Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.7% → 42.8%

Time: 30.3s

Alternatives: 8

Speedup: 16.8×

• could not determine a ground truth

  1. c0 = 8.201244478539281e+307
  2. w = 2.5205075409294953e-280
  3. h = -3.442510329626012e-273
  4. D = 5.702041127470606e-233
  5. d = -1.5093900528711228e+172
  6. M = 5.737760423622554e-176

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: 24.7% 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: 42.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{d_m}{D_m} \cdot \frac{d_m}{D_m}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0}{w \cdot h}\\ t_3 := \frac{\frac{c0}{w}}{h}\\ t_4 := {\left(\frac{d_m}{D_m}\right)}^{2}\\ t_5 := \frac{t_4}{\frac{h}{\frac{c0}{w}}}\\ \mathbf{if}\;c0 \leq -6.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(t_3, t_0, \sqrt{\mathsf{fma}\left(t_3, t_0, M_m\right) \cdot \left(t_3 \cdot t_0 - M_m\right)}\right)\\ \mathbf{elif}\;c0 \leq -2.4 \cdot 10^{+135}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \mathbf{elif}\;c0 \leq -7500:\\ \;\;\;\;t_1 \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(t_2, t_4, M_m\right)}, \frac{d_m}{D_m} \cdot \sqrt{t_2}, t_2 \cdot t_4\right)\\ \mathbf{elif}\;c0 \leq 45:\\ \;\;\;\;\log \left({\left(\sqrt{e^{\frac{c0}{w}}}\right)}^{\left(2 \cdot \left({d_m}^{2} \cdot \frac{c0}{\left(w \cdot h\right) \cdot {D_m}^{2}}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(t_3, t_4, M_m\right)}, \sqrt{t_5 - M_m}, t_5\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (* (/ d_m D_m) (/ d_m D_m)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ c0 (* w h)))
        (t_3 (/ (/ c0 w) h))
        (t_4 (pow (/ d_m D_m) 2.0))
        (t_5 (/ t_4 (/ h (/ c0 w)))))
   (if (<= c0 -6.5e+153)
     (*
      (/ (/ c0 w) 2.0)
      (fma t_3 t_0 (sqrt (* (fma t_3 t_0 M_m) (- (* t_3 t_0) M_m)))))
     (if (<= c0 -2.4e+135)
       (* -0.5 (/ (pow c0 2.0) (/ w 0.0)))
       (if (<= c0 -7500.0)
         (*
          t_1
          (fma
           (sqrt (fma t_2 t_4 M_m))
           (* (/ d_m D_m) (sqrt t_2))
           (* t_2 t_4)))
         (if (<= c0 45.0)
           (log
            (pow
             (sqrt (exp (/ c0 w)))
             (* 2.0 (* (pow d_m 2.0) (/ c0 (* (* w h) (pow D_m 2.0)))))))
           (* t_1 (fma (sqrt (fma t_3 t_4 M_m)) (sqrt (- t_5 M_m)) t_5))))))))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = (d_m / D_m) * (d_m / D_m);
	double t_1 = c0 / (2.0 * w);
	double t_2 = c0 / (w * h);
	double t_3 = (c0 / w) / h;
	double t_4 = pow((d_m / D_m), 2.0);
	double t_5 = t_4 / (h / (c0 / w));
	double tmp;
	if (c0 <= -6.5e+153) {
		tmp = ((c0 / w) / 2.0) * fma(t_3, t_0, sqrt((fma(t_3, t_0, M_m) * ((t_3 * t_0) - M_m))));
	} else if (c0 <= -2.4e+135) {
		tmp = -0.5 * (pow(c0, 2.0) / (w / 0.0));
	} else if (c0 <= -7500.0) {
		tmp = t_1 * fma(sqrt(fma(t_2, t_4, M_m)), ((d_m / D_m) * sqrt(t_2)), (t_2 * t_4));
	} else if (c0 <= 45.0) {
		tmp = log(pow(sqrt(exp((c0 / w))), (2.0 * (pow(d_m, 2.0) * (c0 / ((w * h) * pow(D_m, 2.0)))))));
	} else {
		tmp = t_1 * fma(sqrt(fma(t_3, t_4, M_m)), sqrt((t_5 - M_m)), t_5);
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	t_0 = Float64(Float64(d_m / D_m) * Float64(d_m / D_m))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(c0 / Float64(w * h))
	t_3 = Float64(Float64(c0 / w) / h)
	t_4 = Float64(d_m / D_m) ^ 2.0
	t_5 = Float64(t_4 / Float64(h / Float64(c0 / w)))
	tmp = 0.0
	if (c0 <= -6.5e+153)
		tmp = Float64(Float64(Float64(c0 / w) / 2.0) * fma(t_3, t_0, sqrt(Float64(fma(t_3, t_0, M_m) * Float64(Float64(t_3 * t_0) - M_m)))));
	elseif (c0 <= -2.4e+135)
		tmp = Float64(-0.5 * Float64((c0 ^ 2.0) / Float64(w / 0.0)));
	elseif (c0 <= -7500.0)
		tmp = Float64(t_1 * fma(sqrt(fma(t_2, t_4, M_m)), Float64(Float64(d_m / D_m) * sqrt(t_2)), Float64(t_2 * t_4)));
	elseif (c0 <= 45.0)
		tmp = log((sqrt(exp(Float64(c0 / w))) ^ Float64(2.0 * Float64((d_m ^ 2.0) * Float64(c0 / Float64(Float64(w * h) * (D_m ^ 2.0)))))));
	else
		tmp = Float64(t_1 * fma(sqrt(fma(t_3, t_4, M_m)), sqrt(Float64(t_5 - M_m)), t_5));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(d$95$m / D$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[(h / N[(c0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -6.5e+153], N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(t$95$3 * t$95$0 + N[Sqrt[N[(N[(t$95$3 * t$95$0 + M$95$m), $MachinePrecision] * N[(N[(t$95$3 * t$95$0), $MachinePrecision] - M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, -2.4e+135], N[(-0.5 * N[(N[Power[c0, 2.0], $MachinePrecision] / N[(w / 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, -7500.0], N[(t$95$1 * N[(N[Sqrt[N[(t$95$2 * t$95$4 + M$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 45.0], N[Log[N[Power[N[Sqrt[N[Exp[N[(c0 / w), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(2.0 * N[(N[Power[d$95$m, 2.0], $MachinePrecision] * N[(c0 / N[(N[(w * h), $MachinePrecision] * N[Power[D$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[(N[Sqrt[N[(t$95$3 * t$95$4 + M$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$5 - M$95$m), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c0 < -6.49999999999999972e153

   1. Initial program 33.4%

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

   3. Simplified 51.6%

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

   if -6.49999999999999972e153 < c0 < -2.39999999999999997e135

   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 0.0%

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

   5. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-/l* 0.0%

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

      2. distribute-lft1-in 0.0%

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

      3. metadata-eval 0.0%

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

      4. mul0-lft 83.8%

         \[\leadsto -0.5 \cdot \frac{{c0}^{2}}{\frac{w}{\color{blue}{0}}} \]

   7. Simplified 83.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}} \]

   if -2.39999999999999997e135 < c0 < -7500

   1. Initial program 25.2%

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

   3. Simplified 25.3%

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

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in d around inf 24.8%

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

   if -7500 < c0 < 45

   1. Initial program 18.9%

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

   3. Simplified 28.5%

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

   6. Applied egg-rr 52.4%

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

   7. Taylor expanded in c0 around inf 52.7%

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

      1. associate-/l* 52.8%

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

      2. *-commutative 52.8%

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

      3. *-commutative 52.8%

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

      4. associate-*r* 52.7%

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

      5. associate-/r/ 52.7%

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

      6. associate-*r* 52.8%

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

      7. *-commutative 52.8%

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

      8. *-commutative 52.8%

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

      9. *-commutative 52.8%

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

   9. Simplified 52.8%

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

   if 45 < c0

   1. Initial program 24.1%

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

   3. Simplified 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. Add Preprocessing

   6. Applied egg-rr 45.3%

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

      1. associate-/r* 45.3%

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

      2. associate-*l/ 45.3%

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

      3. associate-/r* 45.3%

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

      4. *-commutative 45.3%

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

      5. associate-*r/ 45.3%

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

      6. associate-/l* 45.3%

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

      7. associate-*l/ 45.3%

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

      8. associate-/r* 47.7%

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

      9. *-commutative 47.7%

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

      10. associate-*r/ 47.7%

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

      11. associate-/l* 47.7%

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

   8. Simplified 47.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, M\right)}, \sqrt{\frac{{\left(\frac{d}{D}\right)}^{2}}{\frac{h}{\frac{c0}{w}}} - M}, \frac{{\left(\frac{d}{D}\right)}^{2}}{\frac{h}{\frac{c0}{w}}}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -6.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, M\right) \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)}\right)\\ \mathbf{elif}\;c0 \leq -2.4 \cdot 10^{+135}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \mathbf{elif}\;c0 \leq -7500:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, M\right)}, \frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}, \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\\ \mathbf{elif}\;c0 \leq 45:\\ \;\;\;\;\log \left({\left(\sqrt{e^{\frac{c0}{w}}}\right)}^{\left(2 \cdot \left({d}^{2} \cdot \frac{c0}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, {\left(\frac{d}{D}\right)}^{2}, M\right)}, \sqrt{\frac{{\left(\frac{d}{D}\right)}^{2}}{\frac{h}{\frac{c0}{w}}} - M}, \frac{{\left(\frac{d}{D}\right)}^{2}}{\frac{h}{\frac{c0}{w}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 44.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0}{w \cdot h}\\ t_2 := {\left(\frac{d_m}{D_m}\right)}^{2}\\ t_3 := \frac{c0 \cdot \left(d_m \cdot d_m\right)}{\left(w \cdot h\right) \cdot \left(D_m \cdot D_m\right)}\\ \mathbf{if}\;t_0 \cdot \left(t_3 + \sqrt{t_3 \cdot t_3 - M_m \cdot M_m}\right) \leq \infty:\\ \;\;\;\;t_0 \cdot \frac{2 \cdot \left(c0 \cdot {d_m}^{2}\right)}{\left(w \cdot h\right) \cdot {D_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\sqrt{e^{\frac{c0}{w}}}\right)}^{\left(\mathsf{fma}\left(t_1, t_2, \sqrt{\mathsf{fma}\left(t_1, t_2, M_m\right) \cdot \mathsf{fma}\left(\frac{\frac{{d_m}^{2}}{w}}{h \cdot D_m}, \frac{c0}{D_m}, -M_m\right)}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ c0 (* w h)))
        (t_2 (pow (/ d_m D_m) 2.0))
        (t_3 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m)))))
   (if (<= (* t_0 (+ t_3 (sqrt (- (* t_3 t_3) (* M_m M_m))))) INFINITY)
     (* t_0 (/ (* 2.0 (* c0 (pow d_m 2.0))) (* (* w h) (pow D_m 2.0))))
     (log
      (pow
       (sqrt (exp (/ c0 w)))
       (fma
        t_1
        t_2
        (sqrt
         (*
          (fma t_1 t_2 M_m)
          (fma (/ (/ (pow d_m 2.0) w) (* h D_m)) (/ c0 D_m) (- M_m))))))))))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = c0 / (w * h);
	double t_2 = pow((d_m / D_m), 2.0);
	double t_3 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double tmp;
	if ((t_0 * (t_3 + sqrt(((t_3 * t_3) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_0 * ((2.0 * (c0 * pow(d_m, 2.0))) / ((w * h) * pow(D_m, 2.0)));
	} else {
		tmp = log(pow(sqrt(exp((c0 / w))), fma(t_1, t_2, sqrt((fma(t_1, t_2, M_m) * fma(((pow(d_m, 2.0) / w) / (h * D_m)), (c0 / D_m), -M_m))))));
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(c0 / Float64(w * h))
	t_2 = Float64(d_m / D_m) ^ 2.0
	t_3 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(t_0 * Float64(Float64(2.0 * Float64(c0 * (d_m ^ 2.0))) / Float64(Float64(w * h) * (D_m ^ 2.0))));
	else
		tmp = log((sqrt(exp(Float64(c0 / w))) ^ fma(t_1, t_2, sqrt(Float64(fma(t_1, t_2, M_m) * fma(Float64(Float64((d_m ^ 2.0) / w) / Float64(h * D_m)), Float64(c0 / D_m), Float64(-M_m)))))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(d$95$m / D$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(2.0 * N[(c0 * N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Power[N[Sqrt[N[Exp[N[(c0 / w), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(t$95$1 * t$95$2 + N[Sqrt[N[(N[(t$95$1 * t$95$2 + M$95$m), $MachinePrecision] * N[(N[(N[(N[Power[d$95$m, 2.0], $MachinePrecision] / w), $MachinePrecision] / N[(h * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(c0 / D$95$m), $MachinePrecision] + (-M$95$m)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 73.9%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in c0 around inf 74.9%

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

      1. associate-*r/ 74.9%

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

   7. Simplified 74.9%

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

   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 20.0%

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

   6. Applied egg-rr 35.1%

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

4. Final simplification 47.5%

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

Alternative 3: 42.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{d_m}{D_m} \cdot \frac{d_m}{D_m}\\ t_1 := \frac{c0}{w \cdot h}\\ t_2 := \frac{\frac{c0}{w}}{h}\\ t_3 := \frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(t_2, t_0, \sqrt{\mathsf{fma}\left(t_2, t_0, M_m\right) \cdot \left(t_2 \cdot t_0 - M_m\right)}\right)\\ t_4 := {\left(\frac{d_m}{D_m}\right)}^{2}\\ \mathbf{if}\;c0 \leq -2.3 \cdot 10^{+151}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;c0 \leq -1.2 \cdot 10^{+135}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \mathbf{elif}\;c0 \leq -120000:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(t_1, t_4, M_m\right)}, \frac{d_m}{D_m} \cdot \sqrt{t_1}, t_1 \cdot t_4\right)\\ \mathbf{elif}\;c0 \leq 8.2 \cdot 10^{-50}:\\ \;\;\;\;\log \left({\left(\sqrt{e^{\frac{c0}{w}}}\right)}^{\left(2 \cdot \left({d_m}^{2} \cdot \frac{c0}{\left(w \cdot h\right) \cdot {D_m}^{2}}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (* (/ d_m D_m) (/ d_m D_m)))
        (t_1 (/ c0 (* w h)))
        (t_2 (/ (/ c0 w) h))
        (t_3
         (*
          (/ (/ c0 w) 2.0)
          (fma t_2 t_0 (sqrt (* (fma t_2 t_0 M_m) (- (* t_2 t_0) M_m))))))
        (t_4 (pow (/ d_m D_m) 2.0)))
   (if (<= c0 -2.3e+151)
     t_3
     (if (<= c0 -1.2e+135)
       (* -0.5 (/ (pow c0 2.0) (/ w 0.0)))
       (if (<= c0 -120000.0)
         (*
          (/ c0 (* 2.0 w))
          (fma
           (sqrt (fma t_1 t_4 M_m))
           (* (/ d_m D_m) (sqrt t_1))
           (* t_1 t_4)))
         (if (<= c0 8.2e-50)
           (log
            (pow
             (sqrt (exp (/ c0 w)))
             (* 2.0 (* (pow d_m 2.0) (/ c0 (* (* w h) (pow D_m 2.0)))))))
           t_3))))))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = (d_m / D_m) * (d_m / D_m);
	double t_1 = c0 / (w * h);
	double t_2 = (c0 / w) / h;
	double t_3 = ((c0 / w) / 2.0) * fma(t_2, t_0, sqrt((fma(t_2, t_0, M_m) * ((t_2 * t_0) - M_m))));
	double t_4 = pow((d_m / D_m), 2.0);
	double tmp;
	if (c0 <= -2.3e+151) {
		tmp = t_3;
	} else if (c0 <= -1.2e+135) {
		tmp = -0.5 * (pow(c0, 2.0) / (w / 0.0));
	} else if (c0 <= -120000.0) {
		tmp = (c0 / (2.0 * w)) * fma(sqrt(fma(t_1, t_4, M_m)), ((d_m / D_m) * sqrt(t_1)), (t_1 * t_4));
	} else if (c0 <= 8.2e-50) {
		tmp = log(pow(sqrt(exp((c0 / w))), (2.0 * (pow(d_m, 2.0) * (c0 / ((w * h) * pow(D_m, 2.0)))))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	t_0 = Float64(Float64(d_m / D_m) * Float64(d_m / D_m))
	t_1 = Float64(c0 / Float64(w * h))
	t_2 = Float64(Float64(c0 / w) / h)
	t_3 = Float64(Float64(Float64(c0 / w) / 2.0) * fma(t_2, t_0, sqrt(Float64(fma(t_2, t_0, M_m) * Float64(Float64(t_2 * t_0) - M_m)))))
	t_4 = Float64(d_m / D_m) ^ 2.0
	tmp = 0.0
	if (c0 <= -2.3e+151)
		tmp = t_3;
	elseif (c0 <= -1.2e+135)
		tmp = Float64(-0.5 * Float64((c0 ^ 2.0) / Float64(w / 0.0)));
	elseif (c0 <= -120000.0)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * fma(sqrt(fma(t_1, t_4, M_m)), Float64(Float64(d_m / D_m) * sqrt(t_1)), Float64(t_1 * t_4)));
	elseif (c0 <= 8.2e-50)
		tmp = log((sqrt(exp(Float64(c0 / w))) ^ Float64(2.0 * Float64((d_m ^ 2.0) * Float64(c0 / Float64(Float64(w * h) * (D_m ^ 2.0)))))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(c0 / w), $MachinePrecision] / 2.0), $MachinePrecision] * N[(t$95$2 * t$95$0 + N[Sqrt[N[(N[(t$95$2 * t$95$0 + M$95$m), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] - M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(d$95$m / D$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[c0, -2.3e+151], t$95$3, If[LessEqual[c0, -1.2e+135], N[(-0.5 * N[(N[Power[c0, 2.0], $MachinePrecision] / N[(w / 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, -120000.0], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(t$95$1 * t$95$4 + M$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 8.2e-50], N[Log[N[Power[N[Sqrt[N[Exp[N[(c0 / w), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(2.0 * N[(N[Power[d$95$m, 2.0], $MachinePrecision] * N[(c0 / N[(N[(w * h), $MachinePrecision] * N[Power[D$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], t$95$3]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c0 < -2.3000000000000001e151 or 8.19999999999999971e-50 < c0

   1. Initial program 27.5%

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

   3. Simplified 48.4%

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

   if -2.3000000000000001e151 < c0 < -1.19999999999999999e135

   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 0.0%

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

   5. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-/l* 0.0%

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

      2. distribute-lft1-in 0.0%

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

      3. metadata-eval 0.0%

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

      4. mul0-lft 83.8%

         \[\leadsto -0.5 \cdot \frac{{c0}^{2}}{\frac{w}{\color{blue}{0}}} \]

   7. Simplified 83.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}} \]

   if -1.19999999999999999e135 < c0 < -1.2e5

   1. Initial program 25.2%

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

   3. Simplified 25.3%

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

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in d around inf 24.8%

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

   if -1.2e5 < c0 < 8.19999999999999971e-50

   1. Initial program 17.4%

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

   3. Simplified 26.7%

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

   6. Applied egg-rr 53.2%

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

   7. Taylor expanded in c0 around inf 53.5%

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

      1. associate-/l* 53.5%

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

      2. *-commutative 53.5%

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

      3. *-commutative 53.5%

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

      4. associate-*r* 53.4%

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

      5. associate-/r/ 53.4%

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

      6. associate-*r* 53.5%

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

      7. *-commutative 53.5%

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

      8. *-commutative 53.5%

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

      9. *-commutative 53.5%

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

   9. Simplified 53.5%

      \[\leadsto \log \left({\left(\sqrt{e^{\frac{c0}{w}}}\right)}^{\color{blue}{\left(2 \cdot \left(\frac{c0}{{D}^{2} \cdot \left(w \cdot h\right)} \cdot {d}^{2}\right)\right)}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -2.3 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, M\right) \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)}\right)\\ \mathbf{elif}\;c0 \leq -1.2 \cdot 10^{+135}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \mathbf{elif}\;c0 \leq -120000:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, M\right)}, \frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}, \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\\ \mathbf{elif}\;c0 \leq 8.2 \cdot 10^{-50}:\\ \;\;\;\;\log \left({\left(\sqrt{e^{\frac{c0}{w}}}\right)}^{\left(2 \cdot \left({d}^{2} \cdot \frac{c0}{\left(w \cdot h\right) \cdot {D}^{2}}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0}{w}}{2} \cdot \mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(\frac{\frac{c0}{w}}{h}, \frac{d}{D} \cdot \frac{d}{D}, M\right) \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.2% accurate, 0.4× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m))))) INFINITY)
     (* t_0 (/ (* 2.0 (* c0 (pow d_m 2.0))) (* (* w h) (pow D_m 2.0))))
     (* t_0 (* c0 0.0)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $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$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(2.0 * N[(c0 * N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(c0 * 0.0), $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 73.9%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in c0 around inf 74.9%

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

      1. associate-*r/ 74.9%

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

   7. Simplified 74.9%

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

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

   1. Initial program 0.0%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in c0 around -inf 1.8%

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

      1. associate-*r* 1.8%

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

      2. neg-mul-1 1.8%

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

      3. distribute-lft1-in 1.8%

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

      4. metadata-eval 1.8%

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

      5. mul0-lft 32.9%

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

      6. distribute-lft-neg-in 32.9%

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

      7. distribute-rgt-neg-in 32.9%

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

      8. metadata-eval 32.9%

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

   7. Simplified 32.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.0%

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

Alternative 5: 50.4% accurate, 0.5× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m)))))))
   (if (<= t_2 INFINITY) t_2 (* t_0 (* c0 0.0)))))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))));
	double tmp;
	if (t_2 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = t_0 * (c0 * 0.0);
	}
	return tmp;
}

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M_m * M_m))));
	double tmp;
	if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = t_0 * (c0 * 0.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $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$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, Infinity], t$95$2, N[(t$95$0 * N[(c0 * 0.0), $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 73.9%

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

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

   1. Initial program 0.0%

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

   3. Simplified 2.3%

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

   5. Taylor expanded in c0 around -inf 1.8%

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

      1. associate-*r* 1.8%

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

      2. neg-mul-1 1.8%

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

      3. distribute-lft1-in 1.8%

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

      4. metadata-eval 1.8%

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

      5. mul0-lft 32.9%

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

      6. distribute-lft-neg-in 32.9%

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

      7. distribute-rgt-neg-in 32.9%

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

      8. metadata-eval 32.9%

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

   7. Simplified 32.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.7%

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

Alternative 6: 32.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := t_0 \cdot \frac{d_m \cdot d_m}{D_m \cdot D_m}\\ \mathbf{if}\;c0 \leq -5 \cdot 10^{+154} \lor \neg \left(c0 \leq 4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t_1 + \sqrt{t_1 \cdot \left(t_0 \cdot \left(\frac{d_m}{D_m} \cdot \frac{d_m}{D_m}\right)\right) - M_m \cdot M_m}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h))) (t_1 (* t_0 (/ (* d_m d_m) (* D_m D_m)))))
   (if (or (<= c0 -5e+154) (not (<= c0 4e-17)))
     (*
      (/ c0 (* 2.0 w))
      (+
       t_1
       (sqrt (- (* t_1 (* t_0 (* (/ d_m D_m) (/ d_m D_m)))) (* M_m M_m)))))
     (* -0.5 (/ (pow c0 2.0) (/ w 0.0))))))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d_m * d_m) / (D_m * D_m));
	double tmp;
	if ((c0 <= -5e+154) || !(c0 <= 4e-17)) {
		tmp = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * (t_0 * ((d_m / D_m) * (d_m / D_m)))) - (M_m * M_m))));
	} else {
		tmp = -0.5 * (pow(c0, 2.0) / (w / 0.0));
	}
	return tmp;
}

Fortran

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
real(8) function code(c0, w, h, d_m, d_m_1, m_m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d_m
    real(8), intent (in) :: d_m_1
    real(8), intent (in) :: m_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = t_0 * ((d_m_1 * d_m_1) / (d_m * d_m))
    if ((c0 <= (-5d+154)) .or. (.not. (c0 <= 4d-17))) then
        tmp = (c0 / (2.0d0 * w)) * (t_1 + sqrt(((t_1 * (t_0 * ((d_m_1 / d_m) * (d_m_1 / d_m)))) - (m_m * m_m))))
    else
        tmp = (-0.5d0) * ((c0 ** 2.0d0) / (w / 0.0d0))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d_m * d_m) / (D_m * D_m));
	double tmp;
	if ((c0 <= -5e+154) || !(c0 <= 4e-17)) {
		tmp = (c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * (t_0 * ((d_m / D_m) * (d_m / D_m)))) - (M_m * M_m))));
	} else {
		tmp = -0.5 * (Math.pow(c0, 2.0) / (w / 0.0));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
M_m = math.fabs(M)
def code(c0, w, h, D_m, d_m, M_m):
	t_0 = c0 / (w * h)
	t_1 = t_0 * ((d_m * d_m) / (D_m * D_m))
	tmp = 0
	if (c0 <= -5e+154) or not (c0 <= 4e-17):
		tmp = (c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * (t_0 * ((d_m / D_m) * (d_m / D_m)))) - (M_m * M_m))))
	else:
		tmp = -0.5 * (math.pow(c0, 2.0) / (w / 0.0))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(t_0 * Float64(Float64(d_m * d_m) / Float64(D_m * D_m)))
	tmp = 0.0
	if ((c0 <= -5e+154) || !(c0 <= 4e-17))
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * Float64(t_0 * Float64(Float64(d_m / D_m) * Float64(d_m / D_m)))) - Float64(M_m * M_m)))));
	else
		tmp = Float64(-0.5 * Float64((c0 ^ 2.0) / Float64(w / 0.0)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
M_m = abs(M);
function tmp_2 = code(c0, w, h, D_m, d_m, M_m)
	t_0 = c0 / (w * h);
	t_1 = t_0 * ((d_m * d_m) / (D_m * D_m));
	tmp = 0.0;
	if ((c0 <= -5e+154) || ~((c0 <= 4e-17)))
		tmp = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * (t_0 * ((d_m / D_m) * (d_m / D_m)))) - (M_m * M_m))));
	else
		tmp = -0.5 * ((c0 ^ 2.0) / (w / 0.0));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(d$95$m * d$95$m), $MachinePrecision] / N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[c0, -5e+154], N[Not[LessEqual[c0, 4e-17]], $MachinePrecision]], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * N[(t$95$0 * N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[Power[c0, 2.0], $MachinePrecision] / N[(w / 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c0 < -5.00000000000000004e154 or 4.00000000000000029e-17 < c0

   1. Initial program 28.1%

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

   3. Simplified 30.4%

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

      1. frac-times 30.4%

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

   6. Applied egg-rr 30.4%

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

   if -5.00000000000000004e154 < c0 < 4.00000000000000029e-17

   1. Initial program 18.0%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in c0 around -inf 5.0%

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

      1. associate-/l* 5.0%

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

      2. distribute-lft1-in 5.0%

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

      3. metadata-eval 5.0%

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

      4. mul0-lft 39.1%

         \[\leadsto -0.5 \cdot \frac{{c0}^{2}}{\frac{w}{\color{blue}{0}}} \]

   7. Simplified 39.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -5 \cdot 10^{+154} \lor \neg \left(c0 \leq 4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := t_0 \cdot \frac{d_m \cdot d_m}{D_m \cdot D_m}\\ t_2 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;c0 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;t_2 \cdot \left(t_1 + \sqrt{t_1 \cdot \left(t_0 \cdot \left(\frac{d_m}{D_m} \cdot \frac{d_m}{D_m}\right)\right) - M_m \cdot M_m}\right)\\ \mathbf{elif}\;c0 \leq 1.02 \cdot 10^{-16}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_1 + \sqrt{t_1 \cdot \frac{d_m \cdot \left(t_0 \cdot \frac{d_m}{D_m}\right)}{D_m} - M_m \cdot M_m}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h)))
        (t_1 (* t_0 (/ (* d_m d_m) (* D_m D_m))))
        (t_2 (/ c0 (* 2.0 w))))
   (if (<= c0 -5e+154)
     (*
      t_2
      (+
       t_1
       (sqrt (- (* t_1 (* t_0 (* (/ d_m D_m) (/ d_m D_m)))) (* M_m M_m)))))
     (if (<= c0 1.02e-16)
       (* -0.5 (/ (pow c0 2.0) (/ w 0.0)))
       (*
        t_2
        (+
         t_1
         (sqrt
          (- (* t_1 (/ (* d_m (* t_0 (/ d_m D_m))) D_m)) (* M_m M_m)))))))))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d_m * d_m) / (D_m * D_m));
	double t_2 = c0 / (2.0 * w);
	double tmp;
	if (c0 <= -5e+154) {
		tmp = t_2 * (t_1 + sqrt(((t_1 * (t_0 * ((d_m / D_m) * (d_m / D_m)))) - (M_m * M_m))));
	} else if (c0 <= 1.02e-16) {
		tmp = -0.5 * (pow(c0, 2.0) / (w / 0.0));
	} else {
		tmp = t_2 * (t_1 + sqrt(((t_1 * ((d_m * (t_0 * (d_m / D_m))) / D_m)) - (M_m * M_m))));
	}
	return tmp;
}

Fortran

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
real(8) function code(c0, w, h, d_m, d_m_1, m_m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d_m
    real(8), intent (in) :: d_m_1
    real(8), intent (in) :: m_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = t_0 * ((d_m_1 * d_m_1) / (d_m * d_m))
    t_2 = c0 / (2.0d0 * w)
    if (c0 <= (-5d+154)) then
        tmp = t_2 * (t_1 + sqrt(((t_1 * (t_0 * ((d_m_1 / d_m) * (d_m_1 / d_m)))) - (m_m * m_m))))
    else if (c0 <= 1.02d-16) then
        tmp = (-0.5d0) * ((c0 ** 2.0d0) / (w / 0.0d0))
    else
        tmp = t_2 * (t_1 + sqrt(((t_1 * ((d_m_1 * (t_0 * (d_m_1 / d_m))) / d_m)) - (m_m * m_m))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d_m * d_m) / (D_m * D_m));
	double t_2 = c0 / (2.0 * w);
	double tmp;
	if (c0 <= -5e+154) {
		tmp = t_2 * (t_1 + Math.sqrt(((t_1 * (t_0 * ((d_m / D_m) * (d_m / D_m)))) - (M_m * M_m))));
	} else if (c0 <= 1.02e-16) {
		tmp = -0.5 * (Math.pow(c0, 2.0) / (w / 0.0));
	} else {
		tmp = t_2 * (t_1 + Math.sqrt(((t_1 * ((d_m * (t_0 * (d_m / D_m))) / D_m)) - (M_m * M_m))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
M_m = math.fabs(M)
def code(c0, w, h, D_m, d_m, M_m):
	t_0 = c0 / (w * h)
	t_1 = t_0 * ((d_m * d_m) / (D_m * D_m))
	t_2 = c0 / (2.0 * w)
	tmp = 0
	if c0 <= -5e+154:
		tmp = t_2 * (t_1 + math.sqrt(((t_1 * (t_0 * ((d_m / D_m) * (d_m / D_m)))) - (M_m * M_m))))
	elif c0 <= 1.02e-16:
		tmp = -0.5 * (math.pow(c0, 2.0) / (w / 0.0))
	else:
		tmp = t_2 * (t_1 + math.sqrt(((t_1 * ((d_m * (t_0 * (d_m / D_m))) / D_m)) - (M_m * M_m))))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(t_0 * Float64(Float64(d_m * d_m) / Float64(D_m * D_m)))
	t_2 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (c0 <= -5e+154)
		tmp = Float64(t_2 * Float64(t_1 + sqrt(Float64(Float64(t_1 * Float64(t_0 * Float64(Float64(d_m / D_m) * Float64(d_m / D_m)))) - Float64(M_m * M_m)))));
	elseif (c0 <= 1.02e-16)
		tmp = Float64(-0.5 * Float64((c0 ^ 2.0) / Float64(w / 0.0)));
	else
		tmp = Float64(t_2 * Float64(t_1 + sqrt(Float64(Float64(t_1 * Float64(Float64(d_m * Float64(t_0 * Float64(d_m / D_m))) / D_m)) - Float64(M_m * M_m)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
M_m = abs(M);
function tmp_2 = code(c0, w, h, D_m, d_m, M_m)
	t_0 = c0 / (w * h);
	t_1 = t_0 * ((d_m * d_m) / (D_m * D_m));
	t_2 = c0 / (2.0 * w);
	tmp = 0.0;
	if (c0 <= -5e+154)
		tmp = t_2 * (t_1 + sqrt(((t_1 * (t_0 * ((d_m / D_m) * (d_m / D_m)))) - (M_m * M_m))));
	elseif (c0 <= 1.02e-16)
		tmp = -0.5 * ((c0 ^ 2.0) / (w / 0.0));
	else
		tmp = t_2 * (t_1 + sqrt(((t_1 * ((d_m * (t_0 * (d_m / D_m))) / D_m)) - (M_m * M_m))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(d$95$m * d$95$m), $MachinePrecision] / N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -5e+154], N[(t$95$2 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * N[(t$95$0 * N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 1.02e-16], N[(-0.5 * N[(N[Power[c0, 2.0], $MachinePrecision] / N[(w / 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * N[(N[(d$95$m * N[(t$95$0 * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D$95$m), $MachinePrecision]), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -5.00000000000000004e154

   1. Initial program 34.3%

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

   3. Simplified 39.5%

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

      1. frac-times 39.5%

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

   6. Applied egg-rr 39.5%

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

   if -5.00000000000000004e154 < c0 < 1.0200000000000001e-16

   1. Initial program 18.0%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in c0 around -inf 5.0%

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

      1. associate-/l* 5.0%

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

      2. distribute-lft1-in 5.0%

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

      3. metadata-eval 5.0%

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

      4. mul0-lft 39.1%

         \[\leadsto -0.5 \cdot \frac{{c0}^{2}}{\frac{w}{\color{blue}{0}}} \]

   7. Simplified 39.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}} \]

   if 1.0200000000000001e-16 < c0

   1. Initial program 25.5%

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

   3. Simplified 26.6%

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

      1. div-inv 26.6%

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

      2. pow2 26.6%

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

      3. pow2 26.6%

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

      4. pow-flip 26.6%

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

      5. metadata-eval 26.6%

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

   6. Applied egg-rr 26.6%

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

      1. *-commutative 26.6%

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

      2. pow2 26.6%

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

      3. metadata-eval 26.6%

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

      4. pow-flip 26.6%

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

      5. pow2 26.6%

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

      6. div-inv 26.6%

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

      7. times-frac 26.6%

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

      8. associate-*r* 26.6%

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

      9. associate-*l/ 26.6%

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

      10. *-commutative 26.6%

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

   8. Applied egg-rr 26.6%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) - M \cdot M}\right)\\ \mathbf{elif}\;c0 \leq 1.02 \cdot 10^{-16}:\\ \;\;\;\;-0.5 \cdot \frac{{c0}^{2}}{\frac{w}{0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \frac{d \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d}{D}\right)}{D} - M \cdot M}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.5% accurate, 16.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right) \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (* (/ c0 (* 2.0 w)) (* c0 0.0)))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	return (c0 / (2.0 * w)) * (c0 * 0.0);
}

Fortran

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
real(8) function code(c0, w, h, d_m, d_m_1, m_m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d_m
    real(8), intent (in) :: d_m_1
    real(8), intent (in) :: m_m
    code = (c0 / (2.0d0 * w)) * (c0 * 0.0d0)
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	return (c0 / (2.0 * w)) * (c0 * 0.0);
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
M_m = math.fabs(M)
def code(c0, w, h, D_m, d_m, M_m):
	return (c0 / (2.0 * w)) * (c0 * 0.0)

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(c0 * 0.0))
end

MATLAB

D_m = abs(D);
d_m = abs(d);
M_m = abs(M);
function tmp = code(c0, w, h, D_m, d_m, M_m)
	tmp = (c0 / (2.0 * w)) * (c0 * 0.0);
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(c0 * 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.1%

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

3. Simplified 23.8%

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

5. Taylor expanded in c0 around -inf 4.6%

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

   1. associate-*r* 4.6%

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

   2. neg-mul-1 4.6%

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

   3. distribute-lft1-in 4.6%

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

   4. metadata-eval 4.6%

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

   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%

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

7. Simplified 26.6%

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

8. Final simplification 26.6%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))))))