylor expanded in c0 around -inf 5.5%

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

   6. Simplified 26.9%

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

   7. Taylor expanded in c0 around 0 35.7%

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

      1. *-commutative 35.7%

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

      2. *-commutative 35.7%

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

      3. associate-*l* 37.9%

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

      4. unpow2 37.9%

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

      5. unpow2 37.9%

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

      6. swap-sqr 49.9%

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

      7. unpow2 49.9%

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

      8. *-commutative 49.9%

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

   9. Simplified 49.9%

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

       1. unpow2 49.9%

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

   11. Applied egg-rr 49.9%

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

   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 82.1%

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

      1. times-frac 78.4%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in c0 around inf 84.1%

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

      1. *-commutative 84.1%

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

      2. *-commutative 84.1%

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

      3. associate-*r* 84.0%

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

      4. associate-/r* 80.5%

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

      5. associate-*l/ 80.5%

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

      6. times-frac 80.3%

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

      7. unpow2 80.3%

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

      8. associate-*r/ 83.7%

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

      9. unpow2 83.7%

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

      10. associate-/l/ 83.8%

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

      11. associate-*r/ 80.4%

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

      12. associate-/r* 80.4%

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

      13. associate-*l/ 83.9%

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

      14. unpow2 83.9%

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

      15. associate-*l/ 87.6%

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

      16. associate-/l* 87.3%

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

      17. *-commutative 87.3%

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

   6. Simplified 87.3%

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

      1. unpow2 87.3%

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

   8. Applied egg-rr 87.3%

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

      1. times-frac 90.8%

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

   10. Applied egg-rr 90.8%

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

   11. Taylor expanded in h around 0 90.9%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{\frac{h}{\frac{d}{D}} \cdot \frac{w}{\frac{d}{D}}}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0 \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty\right):\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{{d}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{\frac{w}{\frac{d}{D}} \cdot \frac{h \cdot D}{d}}\right)\\ \end{array} \]

Alternative 2: 62.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t_0 \cdot \left(2 \cdot {\left(\frac{d}{D} \cdot \sqrt{\frac{1}{\frac{h}{\frac{c0}{w}}}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{{d}^{2}}\\ \end{array} \end{array} \]

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)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (* 2.0 (pow (* (/ d D) (sqrt (/ 1.0 (/ h (/ c0 w))))) 2.0)))
     (* 0.25 (/ (* h (* (* D M) (* D M))) (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 tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (2.0 * pow(((d / D) * sqrt((1.0 / (h / (c0 / w))))), 2.0));
	} else {
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / 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 tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (2.0 * Math.pow(((d / D) * Math.sqrt((1.0 / (h / (c0 / w))))), 2.0));
	} else {
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / 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))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * (2.0 * math.pow(((d / D) * math.sqrt((1.0 / (h / (c0 / w))))), 2.0))
	else:
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / 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)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(2.0 * (Float64(Float64(d / D) * sqrt(Float64(1.0 / Float64(h / Float64(c0 / w))))) ^ 2.0)));
	else
		tmp = Float64(0.25 * Float64(Float64(h * Float64(Float64(D * M) * Float64(D * M))) / (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));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * (2.0 * (((d / D) * sqrt((1.0 / (h / (c0 / w))))) ^ 2.0));
	else
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / (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]}, If[LessEqual[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], Infinity], N[(t$95$0 * N[(2.0 * N[Power[N[(N[(d / D), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h / N[(c0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(h * N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.6%

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

      1. times-frac 69.6%

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

   3. Simplified 70.8%

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

   4. Taylor expanded in c0 around inf 74.0%

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

      1. *-commutative 74.0%

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

      2. *-commutative 74.0%

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

      3. associate-*r* 74.5%

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

      4. associate-/r* 73.2%

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

      5. associate-*l/ 73.2%

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

      6. times-frac 72.5%

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

      7. unpow2 72.5%

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

      8. associate-*r/ 74.8%

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

      9. unpow2 74.8%

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

      10. associate-/l/ 75.5%

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

      11. associate-*r/ 74.4%

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

      12. associate-/r* 73.2%

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

      13. associate-*l/ 74.4%

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

      14. unpow2 74.4%

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

      15. associate-*l/ 76.9%

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

      16. associate-/l* 75.6%

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

      17. *-commutative 75.6%

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

   6. Simplified 75.6%

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

      1. associate-/r/ 74.4%

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

      2. *-commutative 74.4%

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

      3. add-sqr-sqrt 74.0%

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

      4. pow2 74.0%

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

      5. *-commutative 74.0%

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

      6. sqrt-prod 74.0%

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

      7. sqrt-pow1 80.8%

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

      8. metadata-eval 80.8%

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

      9. pow1 80.8%

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

      10. *-commutative 80.8%

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

   8. Applied egg-rr 80.8%

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

      1. clear-num 80.0%

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

      2. inv-pow 80.0%

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

   10. Applied egg-rr 80.0%

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

       1. unpow-1 80.0%

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

       2. associate-/l* 81.2%

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

   12. Simplified 81.2%

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

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

   1. Initial program 0.0%

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

      1. times-frac 0.0%

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

   3. Simplified 1.3%

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

   4. Taylor expanded in c0 around -inf 3.0%

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

   6. Simplified 25.2%

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

   7. Taylor expanded in c0 around 0 34.7%

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

      1. *-commutative 34.7%

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

      2. *-commutative 34.7%

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

      3. associate-*l* 37.0%

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

      4. unpow2 37.0%

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

      5. unpow2 37.0%

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

      6. swap-sqr 48.7%

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

      7. unpow2 48.7%

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

      8. *-commutative 48.7%

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

   9. Simplified 48.7%

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

       1. unpow2 48.7%

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

   11. Applied egg-rr 48.7%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot {\left(\frac{d}{D} \cdot \sqrt{\frac{1}{\frac{h}{\frac{c0}{w}}}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{{d}^{2}}\\ \end{array} \]

Alternative 3: 62.0% accurate, 0.4× speedup

Math

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

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)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (* 2.0 (pow (* (/ d D) (sqrt (/ c0 (* w h)))) 2.0)))
     (* 0.25 (/ (* h (* (* D M) (* D M))) (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 tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (2.0 * pow(((d / D) * sqrt((c0 / (w * h)))), 2.0));
	} else {
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / 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 tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (2.0 * Math.pow(((d / D) * Math.sqrt((c0 / (w * h)))), 2.0));
	} else {
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / 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))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * (2.0 * math.pow(((d / D) * math.sqrt((c0 / (w * h)))), 2.0))
	else:
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / 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)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(2.0 * (Float64(Float64(d / D) * sqrt(Float64(c0 / Float64(w * h)))) ^ 2.0)));
	else
		tmp = Float64(0.25 * Float64(Float64(h * Float64(Float64(D * M) * Float64(D * M))) / (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));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * (2.0 * (((d / D) * sqrt((c0 / (w * h)))) ^ 2.0));
	else
		tmp = 0.25 * ((h * ((D * M) * (D * M))) / (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]}, If[LessEqual[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], Infinity], N[(t$95$0 * N[(2.0 * N[Power[N[(N[(d / D), $MachinePrecision] * N[Sqrt[N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(h * N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.6%

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

      1. times-frac 69.6%

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

   3. Simplified 70.8%

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

   4. Taylor expanded in c0 around inf 74.0%

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

      1. *-commutative 74.0%

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

      2. *-commutative 74.0%

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

      3. associate-*r* 74.5%

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

      4. associate-/r* 73.2%

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

      5. associate-*l/ 73.2%

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

      6. times-frac 72.5%

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

      7. unpow2 72.5%

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

      8. associate-*r/ 74.8%

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

      9. unpow2 74.8%

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

      10. associate-/l/ 75.5%

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

      11. associate-*r/ 74.4%

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

      12. associate-/r* 73.2%

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

      13. associate-*l/ 74.4%

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

      14. unpow2 74.4%

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

      15. associate-*l/ 76.9%

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

      16. associate-/l* 75.6%

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

      17. *-commutative 75.6%

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

   6. Simplified 75.6%

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

      1. associate-/r/ 74.4%

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

      2. *-commutative 74.4%

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

      3. add-sqr-sqrt 74.0%

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

      4. pow2 74.0%

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

      5. *-commutative 74.0%

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

      6. sqrt-prod 74.0%

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

      7. sqrt-pow1 80.8%

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

      8. metadata-eval 80.8%

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

      9. pow1 80.8%

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

      10. *-commutative 80.8%

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

   8. Applied egg-rr 80.8%

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

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

   1. Initial program 0.0%

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

      1. times-frac 0.0%

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

   3. Simplified 1.3%

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

   4. Taylor expanded in c0 around -inf 3.0%

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

   6. Simplified 25.2%

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

   7. Taylor expanded in c0 around 0 34.7%

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

      1. *-commutative 34.7%

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

      2. *-commutative 34.7%

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

      3. associate-*l* 37.0%

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

      4. unpow2 37.0%

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

      5. unpow2 37.0%

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

      6. swap-sqr 48.7%

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

      7. unpow2 48.7%

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

      8. *-commutative 48.7%

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

   9. Simplified 48.7%

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

       1. unpow2 48.7%

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

   11. Applied egg-rr 48.7%

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

4. Final simplification 58.9%

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

Alternative 4: 39.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2.25 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{\frac{h}{d} \cdot \left(D \cdot \left(D \cdot \frac{w}{d}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 2.25e-123)
   0.0
   (* (/ c0 (* 2.0 w)) (* 2.0 (/ c0 (* (/ h d) (* D (* D (/ w d)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 2.25e-123) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / ((h / d) * (D * (D * (w / d))))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.25d-123) then
        tmp = 0.0d0
    else
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (c0 / ((h / d_1) * (d * (d * (w / d_1))))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 2.25e-123) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / ((h / d) * (D * (D * (w / d))))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 2.25e-123:
		tmp = 0.0
	else:
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / ((h / d) * (D * (D * (w / d))))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 2.25e-123)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(c0 / Float64(Float64(h / d) * Float64(D * Float64(D * Float64(w / d)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 2.25e-123)
		tmp = 0.0;
	else
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / ((h / d) * (D * (D * (w / d))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 2.25e-123], 0.0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(c0 / N[(N[(h / d), $MachinePrecision] * N[(D * N[(D * N[(w / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.24999999999999997e-123

   1. Initial program 25.6%

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

      1. times-frac 23.3%

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

   3. Simplified 25.2%

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

   4. Taylor expanded in c0 around -inf 5.1%

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

      1. associate-*r* 5.1%

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

      2. neg-mul-1 5.1%

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

      3. distribute-lft1-in 5.1%

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

      4. metadata-eval 5.1%

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

      5. mul0-lft 31.2%

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

      6. distribute-lft-neg-in 31.2%

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

      7. distribute-rgt-neg-in 31.2%

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

      8. metadata-eval 31.2%

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

      9. mul0-lft 5.1%

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

      10. metadata-eval 5.1%

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

      11. distribute-lft1-in 5.1%

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

      12. distribute-lft-in 4.0%

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

   6. Simplified 31.2%

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

   7. Taylor expanded in c0 around 0 36.3%

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

   if 2.24999999999999997e-123 < M

   1. Initial program 20.5%

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

      1. times-frac 19.4%

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

   3. Simplified 19.4%

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

   4. Taylor expanded in c0 around inf 42.1%

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

      1. *-commutative 42.1%

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

      2. *-commutative 42.1%

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

      3. associate-*r* 42.1%

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

      4. associate-/r* 42.1%

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

      5. associate-*l/ 42.1%

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

      6. times-frac 40.9%

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

      7. unpow2 40.9%

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

      8. associate-*r/ 46.1%

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

      9. unpow2 46.1%

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

      10. associate-/l/ 49.5%

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

      11. associate-*r/ 50.7%

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

      12. associate-/r* 50.7%

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

      13. associate-*l/ 51.9%

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

      14. unpow2 51.9%

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

      15. associate-*l/ 53.2%

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

      16. associate-/l* 53.2%

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

      17. *-commutative 53.2%

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

   6. Simplified 53.2%

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

      1. unpow2 53.2%

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

   8. Applied egg-rr 53.2%

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

      1. times-frac 54.6%

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

   10. Applied egg-rr 54.6%

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

       1. associate-/r/ 54.3%

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

       2. associate-*l* 50.8%

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

       3. associate-/r/ 50.8%

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

   12. Simplified 50.8%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.25 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{\frac{h}{d} \cdot \left(D \cdot \left(D \cdot \frac{w}{d}\right)\right)}\right)\\ \end{array} \]

Alternative 5: 40.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.95 \cdot 10^{-100}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{\frac{h}{\frac{d}{D}} \cdot \frac{w}{\frac{d}{D}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 1.95e-100)
   0.0
   (* (/ c0 (* 2.0 w)) (* 2.0 (/ c0 (* (/ h (/ d D)) (/ w (/ d D))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.95e-100) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / ((h / (d / D)) * (w / (d / D)))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.95d-100) then
        tmp = 0.0d0
    else
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (c0 / ((h / (d_1 / d)) * (w / (d_1 / d)))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.95e-100) {
		tmp = 0.0;
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / ((h / (d / D)) * (w / (d / D)))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 1.95e-100:
		tmp = 0.0
	else:
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / ((h / (d / D)) * (w / (d / D)))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 1.95e-100)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(c0 / Float64(Float64(h / Float64(d / D)) * Float64(w / Float64(d / D))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 1.95e-100)
		tmp = 0.0;
	else
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 / ((h / (d / D)) * (w / (d / D)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 1.95e-100], 0.0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(c0 / N[(N[(h / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(w / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.94999999999999989e-100

   1. Initial program 26.1%

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

      1. times-frac 24.0%

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

   3. Simplified 25.7%

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

   4. Taylor expanded in c0 around -inf 4.9%

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

      1. associate-*r* 4.9%

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

      2. neg-mul-1 4.9%

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

      3. distribute-lft1-in 4.9%

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

      4. metadata-eval 4.9%

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

      5. mul0-lft 31.6%

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

      6. distribute-lft-neg-in 31.6%

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

      7. distribute-rgt-neg-in 31.6%

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

      8. metadata-eval 31.6%

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

      9. mul0-lft 4.9%

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

      10. metadata-eval 4.9%

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

      11. distribute-lft1-in 4.9%

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

      12. distribute-lft-in 3.8%

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

   6. Simplified 31.6%

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

   7. Taylor expanded in c0 around 0 36.4%

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

   if 1.94999999999999989e-100 < M

   1. Initial program 18.8%

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

      1. times-frac 17.5%

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

   3. Simplified 17.5%

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

   4. Taylor expanded in c0 around inf 42.6%

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

      1. *-commutative 42.6%

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

      2. *-commutative 42.6%

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

      3. associate-*r* 42.5%

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

      4. associate-/r* 42.5%

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

      5. associate-*l/ 42.5%

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

      6. times-frac 41.3%

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

      7. unpow2 41.3%

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

      8. associate-*r/ 47.0%

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

      9. unpow2 47.0%

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

      10. associate-/l/ 50.7%

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

      11. associate-*r/ 50.8%

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

      12. associate-/r* 50.7%

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

      13. associate-*l/ 52.1%

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

      14. unpow2 52.1%

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

      15. associate-*l/ 53.4%

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

      16. associate-/l* 53.4%

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

      17. *-commutative 53.4%

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

   6. Simplified 53.4%

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

      1. unpow2 53.4%

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

   8. Applied egg-rr 53.4%

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

      1. times-frac 55.1%

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

   10. Applied egg-rr 55.1%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.95 \cdot 10^{-100}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{c0}{\frac{h}{\frac{d}{D}} \cdot \frac{w}{\frac{d}{D}}}\right)\\ \end{array} \]

Alternative 6: 34.0% 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 23.9%

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

   1. times-frac 22.0%

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

3. Simplified 23.3%

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

4. Taylor expanded in c0 around -inf 3.5%

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

   1. associate-*r* 3.5%

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

   2. neg-mul-1 3.5%

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

   3. distribute-lft1-in 3.5%

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

   4. metadata-eval 3.5%

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

   5. mul0-lft 28.4%

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

   6. distribute-lft-neg-in 28.4%

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

   7. distribute-rgt-neg-in 28.4%

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

   8. metadata-eval 28.4%

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

   9. mul0-lft 3.5%

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

   10. metadata-eval 3.5%

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

   11. distribute-lft1-in 3.5%

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

   12. distribute-lft-in 2.7%

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

6. Simplified 28.4%

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

7. Taylor expanded in c0 around 0 32.3%

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

8. Final simplification 32.3%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023310 
(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))))))