Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.5% → 54.8%

Time: 31.6s

Alternatives: 6

Speedup: 21.6×

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: 24.5% 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: 54.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := {\left(\frac{d}{D}\right)}^{2}\\ t_2 := \sqrt{\mathsf{fma}\left(t\_0, t\_1, M\right)}\\ t_3 := t\_1 \cdot t\_0\\ t_4 := \frac{c0}{2 \cdot w}\\ t_5 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_6 := t\_4 \cdot \left(t\_5 + \sqrt{t\_5 \cdot t\_5 - M \cdot M}\right)\\ \mathbf{if}\;t\_6 \leq -2 \cdot 10^{-101}:\\ \;\;\;\;t\_4 \cdot \mathsf{fma}\left(t\_2, \sqrt{t\_3 - M}, t\_3\right)\\ \mathbf{elif}\;t\_6 \leq 0:\\ \;\;\;\;c0 \cdot \frac{\frac{{M}^{2}}{t\_3 - \sqrt{{t\_3}^{2} - {M}^{2}}}}{2 \cdot w}\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;c0 \cdot \frac{t\_3 + t\_2 \cdot \sqrt{c0 \cdot \frac{t\_1}{w \cdot h} - M}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h)))
        (t_1 (pow (/ d D) 2.0))
        (t_2 (sqrt (fma t_0 t_1 M)))
        (t_3 (* t_1 t_0))
        (t_4 (/ c0 (* 2.0 w)))
        (t_5 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_6 (* t_4 (+ t_5 (sqrt (- (* t_5 t_5) (* M M)))))))
   (if (<= t_6 -2e-101)
     (* t_4 (fma t_2 (sqrt (- t_3 M)) t_3))
     (if (<= t_6 0.0)
       (*
        c0
        (/
         (/ (pow M 2.0) (- t_3 (sqrt (- (pow t_3 2.0) (pow M 2.0)))))
         (* 2.0 w)))
       (if (<= t_6 INFINITY)
         (*
          c0
          (/ (+ t_3 (* t_2 (sqrt (- (* c0 (/ t_1 (* w h))) M)))) (* 2.0 w)))
         (* c0 (/ 0.0 (* 2.0 w))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = pow((d / D), 2.0);
	double t_2 = sqrt(fma(t_0, t_1, M));
	double t_3 = t_1 * t_0;
	double t_4 = c0 / (2.0 * w);
	double t_5 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_6 = t_4 * (t_5 + sqrt(((t_5 * t_5) - (M * M))));
	double tmp;
	if (t_6 <= -2e-101) {
		tmp = t_4 * fma(t_2, sqrt((t_3 - M)), t_3);
	} else if (t_6 <= 0.0) {
		tmp = c0 * ((pow(M, 2.0) / (t_3 - sqrt((pow(t_3, 2.0) - pow(M, 2.0))))) / (2.0 * w));
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = c0 * ((t_3 + (t_2 * sqrt(((c0 * (t_1 / (w * h))) - M)))) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(d / D) ^ 2.0
	t_2 = sqrt(fma(t_0, t_1, M))
	t_3 = Float64(t_1 * t_0)
	t_4 = Float64(c0 / Float64(2.0 * w))
	t_5 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_6 = Float64(t_4 * Float64(t_5 + sqrt(Float64(Float64(t_5 * t_5) - Float64(M * M)))))
	tmp = 0.0
	if (t_6 <= -2e-101)
		tmp = Float64(t_4 * fma(t_2, sqrt(Float64(t_3 - M)), t_3));
	elseif (t_6 <= 0.0)
		tmp = Float64(c0 * Float64(Float64((M ^ 2.0) / Float64(t_3 - sqrt(Float64((t_3 ^ 2.0) - (M ^ 2.0))))) / Float64(2.0 * w)));
	elseif (t_6 <= Inf)
		tmp = Float64(c0 * Float64(Float64(t_3 + Float64(t_2 * sqrt(Float64(Float64(c0 * Float64(t_1 / Float64(w * h))) - M)))) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$0 * t$95$1 + M), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 * N[(t$95$5 + N[Sqrt[N[(N[(t$95$5 * t$95$5), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -2e-101], N[(t$95$4 * N[(t$95$2 * N[Sqrt[N[(t$95$3 - M), $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 0.0], N[(c0 * N[(N[(N[Power[M, 2.0], $MachinePrecision] / N[(t$95$3 - N[Sqrt[N[(N[Power[t$95$3, 2.0], $MachinePrecision] - N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(c0 * N[(N[(t$95$3 + N[(t$95$2 * N[Sqrt[N[(N[(c0 * N[(t$95$1 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -2.0000000000000001e-101

   1. Initial program 57.4%

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

   3. Simplified 52.8%

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

   6. Applied egg-rr 63.4%

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

      1. *-commutative 63.4%

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

      2. fma-undefine 63.4%

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

      3. unsub-neg 63.4%

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

      4. *-commutative 63.4%

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

      5. *-commutative 63.4%

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

   8. Simplified 63.4%

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

   if -2.0000000000000001e-101 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 0.0

   1. Initial program 56.0%

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

   3. Simplified 29.7%

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

   6. Applied egg-rr 46.3%

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

      1. associate--r- 72.5%

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

      2. +-inverses 81.6%

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

      3. *-commutative 81.6%

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

      4. *-commutative 81.6%

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

   8. Simplified 81.6%

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

   if 0.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 88.6%

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

   3. Simplified 90.7%

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

      1. fma-undefine 90.7%

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

      2. associate-*r/ 90.7%

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

      3. *-commutative 90.7%

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

      4. associate-*r* 90.7%

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

      5. associate-*r* 86.3%

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

      6. associate-/l* 86.3%

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

      7. frac-times 86.3%

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

      8. times-frac 90.7%

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

      9. pow2 90.7%

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

   6. Applied egg-rr 93.1%

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

      1. pow1/2 93.1%

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

      2. associate-*l/ 93.1%

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

      3. *-commutative 93.1%

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

      4. associate-*l/ 93.1%

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

      5. *-commutative 93.1%

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

      6. pow2 93.1%

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

      7. frac-times 88.6%

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

      8. pow2 88.6%

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

      9. pow2 88.6%

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

      10. difference-of-squares 88.6%

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

      11. unpow-prod-down 90.8%

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

   8. Applied egg-rr 95.3%

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

      1. unpow1/2 95.3%

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

      2. *-commutative 95.3%

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

      3. unpow1/2 95.3%

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

      4. associate-*l/ 95.3%

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

      5. associate-/l* 95.3%

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

      6. *-commutative 95.3%

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

   10. Simplified 95.3%

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

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

   1. Initial program 0.0%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in c0 around -inf 0.9%

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

      1. mul-1-neg 0.9%

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

      2. distribute-lft-in 0.9%

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

      3. mul-1-neg 0.9%

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

      4. distribute-rgt-neg-in 0.9%

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

      5. associate-/l* 0.2%

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

      6. mul-1-neg 0.2%

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

      7. associate-/l* 0.2%

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

      8. distribute-lft1-in 0.2%

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

      9. metadata-eval 0.2%

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

      10. mul0-lft 38.4%

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

      11. metadata-eval 38.4%

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

   7. Simplified 38.4%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-101}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, M\right)}, \sqrt{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h} - M}, {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;c0 \cdot \frac{\frac{{M}^{2}}{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h} - \sqrt{{\left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, M\right)} \cdot \sqrt{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - M}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ t_1 := \frac{c0}{w \cdot h}\\ t_2 := t\_0 \cdot t\_1\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_4 := \frac{c0}{2 \cdot w} \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{-101}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)}{2 \cdot w}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;c0 \cdot \frac{\frac{{M}^{2}}{t\_2 - \sqrt{{t\_2}^{2} - {M}^{2}}}}{2 \cdot w}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;c0 \cdot \frac{t\_2 + \sqrt{\mathsf{fma}\left(t\_1, t\_0, M\right)} \cdot \sqrt{c0 \cdot \frac{t\_0}{w \cdot h} - M}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (/ d D) 2.0))
        (t_1 (/ c0 (* w h)))
        (t_2 (* t_0 t_1))
        (t_3 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_4 (* (/ c0 (* 2.0 w)) (+ t_3 (sqrt (- (* t_3 t_3) (* M M)))))))
   (if (<= t_4 -2e-101)
     (*
      c0
      (/
       (fma
        c0
        (* d (/ d (* D (* w (* h D)))))
        (/ (* c0 (pow d 2.0)) (* (* w h) (pow D 2.0))))
       (* 2.0 w)))
     (if (<= t_4 0.0)
       (*
        c0
        (/
         (/ (pow M 2.0) (- t_2 (sqrt (- (pow t_2 2.0) (pow M 2.0)))))
         (* 2.0 w)))
       (if (<= t_4 INFINITY)
         (*
          c0
          (/
           (+
            t_2
            (* (sqrt (fma t_1 t_0 M)) (sqrt (- (* c0 (/ t_0 (* w h))) M))))
           (* 2.0 w)))
         (* c0 (/ 0.0 (* 2.0 w))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((d / D), 2.0);
	double t_1 = c0 / (w * h);
	double t_2 = t_0 * t_1;
	double t_3 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_4 = (c0 / (2.0 * w)) * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	double tmp;
	if (t_4 <= -2e-101) {
		tmp = c0 * (fma(c0, (d * (d / (D * (w * (h * D))))), ((c0 * pow(d, 2.0)) / ((w * h) * pow(D, 2.0)))) / (2.0 * w));
	} else if (t_4 <= 0.0) {
		tmp = c0 * ((pow(M, 2.0) / (t_2 - sqrt((pow(t_2, 2.0) - pow(M, 2.0))))) / (2.0 * w));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = c0 * ((t_2 + (sqrt(fma(t_1, t_0, M)) * sqrt(((c0 * (t_0 / (w * h))) - M)))) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / D) ^ 2.0
	t_1 = Float64(c0 / Float64(w * h))
	t_2 = Float64(t_0 * t_1)
	t_3 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_4 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M)))))
	tmp = 0.0
	if (t_4 <= -2e-101)
		tmp = Float64(c0 * Float64(fma(c0, Float64(d * Float64(d / Float64(D * Float64(w * Float64(h * D))))), Float64(Float64(c0 * (d ^ 2.0)) / Float64(Float64(w * h) * (D ^ 2.0)))) / Float64(2.0 * w)));
	elseif (t_4 <= 0.0)
		tmp = Float64(c0 * Float64(Float64((M ^ 2.0) / Float64(t_2 - sqrt(Float64((t_2 ^ 2.0) - (M ^ 2.0))))) / Float64(2.0 * w)));
	elseif (t_4 <= Inf)
		tmp = Float64(c0 * Float64(Float64(t_2 + Float64(sqrt(fma(t_1, t_0, M)) * sqrt(Float64(Float64(c0 * Float64(t_0 / Float64(w * h))) - M)))) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e-101], N[(c0 * N[(N[(c0 * N[(d * N[(d / N[(D * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(c0 * N[(N[(N[Power[M, 2.0], $MachinePrecision] / N[(t$95$2 - N[Sqrt[N[(N[Power[t$95$2, 2.0], $MachinePrecision] - N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(c0 * N[(N[(t$95$2 + N[(N[Sqrt[N[(t$95$1 * t$95$0 + M), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(c0 * N[(t$95$0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -2.0000000000000001e-101

   1. Initial program 57.4%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in c0 around inf 63.0%

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

   if -2.0000000000000001e-101 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 0.0

   1. Initial program 56.0%

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

   3. Simplified 29.7%

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

   6. Applied egg-rr 46.3%

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

      1. associate--r- 72.5%

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

      2. +-inverses 81.6%

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

      3. *-commutative 81.6%

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

      4. *-commutative 81.6%

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

   8. Simplified 81.6%

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

   if 0.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 88.6%

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

   3. Simplified 90.7%

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

      1. fma-undefine 90.7%

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

      2. associate-*r/ 90.7%

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

      3. *-commutative 90.7%

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

      4. associate-*r* 90.7%

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

      5. associate-*r* 86.3%

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

      6. associate-/l* 86.3%

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

      7. frac-times 86.3%

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

      8. times-frac 90.7%

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

      9. pow2 90.7%

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

   6. Applied egg-rr 93.1%

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

      1. pow1/2 93.1%

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

      2. associate-*l/ 93.1%

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

      3. *-commutative 93.1%

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

      4. associate-*l/ 93.1%

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

      5. *-commutative 93.1%

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

      6. pow2 93.1%

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

      7. frac-times 88.6%

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

      8. pow2 88.6%

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

      9. pow2 88.6%

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

      10. difference-of-squares 88.6%

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

      11. unpow-prod-down 90.8%

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

   8. Applied egg-rr 95.3%

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

      1. unpow1/2 95.3%

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

      2. *-commutative 95.3%

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

      3. unpow1/2 95.3%

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

      4. associate-*l/ 95.3%

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

      5. associate-/l* 95.3%

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

      6. *-commutative 95.3%

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

   10. Simplified 95.3%

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

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

   1. Initial program 0.0%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in c0 around -inf 0.9%

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

      1. mul-1-neg 0.9%

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

      2. distribute-lft-in 0.9%

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

      3. mul-1-neg 0.9%

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

      4. distribute-rgt-neg-in 0.9%

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

      5. associate-/l* 0.2%

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

      6. mul-1-neg 0.2%

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

      7. associate-/l* 0.2%

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

      8. distribute-lft1-in 0.2%

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

      9. metadata-eval 0.2%

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

      10. mul0-lft 38.4%

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

      11. metadata-eval 38.4%

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

   7. Simplified 38.4%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-101}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;c0 \cdot \frac{\frac{{M}^{2}}{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h} - \sqrt{{\left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h} + \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, {\left(\frac{d}{D}\right)}^{2}, M\right)} \cdot \sqrt{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - M}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ t_1 := t\_0 \cdot \frac{c0}{w \cdot h}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_3 := \frac{c0}{2 \cdot w} \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-101}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)}{2 \cdot w}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;c0 \cdot \frac{\frac{{M}^{2}}{t\_1 - \sqrt{{t\_1}^{2} - {M}^{2}}}}{2 \cdot w}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;c0 \cdot \frac{t\_1 + \sqrt{c0 \cdot \frac{t\_0}{w \cdot h} - M} \cdot \left(\frac{d}{D} \cdot \sqrt{\frac{\frac{c0}{w}}{h}}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (/ d D) 2.0))
        (t_1 (* t_0 (/ c0 (* w h))))
        (t_2 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_3 (* (/ c0 (* 2.0 w)) (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))))))
   (if (<= t_3 -2e-101)
     (*
      c0
      (/
       (fma
        c0
        (* d (/ d (* D (* w (* h D)))))
        (/ (* c0 (pow d 2.0)) (* (* w h) (pow D 2.0))))
       (* 2.0 w)))
     (if (<= t_3 0.0)
       (*
        c0
        (/
         (/ (pow M 2.0) (- t_1 (sqrt (- (pow t_1 2.0) (pow M 2.0)))))
         (* 2.0 w)))
       (if (<= t_3 INFINITY)
         (*
          c0
          (/
           (+
            t_1
            (*
             (sqrt (- (* c0 (/ t_0 (* w h))) M))
             (* (/ d D) (sqrt (/ (/ c0 w) h)))))
           (* 2.0 w)))
         (* c0 (/ 0.0 (* 2.0 w))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((d / D), 2.0);
	double t_1 = t_0 * (c0 / (w * h));
	double t_2 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_3 = (c0 / (2.0 * w)) * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -2e-101) {
		tmp = c0 * (fma(c0, (d * (d / (D * (w * (h * D))))), ((c0 * pow(d, 2.0)) / ((w * h) * pow(D, 2.0)))) / (2.0 * w));
	} else if (t_3 <= 0.0) {
		tmp = c0 * ((pow(M, 2.0) / (t_1 - sqrt((pow(t_1, 2.0) - pow(M, 2.0))))) / (2.0 * w));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = c0 * ((t_1 + (sqrt(((c0 * (t_0 / (w * h))) - M)) * ((d / D) * sqrt(((c0 / w) / h))))) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / D) ^ 2.0
	t_1 = Float64(t_0 * Float64(c0 / Float64(w * h)))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_3 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))))
	tmp = 0.0
	if (t_3 <= -2e-101)
		tmp = Float64(c0 * Float64(fma(c0, Float64(d * Float64(d / Float64(D * Float64(w * Float64(h * D))))), Float64(Float64(c0 * (d ^ 2.0)) / Float64(Float64(w * h) * (D ^ 2.0)))) / Float64(2.0 * w)));
	elseif (t_3 <= 0.0)
		tmp = Float64(c0 * Float64(Float64((M ^ 2.0) / Float64(t_1 - sqrt(Float64((t_1 ^ 2.0) - (M ^ 2.0))))) / Float64(2.0 * w)));
	elseif (t_3 <= Inf)
		tmp = Float64(c0 * Float64(Float64(t_1 + Float64(sqrt(Float64(Float64(c0 * Float64(t_0 / Float64(w * h))) - M)) * Float64(Float64(d / D) * sqrt(Float64(Float64(c0 / w) / h))))) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-101], N[(c0 * N[(N[(c0 * N[(d * N[(d / N[(D * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(c0 * N[(N[(N[Power[M, 2.0], $MachinePrecision] / N[(t$95$1 - N[Sqrt[N[(N[Power[t$95$1, 2.0], $MachinePrecision] - N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(c0 * N[(N[(t$95$1 + N[(N[Sqrt[N[(N[(c0 * N[(t$95$0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[Sqrt[N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -2.0000000000000001e-101

   1. Initial program 57.4%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in c0 around inf 63.0%

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

   if -2.0000000000000001e-101 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 0.0

   1. Initial program 56.0%

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

   3. Simplified 29.7%

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

   6. Applied egg-rr 46.3%

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

      1. associate--r- 72.5%

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

      2. +-inverses 81.6%

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

      3. *-commutative 81.6%

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

      4. *-commutative 81.6%

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

   8. Simplified 81.6%

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

   if 0.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 88.6%

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

   3. Simplified 90.7%

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

      1. fma-undefine 90.7%

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

      2. associate-*r/ 90.7%

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

      3. *-commutative 90.7%

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

      4. associate-*r* 90.7%

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

      5. associate-*r* 86.3%

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

      6. associate-/l* 86.3%

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

      7. frac-times 86.3%

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

      8. times-frac 90.7%

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

      9. pow2 90.7%

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

   6. Applied egg-rr 93.1%

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

      1. pow1/2 93.1%

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

      2. associate-*l/ 93.1%

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

      3. *-commutative 93.1%

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

      4. associate-*l/ 93.1%

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

      5. *-commutative 93.1%

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

      6. pow2 93.1%

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

      7. frac-times 88.6%

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

      8. pow2 88.6%

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

      9. pow2 88.6%

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

      10. difference-of-squares 88.6%

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

      11. unpow-prod-down 90.8%

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

   8. Applied egg-rr 95.3%

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

      1. unpow1/2 95.3%

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

      2. *-commutative 95.3%

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

      3. unpow1/2 95.3%

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

      4. associate-*l/ 95.3%

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

      5. associate-/l* 95.3%

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

      6. *-commutative 95.3%

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

   10. Simplified 95.3%

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

   11. Taylor expanded in c0 around inf 40.1%

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

       1. associate-/l/ 40.0%

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

   13. Simplified 40.0%

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

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

   1. Initial program 0.0%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in c0 around -inf 0.9%

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

      1. mul-1-neg 0.9%

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

      2. distribute-lft-in 0.9%

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

      3. mul-1-neg 0.9%

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

      4. distribute-rgt-neg-in 0.9%

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

      5. associate-/l* 0.2%

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

      6. mul-1-neg 0.2%

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

      7. associate-/l* 0.2%

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

      8. distribute-lft1-in 0.2%

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

      9. metadata-eval 0.2%

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

      10. mul0-lft 38.4%

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

      11. metadata-eval 38.4%

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

   7. Simplified 38.4%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-101}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \frac{c0 \cdot {d}^{2}}{\left(w \cdot h\right) \cdot {D}^{2}}\right)}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;c0 \cdot \frac{\frac{{M}^{2}}{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h} - \sqrt{{\left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)}^{2} - {M}^{2}}}}{2 \cdot w}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h} + \sqrt{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h} - M} \cdot \left(\frac{d}{D} \cdot \sqrt{\frac{\frac{c0}{w}}{h}}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.8%

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

   3. Simplified 72.2%

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

      1. fma-undefine 75.6%

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

      2. associate-*r/ 73.3%

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

      3. *-commutative 73.3%

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

      4. associate-*r* 71.3%

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

      5. associate-*r* 68.3%

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

      6. associate-/l* 69.4%

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

      7. frac-times 66.5%

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

      8. times-frac 70.8%

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

      9. pow2 70.8%

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

   6. Applied egg-rr 74.2%

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

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

   1. Initial program 0.0%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in c0 around -inf 0.9%

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

      1. mul-1-neg 0.9%

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

      2. distribute-lft-in 0.9%

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

      3. mul-1-neg 0.9%

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

      4. distribute-rgt-neg-in 0.9%

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

      5. associate-/l* 0.2%

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

      6. mul-1-neg 0.2%

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

      7. associate-/l* 0.2%

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

      8. distribute-lft1-in 0.2%

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

      9. metadata-eval 0.2%

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

      10. mul0-lft 38.4%

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

      11. metadata-eval 38.4%

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

   7. Simplified 38.4%

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

4. Final simplification 50.4%

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

Alternative 5: 54.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.8%

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

   3. Simplified 70.0%

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

      1. times-frac 70.0%

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

   6. Applied egg-rr 70.0%

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

      1. times-frac 70.0%

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

   8. Applied egg-rr 70.0%

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

      1. times-frac 70.0%

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

   10. Applied egg-rr 74.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in c0 around -inf 0.9%

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

      1. mul-1-neg 0.9%

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

      2. distribute-lft-in 0.9%

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

      3. mul-1-neg 0.9%

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

      4. distribute-rgt-neg-in 0.9%

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

      5. associate-/l* 0.2%

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

      6. mul-1-neg 0.2%

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

      7. associate-/l* 0.2%

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

      8. distribute-lft1-in 0.2%

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

      9. metadata-eval 0.2%

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

      10. mul0-lft 38.4%

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

      11. metadata-eval 38.4%

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

   7. Simplified 38.4%

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

4. Final simplification 50.4%

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

Alternative 6: 33.5% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ c0 \cdot \frac{0}{2 \cdot w} \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* c0 (/ 0.0 (* 2.0 w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (0.0 / (2.0 * w));
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = c0 * (0.0d0 / (2.0d0 * w))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (0.0 / (2.0 * w));
}

Python

def code(c0, w, h, D, d, M):
	return c0 * (0.0 / (2.0 * w))

Julia

function code(c0, w, h, D, d, M)
	return Float64(c0 * Float64(0.0 / Float64(2.0 * w)))
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.5%

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

3. Simplified 42.1%

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

5. Taylor expanded in c0 around -inf 3.6%

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

   1. mul-1-neg 3.6%

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

   2. distribute-lft-in 3.6%

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

   3. mul-1-neg 3.6%

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

   4. distribute-rgt-neg-in 3.6%

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

   5. associate-/l* 3.2%

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

   6. mul-1-neg 3.2%

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

   7. associate-/l* 3.5%

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

   8. distribute-lft1-in 3.5%

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

   9. metadata-eval 3.5%

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

   10. mul0-lft 29.2%

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

   11. metadata-eval 29.2%

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

7. Simplified 29.2%

   \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(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))))))