Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.8% → 39.2%

Time: 15.5s

Alternatives: 9

Speedup: 3.3×

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: 24.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

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

Alternative 1: 39.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{w + w}\\ t_1 := {\left(\frac{d}{D}\right)}^{2}\\ t_2 := \frac{t\_1}{w}\\ t_3 := \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, t\_1, M\_m\right)}, \sqrt{\mathsf{fma}\left(t\_2, \frac{c0}{h}, -M\_m\right)}, t\_2 \cdot \frac{c0}{h}\right)\\ \mathbf{if}\;c0 \leq -5 \cdot 10^{-135}:\\ \;\;\;\;\left(\left(0.5 \cdot c0\right) \cdot {w}^{-1}\right) \cdot t\_3\\ \mathbf{elif}\;c0 \leq 7.5 \cdot 10^{-133}:\\ \;\;\;\;t\_0 \cdot \left({1}^{0.25} \cdot M\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_3\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (+ w w)))
        (t_1 (pow (/ d D) 2.0))
        (t_2 (/ t_1 w))
        (t_3
         (fma
          (sqrt (fma (/ (/ c0 h) w) t_1 M_m))
          (sqrt (fma t_2 (/ c0 h) (- M_m)))
          (* t_2 (/ c0 h)))))
   (if (<= c0 -5e-135)
     (* (* (* 0.5 c0) (pow w -1.0)) t_3)
     (if (<= c0 7.5e-133) (* t_0 (* (pow 1.0 0.25) M_m)) (* t_0 t_3)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (w + w);
	double t_1 = pow((d / D), 2.0);
	double t_2 = t_1 / w;
	double t_3 = fma(sqrt(fma(((c0 / h) / w), t_1, M_m)), sqrt(fma(t_2, (c0 / h), -M_m)), (t_2 * (c0 / h)));
	double tmp;
	if (c0 <= -5e-135) {
		tmp = ((0.5 * c0) * pow(w, -1.0)) * t_3;
	} else if (c0 <= 7.5e-133) {
		tmp = t_0 * (pow(1.0, 0.25) * M_m);
	} else {
		tmp = t_0 * t_3;
	}
	return tmp;
}

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 / Float64(w + w))
	t_1 = Float64(d / D) ^ 2.0
	t_2 = Float64(t_1 / w)
	t_3 = fma(sqrt(fma(Float64(Float64(c0 / h) / w), t_1, M_m)), sqrt(fma(t_2, Float64(c0 / h), Float64(-M_m))), Float64(t_2 * Float64(c0 / h)))
	tmp = 0.0
	if (c0 <= -5e-135)
		tmp = Float64(Float64(Float64(0.5 * c0) * (w ^ -1.0)) * t_3);
	elseif (c0 <= 7.5e-133)
		tmp = Float64(t_0 * Float64((1.0 ^ 0.25) * M_m));
	else
		tmp = Float64(t_0 * t_3);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / w), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[(c0 / h), $MachinePrecision] / w), $MachinePrecision] * t$95$1 + M$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$2 * N[(c0 / h), $MachinePrecision] + (-M$95$m)), $MachinePrecision]], $MachinePrecision] + N[(t$95$2 * N[(c0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -5e-135], N[(N[(N[(0.5 * c0), $MachinePrecision] * N[Power[w, -1.0], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[c0, 7.5e-133], N[(t$95$0 * N[(N[Power[1.0, 0.25], $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -5.0000000000000002e-135

   1. Initial program 30.6%

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

   4. Applied rewrites 46.1%

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

   6. Applied rewrites 48.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 48.4

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

   8. Applied rewrites 48.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 48.4

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

   10. Applied rewrites 48.4%

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

   if -5.0000000000000002e-135 < c0 < 7.4999999999999999e-133

   1. Initial program 11.9%

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

   3. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 0.0

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

   5. Applied rewrites 0.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt{-1} \cdot M\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 0.0

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

   7. Applied rewrites 0.0%

      \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 40.9%

      \[\leadsto \frac{c0}{w + w} \cdot \left({1}^{0.25} \cdot M\right) \]

   if 7.4999999999999999e-133 < c0

   1. Initial program 16.2%

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

   4. Applied rewrites 30.2%

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

   6. Applied rewrites 32.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 32.6

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

   8. Applied rewrites 32.6%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -5 \cdot 10^{-135}:\\ \;\;\;\;\left(\left(0.5 \cdot c0\right) \cdot {w}^{-1}\right) \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, M\right)}, \sqrt{\mathsf{fma}\left(\frac{{\left(\frac{d}{D}\right)}^{2}}{w}, \frac{c0}{h}, -M\right)}, \frac{{\left(\frac{d}{D}\right)}^{2}}{w} \cdot \frac{c0}{h}\right)\\ \mathbf{elif}\;c0 \leq 7.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \left({1}^{0.25} \cdot M\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, M\right)}, \sqrt{\mathsf{fma}\left(\frac{{\left(\frac{d}{D}\right)}^{2}}{w}, \frac{c0}{h}, -M\right)}, \frac{{\left(\frac{d}{D}\right)}^{2}}{w} \cdot \frac{c0}{h}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 39.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ t_1 := \frac{t\_0}{w}\\ t_2 := \frac{c0}{w + w}\\ \mathbf{if}\;c0 \leq -5 \cdot 10^{-135} \lor \neg \left(c0 \leq 7.5 \cdot 10^{-133}\right):\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, t\_0, M\_m\right)}, \sqrt{\mathsf{fma}\left(t\_1, \frac{c0}{h}, -M\_m\right)}, t\_1 \cdot \frac{c0}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left({1}^{0.25} \cdot M\_m\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (pow (/ d D) 2.0)) (t_1 (/ t_0 w)) (t_2 (/ c0 (+ w w))))
   (if (or (<= c0 -5e-135) (not (<= c0 7.5e-133)))
     (*
      t_2
      (fma
       (sqrt (fma (/ (/ c0 h) w) t_0 M_m))
       (sqrt (fma t_1 (/ c0 h) (- M_m)))
       (* t_1 (/ c0 h))))
     (* t_2 (* (pow 1.0 0.25) M_m)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = pow((d / D), 2.0);
	double t_1 = t_0 / w;
	double t_2 = c0 / (w + w);
	double tmp;
	if ((c0 <= -5e-135) || !(c0 <= 7.5e-133)) {
		tmp = t_2 * fma(sqrt(fma(((c0 / h) / w), t_0, M_m)), sqrt(fma(t_1, (c0 / h), -M_m)), (t_1 * (c0 / h)));
	} else {
		tmp = t_2 * (pow(1.0, 0.25) * M_m);
	}
	return tmp;
}

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(d / D) ^ 2.0
	t_1 = Float64(t_0 / w)
	t_2 = Float64(c0 / Float64(w + w))
	tmp = 0.0
	if ((c0 <= -5e-135) || !(c0 <= 7.5e-133))
		tmp = Float64(t_2 * fma(sqrt(fma(Float64(Float64(c0 / h) / w), t_0, M_m)), sqrt(fma(t_1, Float64(c0 / h), Float64(-M_m))), Float64(t_1 * Float64(c0 / h))));
	else
		tmp = Float64(t_2 * Float64((1.0 ^ 0.25) * M_m));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / w), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[c0, -5e-135], N[Not[LessEqual[c0, 7.5e-133]], $MachinePrecision]], N[(t$95$2 * N[(N[Sqrt[N[(N[(N[(c0 / h), $MachinePrecision] / w), $MachinePrecision] * t$95$0 + M$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$1 * N[(c0 / h), $MachinePrecision] + (-M$95$m)), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 * N[(c0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[Power[1.0, 0.25], $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if c0 < -5.0000000000000002e-135 or 7.4999999999999999e-133 < c0

   1. Initial program 23.8%

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

   4. Applied rewrites 38.7%

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

   6. Applied rewrites 41.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 41.0

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

   8. Applied rewrites 41.0%

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

   if -5.0000000000000002e-135 < c0 < 7.4999999999999999e-133

   1. Initial program 11.9%

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

   3. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 0.0

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

   5. Applied rewrites 0.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt{-1} \cdot M\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 0.0

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

   7. Applied rewrites 0.0%

      \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 40.9%

      \[\leadsto \frac{c0}{w + w} \cdot \left({1}^{0.25} \cdot M\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -5 \cdot 10^{-135} \lor \neg \left(c0 \leq 7.5 \cdot 10^{-133}\right):\\ \;\;\;\;\frac{c0}{w + w} \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, {\left(\frac{d}{D}\right)}^{2}, M\right)}, \sqrt{\mathsf{fma}\left(\frac{{\left(\frac{d}{D}\right)}^{2}}{w}, \frac{c0}{h}, -M\right)}, \frac{{\left(\frac{d}{D}\right)}^{2}}{w} \cdot \frac{c0}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \left({1}^{0.25} \cdot M\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 40.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \left({1}^{0.25} \cdot M\_m\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m))))) INFINITY)
     (* t_0 (/ (* 2.0 (* (* d d) c0)) (* (* (* D D) h) w)))
     (* (/ c0 (+ w w)) (* (pow 1.0 0.25) M_m)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_0 * ((2.0 * ((d * d) * c0)) / (((D * D) * h) * w));
	} else {
		tmp = (c0 / (w + w)) * (pow(1.0, 0.25) * M_m);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((2.0 * ((d * d) * c0)) / (((D * D) * h) * w));
	} else {
		tmp = (c0 / (w + w)) * (Math.pow(1.0, 0.25) * M_m);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= math.inf:
		tmp = t_0 * ((2.0 * ((d * d) * c0)) / (((D * D) * h) * w))
	else:
		tmp = (c0 / (w + w)) * (math.pow(1.0, 0.25) * M_m)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(t_0 * Float64(Float64(2.0 * Float64(Float64(d * d) * c0)) / Float64(Float64(Float64(D * D) * h) * w)));
	else
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64((1.0 ^ 0.25) * M_m));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Inf)
		tmp = t_0 * ((2.0 * ((d * d) * c0)) / (((D * D) * h) * w));
	else
		tmp = (c0 / (w + w)) * ((1.0 ^ 0.25) * M_m);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(2.0 * N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[Power[1.0, 0.25], $MachinePrecision] * M$95$m), $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 70.1%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 71.2

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

   5. Applied rewrites 71.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 0.0

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

   5. Applied rewrites 0.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt{-1} \cdot M\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 0.0

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

   7. Applied rewrites 0.0%

      \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 21.2%

      \[\leadsto \frac{c0}{w + w} \cdot \left({1}^{0.25} \cdot M\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 34.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.96 \cdot 10^{-195}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\left(-M\_m\right) \cdot M\_m}, c0, c0 \cdot \left(\left(d \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot c0\right)\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 1.96e-195)
   (/
    (fma
     (sqrt (* (- M_m) M_m))
     c0
     (* c0 (* (* d (/ d (* (* (* D D) w) h))) c0)))
    (* 2.0 w))
   (* (/ c0 (* 2.0 w)) (/ (* 2.0 (* (* d d) c0)) (* (* (* D D) h) w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 1.96e-195) {
		tmp = fma(sqrt((-M_m * M_m)), c0, (c0 * ((d * (d / (((D * D) * w) * h))) * c0))) / (2.0 * w);
	} else {
		tmp = (c0 / (2.0 * w)) * ((2.0 * ((d * d) * c0)) / (((D * D) * h) * w));
	}
	return tmp;
}

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 1.96e-195)
		tmp = Float64(fma(sqrt(Float64(Float64(-M_m) * M_m)), c0, Float64(c0 * Float64(Float64(d * Float64(d / Float64(Float64(Float64(D * D) * w) * h))) * c0))) / Float64(2.0 * w));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(2.0 * Float64(Float64(d * d) * c0)) / Float64(Float64(Float64(D * D) * h) * w)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 1.96e-195], N[(N[(N[Sqrt[N[((-M$95$m) * M$95$m), $MachinePrecision]], $MachinePrecision] * c0 + N[(c0 * N[(N[(d * N[(d / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.96e-195

   1. Initial program 23.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

   4. Applied rewrites 34.0%

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

   5. Taylor expanded in c0 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 11.4

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

   7. Applied rewrites 11.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 16.0

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

   9. Applied rewrites 16.0%

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

   if 1.96e-195 < M

   1. Initial program 16.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 29.2

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

   5. Applied rewrites 29.2%

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

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.96 \cdot 10^{-195}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\left(-M\right) \cdot M}, c0, c0 \cdot \left(\left(d \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right) \cdot c0\right)\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 33.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    code = (c0 / (2.0d0 * w)) * ((2.0d0 * ((d_1 * d_1) * c0)) / (((d * d) * h) * w))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.4%

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

3. Taylor expanded in c0 around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 29.6

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

5. Applied rewrites 29.6%

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

Alternative 6: 29.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \frac{\frac{c0 \cdot c0}{D \cdot D} \cdot \frac{d \cdot d}{h \cdot w}}{w} \end{array} \]

FPCore

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

C

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

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    code = (((c0 * c0) / (d * d)) * ((d_1 * d_1) / (h * w))) / w
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := N[(N[(N[(N[(c0 * c0), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]

Derivation

1. Initial program 21.4%

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

4. Applied rewrites 33.3%

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

5. Taylor expanded in c0 around inf

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

   1. times-frac N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 24.1

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

7. Applied rewrites 24.1%

   \[\leadsto \frac{\color{blue}{\frac{c0 \cdot c0}{D \cdot D} \cdot \frac{d \cdot d}{h \cdot w}}}{w} \]
8. Add Preprocessing

Alternative 7: 25.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    code = (c0 * c0) * ((d_1 * d_1) / (((d * d) * h) * (w * w)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := N[(N[(c0 * c0), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.4%

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

3. Taylor expanded in c0 around inf

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 23.5

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

5. Applied rewrites 23.5%

   \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}} \]
6. Add Preprocessing

Alternative 8: 2.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ -4 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot h} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (* -4.0 (/ (* (* c0 c0) (* d d)) (* (* D D) h))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	return -4.0 * (((c0 * c0) * (d * d)) / ((D * D) * h));
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    code = (-4.0d0) * (((c0 * c0) * (d_1 * d_1)) / ((d * d) * h))
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	return -4.0 * (((c0 * c0) * (d * d)) / ((D * D) * h));
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	return -4.0 * (((c0 * c0) * (d * d)) / ((D * D) * h))

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	return Float64(-4.0 * Float64(Float64(Float64(c0 * c0) * Float64(d * d)) / Float64(Float64(D * D) * h)))
end

MATLAB

M_m = abs(M);
function tmp = code(c0, w, h, D, d, M_m)
	tmp = -4.0 * (((c0 * c0) * (d * d)) / ((D * D) * h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := N[(-4.0 * N[(N[(N[(c0 * c0), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.4%

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

3. Taylor expanded in c0 around 0

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 0.0

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

5. Applied rewrites 0.0%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt{-1} \cdot M\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 0.0

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

7. Applied rewrites 0.0%

   \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]
8. Step-by-step derivation

9. Applied rewrites 0.0%

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

10. Taylor expanded in c0 around inf

    \[\leadsto \color{blue}{-4 \cdot \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot h}} \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \color{blue}{-4 \cdot \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot h}} \]

    2. lower-/.f64 N/A

       \[\leadsto -4 \cdot \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot h}} \]

    3. lower-*.f64 N/A

       \[\leadsto -4 \cdot \frac{\color{blue}{{c0}^{2} \cdot {d}^{2}}}{{D}^{2} \cdot h} \]

    4. unpow2 N/A

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

    5. lower-*.f64 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 1.8

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

12. Applied rewrites 1.8%

    \[\leadsto \color{blue}{-4 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot h}} \]
13. Add Preprocessing

Alternative 9: 0.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \left(\left(-2 \cdot w\right) \cdot c0\right) \cdot \left(\sqrt{-1} \cdot M\_m\right) \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (* (* (* -2.0 w) c0) (* (sqrt -1.0) M_m)))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	return ((-2.0 * w) * c0) * (sqrt(-1.0) * M_m);
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    code = (((-2.0d0) * w) * c0) * (sqrt((-1.0d0)) * m_m)
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	return ((-2.0 * w) * c0) * (Math.sqrt(-1.0) * M_m);
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	return ((-2.0 * w) * c0) * (math.sqrt(-1.0) * M_m)

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	return Float64(Float64(Float64(-2.0 * w) * c0) * Float64(sqrt(-1.0) * M_m))
end

MATLAB

M_m = abs(M);
function tmp = code(c0, w, h, D, d, M_m)
	tmp = ((-2.0 * w) * c0) * (sqrt(-1.0) * M_m);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := N[(N[(N[(-2.0 * w), $MachinePrecision] * c0), $MachinePrecision] * N[(N[Sqrt[-1.0], $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.4%

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

3. Taylor expanded in c0 around 0

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 0.0

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

5. Applied rewrites 0.0%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt{-1} \cdot M\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 0.0

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

7. Applied rewrites 0.0%

   \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]
8. Step-by-step derivation

9. Applied rewrites 0.0%

   \[\leadsto \color{blue}{\left(\left(-2 \cdot w\right) \cdot c0\right) \cdot \left(\sqrt{-1} \cdot M\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024305 
(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))))))