Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.4% → 56.0%

Time: 31.1s

Alternatives: 13

Speedup: 10.8×

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: 24.4% 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: 56.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := \frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;c0 \cdot \frac{t\_0 + \sqrt{{t\_0}^{2} - {M}^{2}}}{2 \cdot w}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;{\left(d \cdot \frac{c0}{\sqrt{h} \cdot \left(w \cdot D\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot 0}{w}\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 / (w * h)) * ((d / D) ^ 2.0);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = c0 * ((t_0 + sqrt(((t_0 ^ 2.0) - (M ^ 2.0)))) / (2.0 * w));
	elseif (t_2 <= Inf)
		tmp = (d * (c0 / (sqrt(h) * (w * D)))) ^ 2.0;
	else
		tmp = c0 * (-0.5 * ((c0 * 0.0) / 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[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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]}, If[LessEqual[t$95$2, 0.0], 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], If[LessEqual[t$95$2, Infinity], N[Power[N[(d * N[(c0 / N[(N[Sqrt[h], $MachinePrecision] * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(c0 * N[(-0.5 * N[(N[(c0 * 0.0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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))))) < -0.0

   1. Initial program 67.2%

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

   3. Simplified 64.9%

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

      1. fma-undefine 69.7%

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

      2. associate-*r/ 68.1%

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

      3. *-commutative 68.1%

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

      4. associate-*r* 67.3%

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

      5. associate-*r* 62.8%

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

      6. associate-/l* 60.4%

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

      7. frac-times 62.5%

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

      8. times-frac 64.7%

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

      9. pow2 64.7%

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

   6. Applied egg-rr 70.6%

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

   if -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 72.7%

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

   3. Simplified 72.6%

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

   6. Applied egg-rr 81.3%

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

   7. Taylor expanded in c0 around inf 48.2%

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

      1. associate-/l* 50.4%

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

      2. associate-*r* 53.8%

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

   9. Simplified 53.8%

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

   10. Taylor expanded in c0 around 0 48.2%

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

       1. unpow2 48.2%

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

       2. unpow2 48.2%

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

       3. swap-sqr 58.7%

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

       4. associate-*r* 61.1%

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

       5. *-commutative 61.1%

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

       6. unpow2 61.1%

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

       7. rem-square-sqrt 61.0%

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

       8. swap-sqr 65.0%

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

       9. unpow2 65.0%

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

       10. swap-sqr 85.4%

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

       11. times-frac 91.8%

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

       12. associate-*r/ 89.7%

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

       13. associate-*r/ 89.8%

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

       14. unpow2 89.8%

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

   12. Simplified 92.7%

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

   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 19.6%

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

   5. Taylor expanded in c0 around -inf 1.3%

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

      1. associate-/l* 1.3%

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

      2. distribute-lft1-in 1.3%

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

      3. metadata-eval 1.3%

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

      4. associate-/r* 0.7%

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

   7. Simplified 0.7%

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

      1. associate-*r/ 0.7%

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

      2. mul0-lft 41.7%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot \color{blue}{0}}{w}\right) \]

   9. Applied egg-rr 41.7%

      \[\leadsto c0 \cdot \left(-0.5 \cdot \color{blue}{\frac{c0 \cdot 0}{w}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 54.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := \frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-109}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, \frac{d}{\frac{w \cdot \left(h \cdot {D}^{2}\right)}{d}}, \sqrt{\mathsf{fma}\left(c0, t\_0, M\right) \cdot \left(c0 \cdot t\_0 - M\right)}\right)}{2 \cdot w}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;{\left(d \cdot \frac{c0}{\sqrt{h} \cdot \left(w \cdot D\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot 0}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* d (/ d (* D (* w (* h D))))))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -1e-109)
     (*
      c0
      (/
       (fma
        c0
        (/ d (/ (* w (* h (pow D 2.0))) d))
        (sqrt (* (fma c0 t_0 M) (- (* c0 t_0) M))))
       (* 2.0 w)))
     (if (<= t_2 INFINITY)
       (pow (* d (/ c0 (* (sqrt h) (* w D)))) 2.0)
       (* c0 (* -0.5 (/ (* c0 0.0) w)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / (D * (w * (h * D))));
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -1e-109) {
		tmp = c0 * (fma(c0, (d / ((w * (h * pow(D, 2.0))) / d)), sqrt((fma(c0, t_0, M) * ((c0 * t_0) - M)))) / (2.0 * w));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = pow((d * (c0 / (sqrt(h) * (w * D)))), 2.0);
	} else {
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d * Float64(d / Float64(D * Float64(w * Float64(h * D)))))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -1e-109)
		tmp = Float64(c0 * Float64(fma(c0, Float64(d / Float64(Float64(w * Float64(h * (D ^ 2.0))) / d)), sqrt(Float64(fma(c0, t_0, M) * Float64(Float64(c0 * t_0) - M)))) / Float64(2.0 * w)));
	elseif (t_2 <= Inf)
		tmp = Float64(d * Float64(c0 / Float64(sqrt(h) * Float64(w * D)))) ^ 2.0;
	else
		tmp = Float64(c0 * Float64(-0.5 * Float64(Float64(c0 * 0.0) / w)));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d * N[(d / N[(D * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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]}, If[LessEqual[t$95$2, -1e-109], N[(c0 * N[(N[(c0 * N[(d / N[(N[(w * N[(h * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(c0 * t$95$0 + M), $MachinePrecision] * N[(N[(c0 * t$95$0), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Power[N[(d * N[(c0 / N[(N[Sqrt[h], $MachinePrecision] * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(c0 * N[(-0.5 * N[(N[(c0 * 0.0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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))))) < -9.9999999999999999e-110

   1. Initial program 71.7%

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

   3. Simplified 74.4%

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

      1. clear-num 74.4%

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

      2. inv-pow 74.4%

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

      3. *-commutative 74.4%

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

      4. associate-*r* 71.6%

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

      5. associate-*r* 71.7%

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

      6. associate-*l* 74.5%

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

      7. pow2 74.5%

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

   6. Applied egg-rr 74.5%

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

      1. unpow-1 74.5%

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

      2. associate-/l* 74.5%

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

   8. Simplified 74.5%

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

      1. un-div-inv 74.5%

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

      2. associate-*r/ 74.6%

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

   10. Applied egg-rr 74.6%

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

   if -9.9999999999999999e-110 < (*.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 68.8%

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

   3. Simplified 68.6%

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

   6. Applied egg-rr 73.7%

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

   7. Taylor expanded in c0 around inf 43.5%

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

      1. associate-/l* 45.4%

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

      2. associate-*r* 46.3%

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

   9. Simplified 46.3%

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

   10. Taylor expanded in c0 around 0 43.5%

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

       1. unpow2 43.5%

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

       2. unpow2 43.5%

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

       3. swap-sqr 54.5%

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

       4. associate-*r* 54.6%

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

       5. *-commutative 54.6%

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

       6. unpow2 54.6%

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

       7. rem-square-sqrt 54.3%

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

       8. swap-sqr 58.0%

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

       9. unpow2 58.0%

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

       10. swap-sqr 79.3%

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

       11. times-frac 84.9%

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

       12. associate-*r/ 83.1%

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

       13. associate-*r/ 83.1%

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

       14. unpow2 83.1%

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

   12. Simplified 85.6%

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

   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 19.6%

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

   5. Taylor expanded in c0 around -inf 1.3%

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

      1. associate-/l* 1.3%

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

      2. distribute-lft1-in 1.3%

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

      3. metadata-eval 1.3%

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

      4. associate-/r* 0.7%

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

   7. Simplified 0.7%

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

      1. associate-*r/ 0.7%

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

      2. mul0-lft 41.7%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot \color{blue}{0}}{w}\right) \]

   9. Applied egg-rr 41.7%

      \[\leadsto c0 \cdot \left(-0.5 \cdot \color{blue}{\frac{c0 \cdot 0}{w}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 54.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := \frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-109}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{\frac{d}{w}}{h \cdot {D}^{2}}, \sqrt{\mathsf{fma}\left(c0, t\_0, M\right) \cdot \left(c0 \cdot t\_0 - M\right)}\right)}{2 \cdot w}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;{\left(d \cdot \frac{c0}{\sqrt{h} \cdot \left(w \cdot D\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot 0}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* d (/ d (* D (* w (* h D))))))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -1e-109)
     (*
      c0
      (/
       (fma
        c0
        (* d (/ (/ d w) (* h (pow D 2.0))))
        (sqrt (* (fma c0 t_0 M) (- (* c0 t_0) M))))
       (* 2.0 w)))
     (if (<= t_2 INFINITY)
       (pow (* d (/ c0 (* (sqrt h) (* w D)))) 2.0)
       (* c0 (* -0.5 (/ (* c0 0.0) w)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / (D * (w * (h * D))));
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -1e-109) {
		tmp = c0 * (fma(c0, (d * ((d / w) / (h * pow(D, 2.0)))), sqrt((fma(c0, t_0, M) * ((c0 * t_0) - M)))) / (2.0 * w));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = pow((d * (c0 / (sqrt(h) * (w * D)))), 2.0);
	} else {
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d * Float64(d / Float64(D * Float64(w * Float64(h * D)))))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -1e-109)
		tmp = Float64(c0 * Float64(fma(c0, Float64(d * Float64(Float64(d / w) / Float64(h * (D ^ 2.0)))), sqrt(Float64(fma(c0, t_0, M) * Float64(Float64(c0 * t_0) - M)))) / Float64(2.0 * w)));
	elseif (t_2 <= Inf)
		tmp = Float64(d * Float64(c0 / Float64(sqrt(h) * Float64(w * D)))) ^ 2.0;
	else
		tmp = Float64(c0 * Float64(-0.5 * Float64(Float64(c0 * 0.0) / w)));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d * N[(d / N[(D * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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]}, If[LessEqual[t$95$2, -1e-109], N[(c0 * N[(N[(c0 * N[(d * N[(N[(d / w), $MachinePrecision] / N[(h * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(c0 * t$95$0 + M), $MachinePrecision] * N[(N[(c0 * t$95$0), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Power[N[(d * N[(c0 / N[(N[Sqrt[h], $MachinePrecision] * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(c0 * N[(-0.5 * N[(N[(c0 * 0.0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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))))) < -9.9999999999999999e-110

   1. Initial program 71.7%

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

   3. Simplified 74.4%

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

      1. *-un-lft-identity 74.4%

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

      2. *-commutative 74.4%

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

      3. associate-*r* 71.6%

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

      4. associate-*r* 71.6%

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

      5. associate-*l* 74.5%

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

      6. pow2 74.5%

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

   6. Applied egg-rr 74.5%

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

      1. *-lft-identity 74.5%

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

      2. associate-/r* 74.5%

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

   8. Simplified 74.5%

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

   if -9.9999999999999999e-110 < (*.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 68.8%

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

   3. Simplified 68.6%

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

   6. Applied egg-rr 73.7%

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

   7. Taylor expanded in c0 around inf 43.5%

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

      1. associate-/l* 45.4%

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

      2. associate-*r* 46.3%

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

   9. Simplified 46.3%

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

   10. Taylor expanded in c0 around 0 43.5%

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

       1. unpow2 43.5%

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

       2. unpow2 43.5%

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

       3. swap-sqr 54.5%

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

       4. associate-*r* 54.6%

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

       5. *-commutative 54.6%

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

       6. unpow2 54.6%

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

       7. rem-square-sqrt 54.3%

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

       8. swap-sqr 58.0%

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

       9. unpow2 58.0%

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

       10. swap-sqr 79.3%

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

       11. times-frac 84.9%

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

       12. associate-*r/ 83.1%

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

       13. associate-*r/ 83.1%

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

       14. unpow2 83.1%

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

   12. Simplified 85.6%

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

   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 19.6%

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

   5. Taylor expanded in c0 around -inf 1.3%

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

      1. associate-/l* 1.3%

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

      2. distribute-lft1-in 1.3%

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

      3. metadata-eval 1.3%

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

      4. associate-/r* 0.7%

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

   7. Simplified 0.7%

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

      1. associate-*r/ 0.7%

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

      2. mul0-lft 41.7%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot \color{blue}{0}}{w}\right) \]

   9. Applied egg-rr 41.7%

      \[\leadsto c0 \cdot \left(-0.5 \cdot \color{blue}{\frac{c0 \cdot 0}{w}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 54.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := \frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-109}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, t\_0, \sqrt{\mathsf{fma}\left(c0, t\_0, M\right) \cdot \left(c0 \cdot t\_0 - M\right)}\right)}{2 \cdot w}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;{\left(d \cdot \frac{c0}{\sqrt{h} \cdot \left(w \cdot D\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot 0}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* d (/ d (* D (* w (* h D))))))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -1e-109)
     (*
      c0
      (/ (fma c0 t_0 (sqrt (* (fma c0 t_0 M) (- (* c0 t_0) M)))) (* 2.0 w)))
     (if (<= t_2 INFINITY)
       (pow (* d (/ c0 (* (sqrt h) (* w D)))) 2.0)
       (* c0 (* -0.5 (/ (* c0 0.0) w)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / (D * (w * (h * D))));
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -1e-109) {
		tmp = c0 * (fma(c0, t_0, sqrt((fma(c0, t_0, M) * ((c0 * t_0) - M)))) / (2.0 * w));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = pow((d * (c0 / (sqrt(h) * (w * D)))), 2.0);
	} else {
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d * Float64(d / Float64(D * Float64(w * Float64(h * D)))))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -1e-109)
		tmp = Float64(c0 * Float64(fma(c0, t_0, sqrt(Float64(fma(c0, t_0, M) * Float64(Float64(c0 * t_0) - M)))) / Float64(2.0 * w)));
	elseif (t_2 <= Inf)
		tmp = Float64(d * Float64(c0 / Float64(sqrt(h) * Float64(w * D)))) ^ 2.0;
	else
		tmp = Float64(c0 * Float64(-0.5 * Float64(Float64(c0 * 0.0) / w)));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d * N[(d / N[(D * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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]}, If[LessEqual[t$95$2, -1e-109], N[(c0 * N[(N[(c0 * t$95$0 + N[Sqrt[N[(N[(c0 * t$95$0 + M), $MachinePrecision] * N[(N[(c0 * t$95$0), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Power[N[(d * N[(c0 / N[(N[Sqrt[h], $MachinePrecision] * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(c0 * N[(-0.5 * N[(N[(c0 * 0.0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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))))) < -9.9999999999999999e-110

   1. Initial program 71.7%

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

   3. Simplified 74.4%

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

   if -9.9999999999999999e-110 < (*.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 68.8%

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

   3. Simplified 68.6%

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

   6. Applied egg-rr 73.7%

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

   7. Taylor expanded in c0 around inf 43.5%

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

      1. associate-/l* 45.4%

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

      2. associate-*r* 46.3%

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

   9. Simplified 46.3%

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

   10. Taylor expanded in c0 around 0 43.5%

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

       1. unpow2 43.5%

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

       2. unpow2 43.5%

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

       3. swap-sqr 54.5%

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

       4. associate-*r* 54.6%

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

       5. *-commutative 54.6%

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

       6. unpow2 54.6%

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

       7. rem-square-sqrt 54.3%

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

       8. swap-sqr 58.0%

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

       9. unpow2 58.0%

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

       10. swap-sqr 79.3%

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

       11. times-frac 84.9%

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

       12. associate-*r/ 83.1%

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

       13. associate-*r/ 83.1%

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

       14. unpow2 83.1%

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

   12. Simplified 85.6%

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

   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 19.6%

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

   5. Taylor expanded in c0 around -inf 1.3%

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

      1. associate-/l* 1.3%

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

      2. distribute-lft1-in 1.3%

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

      3. metadata-eval 1.3%

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

      4. associate-/r* 0.7%

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

   7. Simplified 0.7%

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

      1. associate-*r/ 0.7%

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

      2. mul0-lft 41.7%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot \color{blue}{0}}{w}\right) \]

   9. Applied egg-rr 41.7%

      \[\leadsto c0 \cdot \left(-0.5 \cdot \color{blue}{\frac{c0 \cdot 0}{w}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 56.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;{\left(d \cdot \frac{c0}{\sqrt{h} \cdot \left(w \cdot D\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot 0}{w}\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 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))))) < -0.0

   1. Initial program 67.2%

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

   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 72.7%

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

   3. Simplified 72.6%

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

   6. Applied egg-rr 81.3%

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

   7. Taylor expanded in c0 around inf 48.2%

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

      1. associate-/l* 50.4%

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

      2. associate-*r* 53.8%

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

   9. Simplified 53.8%

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

   10. Taylor expanded in c0 around 0 48.2%

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

       1. unpow2 48.2%

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

       2. unpow2 48.2%

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

       3. swap-sqr 58.7%

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

       4. associate-*r* 61.1%

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

       5. *-commutative 61.1%

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

       6. unpow2 61.1%

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

       7. rem-square-sqrt 61.0%

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

       8. swap-sqr 65.0%

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

       9. unpow2 65.0%

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

       10. swap-sqr 85.4%

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

       11. times-frac 91.8%

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

       12. associate-*r/ 89.7%

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

       13. associate-*r/ 89.8%

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

       14. unpow2 89.8%

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

   12. Simplified 92.7%

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

   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 19.6%

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

   5. Taylor expanded in c0 around -inf 1.3%

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

      1. associate-/l* 1.3%

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

      2. distribute-lft1-in 1.3%

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

      3. metadata-eval 1.3%

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

      4. associate-/r* 0.7%

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

   7. Simplified 0.7%

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

      1. associate-*r/ 0.7%

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

      2. mul0-lft 41.7%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot \color{blue}{0}}{w}\right) \]

   9. Applied egg-rr 41.7%

      \[\leadsto c0 \cdot \left(-0.5 \cdot \color{blue}{\frac{c0 \cdot 0}{w}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 56.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;{\left(c0 \cdot \frac{d}{w \cdot \left(D \cdot \sqrt{h}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot 0}{w}\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 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))))) < -0.0

   1. Initial program 67.2%

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

   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 72.7%

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

   3. Simplified 72.6%

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

   6. Applied egg-rr 81.3%

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

   7. Taylor expanded in c0 around inf 48.2%

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

      1. associate-/l* 50.4%

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

      2. associate-*r* 53.8%

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

   9. Simplified 53.8%

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

       1. add-sqr-sqrt 53.7%

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

       2. pow2 53.7%

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

   11. Applied egg-rr 89.8%

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

   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 19.6%

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

   5. Taylor expanded in c0 around -inf 1.3%

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

      1. associate-/l* 1.3%

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

      2. distribute-lft1-in 1.3%

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

      3. metadata-eval 1.3%

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

      4. associate-/r* 0.7%

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

   7. Simplified 0.7%

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

      1. associate-*r/ 0.7%

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

      2. mul0-lft 41.7%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot \color{blue}{0}}{w}\right) \]

   9. Applied egg-rr 41.7%

      \[\leadsto c0 \cdot \left(-0.5 \cdot \color{blue}{\frac{c0 \cdot 0}{w}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 54.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot 0}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 INFINITY) t_1 (* c0 (* -0.5 (/ (* c0 0.0) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w));
	}
	return tmp;
}

Java

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

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(c0 * N[(-0.5 * N[(N[(c0 * 0.0), $MachinePrecision] / 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 70.0%

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

   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 19.6%

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

   5. Taylor expanded in c0 around -inf 1.3%

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

      1. associate-/l* 1.3%

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

      2. distribute-lft1-in 1.3%

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

      3. metadata-eval 1.3%

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

      4. associate-/r* 0.7%

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

   7. Simplified 0.7%

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

      1. associate-*r/ 0.7%

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

      2. mul0-lft 41.7%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot \color{blue}{0}}{w}\right) \]

   9. Applied egg-rr 41.7%

      \[\leadsto c0 \cdot \left(-0.5 \cdot \color{blue}{\frac{c0 \cdot 0}{w}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 36.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ \mathbf{if}\;d \leq 2.5 \cdot 10^{+25}:\\ \;\;\;\;c0 \cdot \frac{\frac{c0}{w \cdot h} \cdot t\_0}{2 \cdot w}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+117} \lor \neg \left(d \leq 8.5 \cdot 10^{+270}\right):\\ \;\;\;\;c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot 0}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{c0 \cdot \frac{t\_0}{w \cdot h}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (/ d D) 2.0)))
   (if (<= d 2.5e+25)
     (* c0 (/ (* (/ c0 (* w h)) t_0) (* 2.0 w)))
     (if (or (<= d 4e+117) (not (<= d 8.5e+270)))
       (* c0 (* -0.5 (/ (* c0 0.0) w)))
       (* c0 (/ (* c0 (/ t_0 (* w h))) (* 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 tmp;
	if (d <= 2.5e+25) {
		tmp = c0 * (((c0 / (w * h)) * t_0) / (2.0 * w));
	} else if ((d <= 4e+117) || !(d <= 8.5e+270)) {
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w));
	} else {
		tmp = c0 * ((c0 * (t_0 / (w * h))) / (2.0 * w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_1 / d) ** 2.0d0
    if (d_1 <= 2.5d+25) then
        tmp = c0 * (((c0 / (w * h)) * t_0) / (2.0d0 * w))
    else if ((d_1 <= 4d+117) .or. (.not. (d_1 <= 8.5d+270))) then
        tmp = c0 * ((-0.5d0) * ((c0 * 0.0d0) / w))
    else
        tmp = c0 * ((c0 * (t_0 / (w * h))) / (2.0d0 * w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = Math.pow((d / D), 2.0);
	double tmp;
	if (d <= 2.5e+25) {
		tmp = c0 * (((c0 / (w * h)) * t_0) / (2.0 * w));
	} else if ((d <= 4e+117) || !(d <= 8.5e+270)) {
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w));
	} else {
		tmp = c0 * ((c0 * (t_0 / (w * h))) / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow((d / D), 2.0)
	tmp = 0
	if d <= 2.5e+25:
		tmp = c0 * (((c0 / (w * h)) * t_0) / (2.0 * w))
	elif (d <= 4e+117) or not (d <= 8.5e+270):
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w))
	else:
		tmp = c0 * ((c0 * (t_0 / (w * h))) / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / D) ^ 2.0
	tmp = 0.0
	if (d <= 2.5e+25)
		tmp = Float64(c0 * Float64(Float64(Float64(c0 / Float64(w * h)) * t_0) / Float64(2.0 * w)));
	elseif ((d <= 4e+117) || !(d <= 8.5e+270))
		tmp = Float64(c0 * Float64(-0.5 * Float64(Float64(c0 * 0.0) / w)));
	else
		tmp = Float64(c0 * Float64(Float64(c0 * Float64(t_0 / Float64(w * h))) / 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;
	tmp = 0.0;
	if (d <= 2.5e+25)
		tmp = c0 * (((c0 / (w * h)) * t_0) / (2.0 * w));
	elseif ((d <= 4e+117) || ~((d <= 8.5e+270)))
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w));
	else
		tmp = c0 * ((c0 * (t_0 / (w * h))) / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, 2.5e+25], N[(c0 * N[(N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[d, 4e+117], N[Not[LessEqual[d, 8.5e+270]], $MachinePrecision]], N[(c0 * N[(-0.5 * N[(N[(c0 * 0.0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(c0 * N[(t$95$0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < 2.50000000000000012e25

   1. Initial program 23.6%

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

   3. Simplified 35.8%

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

   5. Taylor expanded in w around -inf 3.1%

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

      1. distribute-lft-out 3.1%

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

      2. distribute-lft1-in 3.1%

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

      3. metadata-eval 3.1%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{-1 \cdot \left(\frac{\color{blue}{0} \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}\right)}{2 \cdot w} \]

      4. associate-*r* 3.1%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\color{blue}{\left(M \cdot c0\right) \cdot {d}^{2}}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}\right)}{2 \cdot w} \]

   7. Simplified 3.1%

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

      1. add-sqr-sqrt 3.1%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{\sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)} \cdot \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}}}\right)}{2 \cdot w} \]

      2. pow1/2 3.1%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5}} \cdot \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}}\right)}{2 \cdot w} \]

      3. pow1/2 3.1%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5} \cdot \color{blue}{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5}}}\right)}{2 \cdot w} \]

      4. pow-prod-down 6.4%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{{\left(\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right) \cdot \left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)\right)}^{0.5}}}\right)}{2 \cdot w} \]

   9. Applied egg-rr 36.6%

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

       1. unpow1/2 36.6%

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

       2. div0 36.6%

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

   11. Simplified 36.6%

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

   12. Taylor expanded in c0 around inf 28.9%

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

       1. *-commutative 28.9%

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

       2. times-frac 28.3%

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

       3. unpow2 28.3%

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

       4. unpow2 28.3%

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

       5. times-frac 38.8%

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

       6. unpow2 38.8%

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

   14. Simplified 38.8%

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

   if 2.50000000000000012e25 < d < 4.0000000000000002e117 or 8.50000000000000063e270 < d

   1. Initial program 9.8%

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

   3. Simplified 22.1%

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

   5. Taylor expanded in c0 around -inf 6.3%

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

      1. associate-/l* 6.3%

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

      2. distribute-lft1-in 6.3%

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

      3. metadata-eval 6.3%

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

      4. associate-/r* 3.3%

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

   7. Simplified 3.3%

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

      1. associate-*r/ 3.3%

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

      2. mul0-lft 56.7%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot \color{blue}{0}}{w}\right) \]

   9. Applied egg-rr 56.7%

      \[\leadsto c0 \cdot \left(-0.5 \cdot \color{blue}{\frac{c0 \cdot 0}{w}}\right) \]

   if 4.0000000000000002e117 < d < 8.50000000000000063e270

   1. Initial program 28.5%

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

   3. Simplified 46.1%

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

   5. Taylor expanded in w around -inf 0.0%

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

      1. distribute-lft-out 0.0%

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

      2. distribute-lft1-in 0.0%

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

      3. metadata-eval 0.0%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{-1 \cdot \left(\frac{\color{blue}{0} \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}\right)}{2 \cdot w} \]

      4. associate-*r* 2.2%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\color{blue}{\left(M \cdot c0\right) \cdot {d}^{2}}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}\right)}{2 \cdot w} \]

   7. Simplified 2.2%

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

      1. add-sqr-sqrt 2.2%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{\sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)} \cdot \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}}}\right)}{2 \cdot w} \]

      2. pow1/2 2.2%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5}} \cdot \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}}\right)}{2 \cdot w} \]

      3. pow1/2 2.2%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5} \cdot \color{blue}{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5}}}\right)}{2 \cdot w} \]

      4. pow-prod-down 2.2%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{{\left(\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right) \cdot \left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)\right)}^{0.5}}}\right)}{2 \cdot w} \]

   9. Applied egg-rr 51.3%

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

       1. unpow1/2 51.3%

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

       2. div0 51.3%

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

   11. Simplified 51.3%

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

   12. Taylor expanded in c0 around inf 40.0%

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

       1. associate-/r* 40.1%

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

       2. associate-/l* 40.1%

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

       3. unpow2 40.1%

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

       4. unpow2 40.1%

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

       5. times-frac 47.1%

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

       6. unpow2 47.1%

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

       7. associate-*r/ 49.3%

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

   14. Simplified 49.3%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.5 \cdot 10^{+25}:\\ \;\;\;\;c0 \cdot \frac{\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}}{2 \cdot w}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+117} \lor \neg \left(d \leq 8.5 \cdot 10^{+270}\right):\\ \;\;\;\;c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot 0}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 37.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.12 \cdot 10^{+24} \lor \neg \left(d \leq 2 \cdot 10^{+118}\right) \land d \leq 2.5 \cdot 10^{+270}:\\ \;\;\;\;c0 \cdot \frac{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot 0}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= d 1.12e+24) (and (not (<= d 2e+118)) (<= d 2.5e+270)))
   (* c0 (/ (* c0 (/ (pow (/ d D) 2.0) (* w h))) (* 2.0 w)))
   (* c0 (* -0.5 (/ (* c0 0.0) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d <= 1.12e+24) || (!(d <= 2e+118) && (d <= 2.5e+270))) {
		tmp = c0 * ((c0 * (pow((d / D), 2.0) / (w * h))) / (2.0 * w));
	} else {
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d_1 <= 1.12d+24) .or. (.not. (d_1 <= 2d+118)) .and. (d_1 <= 2.5d+270)) then
        tmp = c0 * ((c0 * (((d_1 / d) ** 2.0d0) / (w * h))) / (2.0d0 * w))
    else
        tmp = c0 * ((-0.5d0) * ((c0 * 0.0d0) / w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d <= 1.12e+24) || (!(d <= 2e+118) && (d <= 2.5e+270))) {
		tmp = c0 * ((c0 * (Math.pow((d / D), 2.0) / (w * h))) / (2.0 * w));
	} else {
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d <= 1.12e+24) or (not (d <= 2e+118) and (d <= 2.5e+270)):
		tmp = c0 * ((c0 * (math.pow((d / D), 2.0) / (w * h))) / (2.0 * w))
	else:
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((d <= 1.12e+24) || (!(d <= 2e+118) && (d <= 2.5e+270)))
		tmp = Float64(c0 * Float64(Float64(c0 * Float64((Float64(d / D) ^ 2.0) / Float64(w * h))) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(-0.5 * Float64(Float64(c0 * 0.0) / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d <= 1.12e+24) || (~((d <= 2e+118)) && (d <= 2.5e+270)))
		tmp = c0 * ((c0 * (((d / D) ^ 2.0) / (w * h))) / (2.0 * w));
	else
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[d, 1.12e+24], And[N[Not[LessEqual[d, 2e+118]], $MachinePrecision], LessEqual[d, 2.5e+270]]], N[(c0 * N[(N[(c0 * N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(-0.5 * N[(N[(c0 * 0.0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.12e24 or 1.99999999999999993e118 < d < 2.49999999999999988e270

   1. Initial program 24.6%

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

   3. Simplified 37.9%

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

   5. Taylor expanded in w around -inf 2.5%

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

      1. distribute-lft-out 2.5%

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

      2. distribute-lft1-in 2.5%

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

      3. metadata-eval 2.5%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{-1 \cdot \left(\frac{\color{blue}{0} \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}\right)}{2 \cdot w} \]

      4. associate-*r* 2.9%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\color{blue}{\left(M \cdot c0\right) \cdot {d}^{2}}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}\right)}{2 \cdot w} \]

   7. Simplified 2.9%

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

      1. add-sqr-sqrt 2.9%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{\sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)} \cdot \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}}}\right)}{2 \cdot w} \]

      2. pow1/2 2.9%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5}} \cdot \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}}\right)}{2 \cdot w} \]

      3. pow1/2 2.9%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5} \cdot \color{blue}{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5}}}\right)}{2 \cdot w} \]

      4. pow-prod-down 5.5%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{{\left(\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right) \cdot \left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)\right)}^{0.5}}}\right)}{2 \cdot w} \]

   9. Applied egg-rr 39.7%

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

       1. unpow1/2 39.7%

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

       2. div0 39.7%

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

   11. Simplified 39.7%

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

   12. Taylor expanded in c0 around inf 31.2%

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

       1. associate-/r* 30.7%

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

       2. associate-/l* 31.2%

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

       3. unpow2 31.2%

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

       4. unpow2 31.2%

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

       5. times-frac 40.4%

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

       6. unpow2 40.4%

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

       7. associate-*r/ 41.3%

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

   14. Simplified 41.3%

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

   if 1.12e24 < d < 1.99999999999999993e118 or 2.49999999999999988e270 < d

   1. Initial program 9.8%

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

   3. Simplified 22.1%

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

   5. Taylor expanded in c0 around -inf 6.3%

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

      1. associate-/l* 6.3%

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

      2. distribute-lft1-in 6.3%

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

      3. metadata-eval 6.3%

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

      4. associate-/r* 3.3%

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

   7. Simplified 3.3%

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

      1. associate-*r/ 3.3%

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

      2. mul0-lft 56.7%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot \color{blue}{0}}{w}\right) \]

   9. Applied egg-rr 56.7%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.12 \cdot 10^{+24} \lor \neg \left(d \leq 2 \cdot 10^{+118}\right) \land d \leq 2.5 \cdot 10^{+270}:\\ \;\;\;\;c0 \cdot \frac{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot 0}{w}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 8.6 \cdot 10^{+24} \lor \neg \left(d \leq 4.9 \cdot 10^{+117}\right):\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot 0}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= d 8.6e+24) (not (<= d 4.9e+117)))
   (* c0 (/ (fma c0 (* d (/ d (* D (* w (* h D))))) M) (* 2.0 w)))
   (* c0 (* -0.5 (/ (* c0 0.0) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d <= 8.6e+24) || !(d <= 4.9e+117)) {
		tmp = c0 * (fma(c0, (d * (d / (D * (w * (h * D))))), M) / (2.0 * w));
	} else {
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((d <= 8.6e+24) || !(d <= 4.9e+117))
		tmp = Float64(c0 * Float64(fma(c0, Float64(d * Float64(d / Float64(D * Float64(w * Float64(h * D))))), M) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(-0.5 * Float64(Float64(c0 * 0.0) / w)));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[d, 8.6e+24], N[Not[LessEqual[d, 4.9e+117]], $MachinePrecision]], N[(c0 * N[(N[(c0 * N[(d * N[(d / N[(D * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + M), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(-0.5 * N[(N[(c0 * 0.0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 8.59999999999999975e24 or 4.9000000000000001e117 < d

   1. Initial program 23.5%

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

   3. Simplified 37.1%

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

   5. Taylor expanded in w around -inf 2.3%

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

      1. distribute-lft-out 2.3%

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

      2. distribute-lft1-in 2.3%

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

      3. metadata-eval 2.3%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{-1 \cdot \left(\frac{\color{blue}{0} \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}\right)}{2 \cdot w} \]

      4. associate-*r* 2.8%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\color{blue}{\left(M \cdot c0\right) \cdot {d}^{2}}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}\right)}{2 \cdot w} \]

   7. Simplified 2.8%

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

      1. add-sqr-sqrt 2.8%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{\sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)} \cdot \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}}}\right)}{2 \cdot w} \]

      2. pow1/2 2.8%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5}} \cdot \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}}\right)}{2 \cdot w} \]

      3. pow1/2 2.8%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5} \cdot \color{blue}{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5}}}\right)}{2 \cdot w} \]

      4. pow-prod-down 5.3%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{{\left(\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right) \cdot \left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)\right)}^{0.5}}}\right)}{2 \cdot w} \]

   9. Applied egg-rr 39.0%

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

       1. unpow1/2 39.0%

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

       2. div0 39.0%

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

   11. Simplified 39.0%

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

   12. Taylor expanded in M around 0 41.3%

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

   if 8.59999999999999975e24 < d < 4.9000000000000001e117

   1. Initial program 14.0%

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

   3. Simplified 23.0%

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

   5. Taylor expanded in c0 around -inf 9.1%

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

      1. associate-/l* 9.1%

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

      2. distribute-lft1-in 9.1%

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

      3. metadata-eval 9.1%

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

      4. associate-/r* 4.7%

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

   7. Simplified 4.7%

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

      1. associate-*r/ 4.7%

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

      2. mul0-lft 53.9%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot \color{blue}{0}}{w}\right) \]

   9. Applied egg-rr 53.9%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 8.6 \cdot 10^{+24} \lor \neg \left(d \leq 4.9 \cdot 10^{+117}\right):\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot 0}{w}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.3% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 4.8 \cdot 10^{+142}:\\ \;\;\;\;c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot 0}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(M \cdot \frac{0.5}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 4.8e+142) (* c0 (* -0.5 (/ (* c0 0.0) w))) (* c0 (* M (/ 0.5 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 4.8e+142) {
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w));
	} else {
		tmp = c0 * (M * (0.5 / w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 4.8d+142) then
        tmp = c0 * ((-0.5d0) * ((c0 * 0.0d0) / w))
    else
        tmp = c0 * (m * (0.5d0 / w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 4.8e+142) {
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w));
	} else {
		tmp = c0 * (M * (0.5 / w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 4.8e+142:
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w))
	else:
		tmp = c0 * (M * (0.5 / w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 4.8e+142)
		tmp = Float64(c0 * Float64(-0.5 * Float64(Float64(c0 * 0.0) / w)));
	else
		tmp = Float64(c0 * Float64(M * Float64(0.5 / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 4.8e+142)
		tmp = c0 * (-0.5 * ((c0 * 0.0) / w));
	else
		tmp = c0 * (M * (0.5 / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 4.8e+142], N[(c0 * N[(-0.5 * N[(N[(c0 * 0.0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(M * N[(0.5 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.7999999999999998e142

   1. Initial program 25.7%

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

   3. Simplified 35.7%

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

   5. Taylor expanded in c0 around -inf 2.6%

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

      1. associate-/l* 2.6%

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

      2. distribute-lft1-in 2.6%

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

      3. metadata-eval 2.6%

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

      4. associate-/r* 2.1%

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

   7. Simplified 2.1%

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

      1. associate-*r/ 2.1%

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

      2. mul0-lft 31.5%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot \color{blue}{0}}{w}\right) \]

   9. Applied egg-rr 31.5%

      \[\leadsto c0 \cdot \left(-0.5 \cdot \color{blue}{\frac{c0 \cdot 0}{w}}\right) \]

   if 4.7999999999999998e142 < 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 37.2%

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

   5. Taylor expanded in w around -inf 0.0%

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

      1. distribute-lft-out 0.0%

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

      2. distribute-lft1-in 0.0%

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

      3. metadata-eval 0.0%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{-1 \cdot \left(\frac{\color{blue}{0} \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}\right)}{2 \cdot w} \]

      4. associate-*r* 0.0%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\color{blue}{\left(M \cdot c0\right) \cdot {d}^{2}}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}\right)}{2 \cdot w} \]

   7. Simplified 0.0%

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

      1. add-sqr-sqrt 0.0%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{\sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)} \cdot \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}}}\right)}{2 \cdot w} \]

      2. pow1/2 0.0%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5}} \cdot \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}}\right)}{2 \cdot w} \]

      3. pow1/2 0.0%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5} \cdot \color{blue}{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5}}}\right)}{2 \cdot w} \]

      4. pow-prod-down 0.0%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{{\left(\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right) \cdot \left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)\right)}^{0.5}}}\right)}{2 \cdot w} \]

   9. Applied egg-rr 37.2%

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

       1. unpow1/2 37.2%

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

       2. div0 37.2%

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

   11. Simplified 37.2%

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

   12. Taylor expanded in c0 around 0 28.4%

       \[\leadsto c0 \cdot \color{blue}{\left(0.5 \cdot \frac{M}{w}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 28.4%

          \[\leadsto c0 \cdot \color{blue}{\frac{0.5 \cdot M}{w}} \]

       2. *-commutative 28.4%

          \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot 0.5}}{w} \]

       3. associate-*r/ 28.4%

          \[\leadsto c0 \cdot \color{blue}{\left(M \cdot \frac{0.5}{w}\right)} \]

   14. Simplified 28.4%

       \[\leadsto c0 \cdot \color{blue}{\left(M \cdot \frac{0.5}{w}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 18.7% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ c0 \cdot \left(M \cdot \frac{0.5}{w}\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* c0 (* M (/ 0.5 w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (M * (0.5 / 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 * (m * (0.5d0 / w))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (M * (0.5 / w));
}

Python

def code(c0, w, h, D, d, M):
	return c0 * (M * (0.5 / w))

Julia

function code(c0, w, h, D, d, M)
	return Float64(c0 * Float64(M * Float64(0.5 / w)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = c0 * (M * (0.5 / w));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(c0 * N[(M * N[(0.5 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.7%

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

3. Simplified 35.9%

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

5. Taylor expanded in w around -inf 2.6%

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

   1. distribute-lft-out 2.6%

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

   2. distribute-lft1-in 2.6%

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

   3. metadata-eval 2.6%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{-1 \cdot \left(\frac{\color{blue}{0} \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}\right)}{2 \cdot w} \]

   4. associate-*r* 3.0%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\color{blue}{\left(M \cdot c0\right) \cdot {d}^{2}}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}\right)}{2 \cdot w} \]

7. Simplified 3.0%

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

   1. add-sqr-sqrt 3.0%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{\sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)} \cdot \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}}}\right)}{2 \cdot w} \]

   2. pow1/2 3.0%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5}} \cdot \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}}\right)}{2 \cdot w} \]

   3. pow1/2 3.0%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5} \cdot \color{blue}{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5}}}\right)}{2 \cdot w} \]

   4. pow-prod-down 5.3%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{{\left(\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right) \cdot \left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)\right)}^{0.5}}}\right)}{2 \cdot w} \]

9. Applied egg-rr 38.0%

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

    1. unpow1/2 38.0%

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

    2. div0 38.0%

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

11. Simplified 38.0%

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

12. Taylor expanded in c0 around 0 15.6%

    \[\leadsto c0 \cdot \color{blue}{\left(0.5 \cdot \frac{M}{w}\right)} \]
13. Step-by-step derivation

    1. associate-*r/ 15.6%

       \[\leadsto c0 \cdot \color{blue}{\frac{0.5 \cdot M}{w}} \]

    2. *-commutative 15.6%

       \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot 0.5}}{w} \]

    3. associate-*r/ 15.6%

       \[\leadsto c0 \cdot \color{blue}{\left(M \cdot \frac{0.5}{w}\right)} \]

14. Simplified 15.6%

    \[\leadsto c0 \cdot \color{blue}{\left(M \cdot \frac{0.5}{w}\right)} \]
15. Add Preprocessing

Alternative 13: 18.2% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ M \cdot \frac{c0 \cdot 0.5}{w} \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* M (/ (* c0 0.5) w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return M * ((c0 * 0.5) / 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 = m * ((c0 * 0.5d0) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return M * ((c0 * 0.5) / w);
}

Python

def code(c0, w, h, D, d, M):
	return M * ((c0 * 0.5) / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(M * Float64(Float64(c0 * 0.5) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = M * ((c0 * 0.5) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(M * N[(N[(c0 * 0.5), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.7%

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

3. Simplified 35.9%

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

5. Taylor expanded in w around -inf 2.6%

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

   1. distribute-lft-out 2.6%

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

   2. distribute-lft1-in 2.6%

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

   3. metadata-eval 2.6%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{-1 \cdot \left(\frac{\color{blue}{0} \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}\right)}{2 \cdot w} \]

   4. associate-*r* 3.0%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\color{blue}{\left(M \cdot c0\right) \cdot {d}^{2}}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}\right)}{2 \cdot w} \]

7. Simplified 3.0%

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

   1. add-sqr-sqrt 3.0%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{\sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)} \cdot \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}}}\right)}{2 \cdot w} \]

   2. pow1/2 3.0%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5}} \cdot \sqrt{-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)}}\right)}{2 \cdot w} \]

   3. pow1/2 3.0%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5} \cdot \color{blue}{{\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)}^{0.5}}}\right)}{2 \cdot w} \]

   4. pow-prod-down 5.3%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \sqrt{\color{blue}{{\left(\left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right) \cdot \left(-1 \cdot \left(\frac{0 \cdot \frac{\left(M \cdot c0\right) \cdot {d}^{2}}{{D}^{2} \cdot h}}{w} + {M}^{2}\right)\right)\right)}^{0.5}}}\right)}{2 \cdot w} \]

9. Applied egg-rr 38.0%

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

    1. unpow1/2 38.0%

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

    2. div0 38.0%

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

11. Simplified 38.0%

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

12. Taylor expanded in c0 around 0 13.9%

    \[\leadsto \color{blue}{0.5 \cdot \frac{M \cdot c0}{w}} \]
13. Step-by-step derivation

    1. *-commutative 13.9%

       \[\leadsto \color{blue}{\frac{M \cdot c0}{w} \cdot 0.5} \]

    2. associate-/l* 13.8%

       \[\leadsto \color{blue}{\left(M \cdot \frac{c0}{w}\right)} \cdot 0.5 \]

    3. associate-*r* 13.8%

       \[\leadsto \color{blue}{M \cdot \left(\frac{c0}{w} \cdot 0.5\right)} \]

    4. associate-*l/ 13.8%

       \[\leadsto M \cdot \color{blue}{\frac{c0 \cdot 0.5}{w}} \]

14. Simplified 13.8%

    \[\leadsto \color{blue}{M \cdot \frac{c0 \cdot 0.5}{w}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024113 
(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))))))