Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A

Percentage Accurate: 76.8% → 99.5%

Time: 12.3s

Alternatives: 11

Speedup: 1.5×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\left(\frac{8}{3} \cdot t\_0\right) \cdot t\_0}{\sin x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / sin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = (((8.0d0 / 3.0d0) * t_0) * t_0) / sin(x)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / Math.sin(x);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return (((8.0 / 3.0) * t_0) * t_0) / math.sin(x)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = (((8.0 / 3.0) * t_0) * t_0) / sin(x);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Initial Program: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\left(\frac{8}{3} \cdot t\_0\right) \cdot t\_0}{\sin x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / sin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = (((8.0d0 / 3.0d0) * t_0) * t_0) / sin(x)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / Math.sin(x);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return (((8.0 / 3.0) * t_0) * t_0) / math.sin(x)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = (((8.0 / 3.0) * t_0) * t_0) / sin(x);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x\_m}{\frac{1}{\mathsf{fma}\left(x\_m, x\_m \cdot 0.05555555555555555, 0.6666666666666666\right)}}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(0.5 \cdot x\_m\right)}^{2}}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 4e-7)
    (/ x_m (/ 1.0 (fma x_m (* x_m 0.05555555555555555) 0.6666666666666666)))
    (* 2.6666666666666665 (/ (pow (sin (* 0.5 x_m)) 2.0) (sin x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 4e-7) {
		tmp = x_m / (1.0 / fma(x_m, (x_m * 0.05555555555555555), 0.6666666666666666));
	} else {
		tmp = 2.6666666666666665 * (pow(sin((0.5 * x_m)), 2.0) / sin(x_m));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 4e-7)
		tmp = Float64(x_m / Float64(1.0 / fma(x_m, Float64(x_m * 0.05555555555555555), 0.6666666666666666)));
	else
		tmp = Float64(2.6666666666666665 * Float64((sin(Float64(0.5 * x_m)) ^ 2.0) / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 4e-7], N[(x$95$m / N[(1.0 / N[(x$95$m * N[(x$95$m * 0.05555555555555555), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999998e-7

   1. Initial program 66.6%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + \frac{1}{18} \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + \frac{1}{18} \cdot {x}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{18} \cdot {x}^{2} + \frac{2}{3}\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{18}, {x}^{2}, \frac{2}{3}\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\frac{1}{18}, \color{blue}{x \cdot x}, \frac{2}{3}\right) \]

      5. *-lowering-*.f64 65.5

         \[\leadsto x \cdot \mathsf{fma}\left(0.05555555555555555, \color{blue}{x \cdot x}, 0.6666666666666666\right) \]

   5. Simplified 65.5%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.05555555555555555, x \cdot x, 0.6666666666666666\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto x \cdot \color{blue}{\frac{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}}}} \]

      6. flip-+ N/A

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{\frac{1}{18} \cdot \left(x \cdot x\right) + \frac{2}{3}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{1}{18} \cdot \left(x \cdot x\right) + \frac{2}{3}}}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{\left(\frac{1}{18} \cdot x\right) \cdot x} + \frac{2}{3}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{x \cdot \left(\frac{1}{18} \cdot x\right)} + \frac{2}{3}}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\frac{1}{\color{blue}{\mathsf{fma}\left(x, \frac{1}{18} \cdot x, \frac{2}{3}\right)}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{x}{\frac{1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{18}}, \frac{2}{3}\right)}} \]

      12. *-lowering-*.f64 65.7

          \[\leadsto \frac{x}{\frac{1}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.05555555555555555}, 0.6666666666666666\right)}} \]

   7. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(x, x \cdot 0.05555555555555555, 0.6666666666666666\right)}}} \]

   if 3.9999999999999998e-7 < x

   1. Initial program 99.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{8}{3} \cdot \frac{{\sin \left(\frac{1}{2} \cdot x\right)}^{2}}{\sin x}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{8}{3} \cdot \frac{{\sin \left(\frac{1}{2} \cdot x\right)}^{2}}{\sin x}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{8}{3} \cdot \color{blue}{\frac{{\sin \left(\frac{1}{2} \cdot x\right)}^{2}}{\sin x}} \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{8}{3} \cdot \frac{\color{blue}{{\sin \left(\frac{1}{2} \cdot x\right)}^{2}}}{\sin x} \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{8}{3} \cdot \frac{{\color{blue}{\sin \left(\frac{1}{2} \cdot x\right)}}^{2}}{\sin x} \]

      5. *-commutative N/A

         \[\leadsto \frac{8}{3} \cdot \frac{{\sin \color{blue}{\left(x \cdot \frac{1}{2}\right)}}^{2}}{\sin x} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{8}{3} \cdot \frac{{\sin \color{blue}{\left(x \cdot \frac{1}{2}\right)}}^{2}}{\sin x} \]

      7. sin-lowering-sin.f64 98.9

         \[\leadsto 2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\color{blue}{\sin x}} \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\frac{1}{\mathsf{fma}\left(x, x \cdot 0.05555555555555555, 0.6666666666666666\right)}}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(0.5 \cdot x\right)}^{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot x\_m\right)\\ x\_s \cdot \left(\frac{t\_0}{\sin x\_m} \cdot \frac{t\_0}{0.375}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 x_m)))) (* x_s (* (/ t_0 (sin x_m)) (/ t_0 0.375)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((0.5 * x_m));
	return x_s * ((t_0 / sin(x_m)) * (t_0 / 0.375));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    t_0 = sin((0.5d0 * x_m))
    code = x_s * ((t_0 / sin(x_m)) * (t_0 / 0.375d0))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin((0.5 * x_m));
	return x_s * ((t_0 / Math.sin(x_m)) * (t_0 / 0.375));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin((0.5 * x_m))
	return x_s * ((t_0 / math.sin(x_m)) * (t_0 / 0.375))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = sin(Float64(0.5 * x_m))
	return Float64(x_s * Float64(Float64(t_0 / sin(x_m)) * Float64(t_0 / 0.375)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	t_0 = sin((0.5 * x_m));
	tmp = x_s * ((t_0 / sin(x_m)) * (t_0 / 0.375));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * N[(N[(t$95$0 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.2%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   2. associate-*l* N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin x}{\frac{8}{3}}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}}} \]

   5. frac-2neg N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\frac{\mathsf{neg}\left(\sin x\right)}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   6. neg-mul-1 N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\color{blue}{-1 \cdot \sin x}}{\mathsf{neg}\left(\frac{8}{3}\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\color{blue}{\sin x \cdot -1}}{\mathsf{neg}\left(\frac{8}{3}\right)}} \]

   8. associate-/l* N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x \cdot \frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   9. times-frac N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \frac{\sin \left(0.5 \cdot x\right)}{0.375}} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot x\_m\right)\\ x\_s \cdot \left(t\_0 \cdot \frac{\mathsf{fma}\left(t\_0, -2.6666666666666665, 0\right)}{0 - \sin x\_m}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 x_m))))
   (* x_s (* t_0 (/ (fma t_0 -2.6666666666666665 0.0) (- 0.0 (sin x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((0.5 * x_m));
	return x_s * (t_0 * (fma(t_0, -2.6666666666666665, 0.0) / (0.0 - sin(x_m))));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = sin(Float64(0.5 * x_m))
	return Float64(x_s * Float64(t_0 * Float64(fma(t_0, -2.6666666666666665, 0.0) / Float64(0.0 - sin(x_m)))))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * N[(t$95$0 * N[(N[(t$95$0 * -2.6666666666666665 + 0.0), $MachinePrecision] / N[(0.0 - N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.2%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   2. associate-*l* N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin x}{\frac{8}{3}}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}}} \]

   5. frac-2neg N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\frac{\mathsf{neg}\left(\sin x\right)}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   6. neg-mul-1 N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\color{blue}{-1 \cdot \sin x}}{\mathsf{neg}\left(\frac{8}{3}\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\color{blue}{\sin x \cdot -1}}{\mathsf{neg}\left(\frac{8}{3}\right)}} \]

   8. associate-/l* N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x \cdot \frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   9. times-frac N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \frac{\sin \left(0.5 \cdot x\right)}{0.375}} \]
5. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{1}{2} \cdot x\right)}{\sin x} \cdot \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\frac{3}{8}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{1}{2} \cdot x\right)}{\sin x}} \cdot \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\frac{3}{8}} \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\frac{1}{2} \cdot x\right)}}{\sin x} \cdot \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\frac{3}{8}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\frac{1}{2} \cdot x\right)}}{\sin x} \cdot \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\frac{3}{8}} \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{\sin x}} \cdot \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\frac{3}{8}} \]

   6. div-inv N/A

      \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\sin x} \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{1}{\frac{3}{8}}\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\sin x} \cdot \left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\frac{8}{3}}\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\sin x} \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}\right)} \]

   9. sin-lowering-sin.f64 N/A

      \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\sin x} \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot x\right)} \cdot \frac{8}{3}\right) \]

   10. *-lowering-*.f64 99.3

       \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \left(\sin \color{blue}{\left(0.5 \cdot x\right)} \cdot 2.6666666666666665\right) \]

6. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \left(\sin \left(0.5 \cdot x\right) \cdot 2.6666666666666665\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}\right) \cdot \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\sin x}} \]

   2. clear-num N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{\sin \left(\frac{1}{2} \cdot x\right)}}} \]

   3. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(\frac{1}{2} \cdot x\right)}}} \]

   4. frac-2neg N/A

      \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}}{\color{blue}{\frac{\mathsf{neg}\left(\sin x\right)}{\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot x\right)\right)}}} \]

   5. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}}{\mathsf{neg}\left(\sin x\right)} \cdot \left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot x\right)\right)\right)} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}}{\mathsf{neg}\left(\sin x\right)} \cdot \left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot x\right)\right)\right)} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}}{\mathsf{neg}\left(\sin x\right)}} \cdot \left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot x\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}}}{\mathsf{neg}\left(\sin x\right)} \cdot \left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot x\right)\right)\right) \]

   9. sin-lowering-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\frac{1}{2} \cdot x\right)} \cdot \frac{8}{3}}{\mathsf{neg}\left(\sin x\right)} \cdot \left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot x\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\sin \color{blue}{\left(\frac{1}{2} \cdot x\right)} \cdot \frac{8}{3}}{\mathsf{neg}\left(\sin x\right)} \cdot \left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot x\right)\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}}{\color{blue}{0 - \sin x}} \cdot \left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot x\right)\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}}{\color{blue}{0 - \sin x}} \cdot \left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot x\right)\right)\right) \]

   13. sin-lowering-sin.f64 N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}}{0 - \color{blue}{\sin x}} \cdot \left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot x\right)\right)\right) \]

   14. neg-sub0 N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}}{0 - \sin x} \cdot \color{blue}{\left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right)} \]

   15. --lowering--.f64 N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}}{0 - \sin x} \cdot \color{blue}{\left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right)} \]

   16. sin-lowering-sin.f64 N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}}{0 - \sin x} \cdot \left(0 - \color{blue}{\sin \left(\frac{1}{2} \cdot x\right)}\right) \]

   17. *-lowering-*.f64 99.3

       \[\leadsto \frac{\sin \left(0.5 \cdot x\right) \cdot 2.6666666666666665}{0 - \sin x} \cdot \left(0 - \sin \color{blue}{\left(0.5 \cdot x\right)}\right) \]

8. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right) \cdot 2.6666666666666665}{0 - \sin x} \cdot \left(0 - \sin \left(0.5 \cdot x\right)\right)} \]
9. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}}{\color{blue}{\mathsf{neg}\left(\sin x\right)}} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   2. distribute-frac-neg2 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}}{\sin x}\right)\right)} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   3. distribute-frac-neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}\right)}{\sin x}} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}\right)}{\sin x}} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{8}{3} \cdot \sin \left(\frac{1}{2} \cdot x\right)}\right)}{\sin x} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{8}{3}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right)}}{\sin x} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   7. +-lft-identity N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{8}{3}\right)\right) \cdot \color{blue}{\left(0 + \sin \left(\frac{1}{2} \cdot x\right)\right)}}{\sin x} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{8}{3}\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot x\right) + 0\right)}}{\sin x} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   9. distribute-rgt-in N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\frac{1}{2} \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{8}{3}\right)\right) + 0 \cdot \left(\mathsf{neg}\left(\frac{8}{3}\right)\right)}}{\sin x} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{8}{3}\right)\right) + 0 \cdot \color{blue}{\frac{-8}{3}}}{\sin x} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{8}{3}\right)\right) + \color{blue}{0}}{\sin x} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot x\right), \mathsf{neg}\left(\frac{8}{3}\right), 0\right)}}{\sin x} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   13. sin-lowering-sin.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin \left(\frac{1}{2} \cdot x\right)}, \mathsf{neg}\left(\frac{8}{3}\right), 0\right)}{\sin x} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot x\right)}, \mathsf{neg}\left(\frac{8}{3}\right), 0\right)}{\sin x} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot x\right), \color{blue}{\frac{-8}{3}}, 0\right)}{\sin x} \cdot \left(0 - \sin \left(\frac{1}{2} \cdot x\right)\right) \]

   16. sin-lowering-sin.f64 99.3

       \[\leadsto \frac{\mathsf{fma}\left(\sin \left(0.5 \cdot x\right), -2.6666666666666665, 0\right)}{\color{blue}{\sin x}} \cdot \left(0 - \sin \left(0.5 \cdot x\right)\right) \]

10. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin \left(0.5 \cdot x\right), -2.6666666666666665, 0\right)}{\sin x}} \cdot \left(0 - \sin \left(0.5 \cdot x\right)\right) \]

11. Final simplification 99.3%

    \[\leadsto \sin \left(0.5 \cdot x\right) \cdot \frac{\mathsf{fma}\left(\sin \left(0.5 \cdot x\right), -2.6666666666666665, 0\right)}{0 - \sin x} \]
12. Add Preprocessing

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot x\_m\right)\\ x\_s \cdot \left(\frac{t\_0}{\sin x\_m} \cdot \left(t\_0 \cdot 2.6666666666666665\right)\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 x_m))))
   (* x_s (* (/ t_0 (sin x_m)) (* t_0 2.6666666666666665)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((0.5 * x_m));
	return x_s * ((t_0 / sin(x_m)) * (t_0 * 2.6666666666666665));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    t_0 = sin((0.5d0 * x_m))
    code = x_s * ((t_0 / sin(x_m)) * (t_0 * 2.6666666666666665d0))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin((0.5 * x_m));
	return x_s * ((t_0 / Math.sin(x_m)) * (t_0 * 2.6666666666666665));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin((0.5 * x_m))
	return x_s * ((t_0 / math.sin(x_m)) * (t_0 * 2.6666666666666665))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = sin(Float64(0.5 * x_m))
	return Float64(x_s * Float64(Float64(t_0 / sin(x_m)) * Float64(t_0 * 2.6666666666666665)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	t_0 = sin((0.5 * x_m));
	tmp = x_s * ((t_0 / sin(x_m)) * (t_0 * 2.6666666666666665));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * N[(N[(t$95$0 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.2%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \]

4. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \left(2.6666666666666665 \cdot \sin \left(0.5 \cdot x\right)\right)} \]

5. Final simplification 99.3%

   \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \left(\sin \left(0.5 \cdot x\right) \cdot 2.6666666666666665\right) \]
6. Add Preprocessing

Alternative 5: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0045:\\ \;\;\;\;\frac{x\_m}{\frac{1}{\mathsf{fma}\left(x\_m, x\_m \cdot 0.05555555555555555, 0.6666666666666666\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\cos x\_m, -0.5, 0.5\right)}{0.375}}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0045)
    (/ x_m (/ 1.0 (fma x_m (* x_m 0.05555555555555555) 0.6666666666666666)))
    (/ (/ (fma (cos x_m) -0.5 0.5) 0.375) (sin x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0045) {
		tmp = x_m / (1.0 / fma(x_m, (x_m * 0.05555555555555555), 0.6666666666666666));
	} else {
		tmp = (fma(cos(x_m), -0.5, 0.5) / 0.375) / sin(x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0045)
		tmp = Float64(x_m / Float64(1.0 / fma(x_m, Float64(x_m * 0.05555555555555555), 0.6666666666666666)));
	else
		tmp = Float64(Float64(fma(cos(x_m), -0.5, 0.5) / 0.375) / sin(x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0045], N[(x$95$m / N[(1.0 / N[(x$95$m * N[(x$95$m * 0.05555555555555555), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x$95$m], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] / 0.375), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.00449999999999999966

   1. Initial program 66.6%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + \frac{1}{18} \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + \frac{1}{18} \cdot {x}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{18} \cdot {x}^{2} + \frac{2}{3}\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{18}, {x}^{2}, \frac{2}{3}\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\frac{1}{18}, \color{blue}{x \cdot x}, \frac{2}{3}\right) \]

      5. *-lowering-*.f64 65.5

         \[\leadsto x \cdot \mathsf{fma}\left(0.05555555555555555, \color{blue}{x \cdot x}, 0.6666666666666666\right) \]

   5. Simplified 65.5%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.05555555555555555, x \cdot x, 0.6666666666666666\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto x \cdot \color{blue}{\frac{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}}}} \]

      6. flip-+ N/A

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{\frac{1}{18} \cdot \left(x \cdot x\right) + \frac{2}{3}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{1}{18} \cdot \left(x \cdot x\right) + \frac{2}{3}}}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{\left(\frac{1}{18} \cdot x\right) \cdot x} + \frac{2}{3}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{x \cdot \left(\frac{1}{18} \cdot x\right)} + \frac{2}{3}}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\frac{1}{\color{blue}{\mathsf{fma}\left(x, \frac{1}{18} \cdot x, \frac{2}{3}\right)}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{x}{\frac{1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{18}}, \frac{2}{3}\right)}} \]

      12. *-lowering-*.f64 65.7

          \[\leadsto \frac{x}{\frac{1}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.05555555555555555}, 0.6666666666666666\right)}} \]

   7. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(x, x \cdot 0.05555555555555555, 0.6666666666666666\right)}}} \]

   if 0.00449999999999999966 < x

   1. Initial program 99.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. /-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{1}}}{\sin x} \]

      2. clear-num N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{1}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}}}{\sin x} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{1}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}}}{\sin x} \]

      4. frac-2neg N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}}}{\sin x} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{\frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}}{\sin x} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}\right)}}}{\sin x} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\frac{1}{\frac{-1}{\color{blue}{\left(\mathsf{neg}\left(\frac{8}{3}\right)\right) \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}}}{\sin x} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}}}{\sin x} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}}}{\sin x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{\frac{\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{8}{3}}\right)}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}}{\sin x} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{\frac{\frac{-1}{\color{blue}{\frac{-8}{3}}}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}}{\sin x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{\frac{\color{blue}{\frac{3}{8}}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}}{\sin x} \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{\frac{1}{\frac{\frac{3}{8}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)}}}}{\sin x} \]

      14. sub-neg N/A

          \[\leadsto \frac{\frac{1}{\frac{\frac{3}{8}}{\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)}}}}{\sin x} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{1}{\frac{\frac{3}{8}}{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) + \frac{1}{2}}}}}{\sin x} \]

   4. Applied egg-rr 98.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{0.375}{\mathsf{fma}\left(\cos x, -0.5, 0.5\right)}}}}{\sin x} \]
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\color{blue}{\frac{\cos x \cdot \frac{-1}{2} + \frac{1}{2}}{\frac{3}{8}}}}{\sin x} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\cos x \cdot \frac{-1}{2} + \frac{1}{2}}{\frac{3}{8}}}}{\sin x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{2}, \frac{1}{2}\right)}}{\frac{3}{8}}}{\sin x} \]

      4. cos-lowering-cos.f64 98.6

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{\cos x}, -0.5, 0.5\right)}{0.375}}{\sin x} \]

   6. Applied egg-rr 98.6%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\cos x, -0.5, 0.5\right)}{0.375}}}{\sin x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0045:\\ \;\;\;\;\frac{x\_m}{\frac{1}{\mathsf{fma}\left(x\_m, x\_m \cdot 0.05555555555555555, 0.6666666666666666\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x\_m, -0.5, 0.5\right)}{\sin x\_m \cdot 0.375}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0045)
    (/ x_m (/ 1.0 (fma x_m (* x_m 0.05555555555555555) 0.6666666666666666)))
    (/ (fma (cos x_m) -0.5 0.5) (* (sin x_m) 0.375)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0045) {
		tmp = x_m / (1.0 / fma(x_m, (x_m * 0.05555555555555555), 0.6666666666666666));
	} else {
		tmp = fma(cos(x_m), -0.5, 0.5) / (sin(x_m) * 0.375);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0045)
		tmp = Float64(x_m / Float64(1.0 / fma(x_m, Float64(x_m * 0.05555555555555555), 0.6666666666666666)));
	else
		tmp = Float64(fma(cos(x_m), -0.5, 0.5) / Float64(sin(x_m) * 0.375));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0045], N[(x$95$m / N[(1.0 / N[(x$95$m * N[(x$95$m * 0.05555555555555555), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x$95$m], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] / N[(N[Sin[x$95$m], $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.00449999999999999966

   1. Initial program 66.6%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + \frac{1}{18} \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + \frac{1}{18} \cdot {x}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{18} \cdot {x}^{2} + \frac{2}{3}\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{18}, {x}^{2}, \frac{2}{3}\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\frac{1}{18}, \color{blue}{x \cdot x}, \frac{2}{3}\right) \]

      5. *-lowering-*.f64 65.5

         \[\leadsto x \cdot \mathsf{fma}\left(0.05555555555555555, \color{blue}{x \cdot x}, 0.6666666666666666\right) \]

   5. Simplified 65.5%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.05555555555555555, x \cdot x, 0.6666666666666666\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto x \cdot \color{blue}{\frac{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}}}} \]

      6. flip-+ N/A

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{\frac{1}{18} \cdot \left(x \cdot x\right) + \frac{2}{3}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{1}{18} \cdot \left(x \cdot x\right) + \frac{2}{3}}}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{\left(\frac{1}{18} \cdot x\right) \cdot x} + \frac{2}{3}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{x \cdot \left(\frac{1}{18} \cdot x\right)} + \frac{2}{3}}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\frac{1}{\color{blue}{\mathsf{fma}\left(x, \frac{1}{18} \cdot x, \frac{2}{3}\right)}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{x}{\frac{1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{18}}, \frac{2}{3}\right)}} \]

      12. *-lowering-*.f64 65.7

          \[\leadsto \frac{x}{\frac{1}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.05555555555555555}, 0.6666666666666666\right)}} \]

   7. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(x, x \cdot 0.05555555555555555, 0.6666666666666666\right)}}} \]

   if 0.00449999999999999966 < x

   1. Initial program 99.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

      2. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin x}{\frac{8}{3}}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

      4. clear-num N/A

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}}} \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x, -0.5, 0.5\right)}{\sin x \cdot 0.375}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0045:\\ \;\;\;\;\frac{x\_m}{\frac{1}{\mathsf{fma}\left(x\_m, x\_m \cdot 0.05555555555555555, 0.6666666666666666\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x\_m, -1.3333333333333333, 1.3333333333333333\right)}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0045)
    (/ x_m (/ 1.0 (fma x_m (* x_m 0.05555555555555555) 0.6666666666666666)))
    (/ (fma (cos x_m) -1.3333333333333333 1.3333333333333333) (sin x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0045) {
		tmp = x_m / (1.0 / fma(x_m, (x_m * 0.05555555555555555), 0.6666666666666666));
	} else {
		tmp = fma(cos(x_m), -1.3333333333333333, 1.3333333333333333) / sin(x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0045)
		tmp = Float64(x_m / Float64(1.0 / fma(x_m, Float64(x_m * 0.05555555555555555), 0.6666666666666666)));
	else
		tmp = Float64(fma(cos(x_m), -1.3333333333333333, 1.3333333333333333) / sin(x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0045], N[(x$95$m / N[(1.0 / N[(x$95$m * N[(x$95$m * 0.05555555555555555), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x$95$m], $MachinePrecision] * -1.3333333333333333 + 1.3333333333333333), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.00449999999999999966

   1. Initial program 66.6%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + \frac{1}{18} \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + \frac{1}{18} \cdot {x}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{18} \cdot {x}^{2} + \frac{2}{3}\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{18}, {x}^{2}, \frac{2}{3}\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\frac{1}{18}, \color{blue}{x \cdot x}, \frac{2}{3}\right) \]

      5. *-lowering-*.f64 65.5

         \[\leadsto x \cdot \mathsf{fma}\left(0.05555555555555555, \color{blue}{x \cdot x}, 0.6666666666666666\right) \]

   5. Simplified 65.5%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.05555555555555555, x \cdot x, 0.6666666666666666\right)} \]
   6. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto x \cdot \color{blue}{\frac{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{18} \cdot \left(x \cdot x\right)\right) - \frac{2}{3} \cdot \frac{2}{3}}{\frac{1}{18} \cdot \left(x \cdot x\right) - \frac{2}{3}}}}} \]

      6. flip-+ N/A

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{\frac{1}{18} \cdot \left(x \cdot x\right) + \frac{2}{3}}}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{1}{18} \cdot \left(x \cdot x\right) + \frac{2}{3}}}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{\left(\frac{1}{18} \cdot x\right) \cdot x} + \frac{2}{3}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{x \cdot \left(\frac{1}{18} \cdot x\right)} + \frac{2}{3}}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\frac{1}{\color{blue}{\mathsf{fma}\left(x, \frac{1}{18} \cdot x, \frac{2}{3}\right)}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{x}{\frac{1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{18}}, \frac{2}{3}\right)}} \]

      12. *-lowering-*.f64 65.7

          \[\leadsto \frac{x}{\frac{1}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.05555555555555555}, 0.6666666666666666\right)}} \]

   7. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(x, x \cdot 0.05555555555555555, 0.6666666666666666\right)}}} \]

   if 0.00449999999999999966 < x

   1. Initial program 99.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}{\sin x} \]

      2. sqr-sin-a N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)}}{\sin x} \]

      3. sub-neg N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right)}}{\sin x} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{8}{3} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}}{\sin x} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\frac{8}{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{4}{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{\frac{3}{8}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{1}{2}}{\color{blue}{\frac{-1}{\frac{-8}{3}}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{-1}{\color{blue}{\mathsf{neg}\left(\frac{8}{3}\right)}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{8}{3}}\right)}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{\frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}}{\sin x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{8}{3}}\right)}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{\frac{-1}{\color{blue}{\frac{-8}{3}}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{2}}{\color{blue}{\frac{3}{8}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{4}{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{4}{3} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}}{\sin x} \]

   4. Applied egg-rr 98.3%

      \[\leadsto \frac{\color{blue}{1.3333333333333333 + \left(\cos x \cdot -0.5\right) \cdot 2.6666666666666665}}{\sin x} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cos x \cdot \frac{-1}{2}\right) \cdot \frac{8}{3} + \frac{4}{3}}}{\sin x} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\cos x \cdot \left(\frac{-1}{2} \cdot \frac{8}{3}\right)} + \frac{4}{3}}{\sin x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{2} \cdot \frac{8}{3}, \frac{4}{3}\right)}}{\sin x} \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\cos x}, \frac{-1}{2} \cdot \frac{8}{3}, \frac{4}{3}\right)}{\sin x} \]

      5. metadata-eval 98.6

         \[\leadsto \frac{\mathsf{fma}\left(\cos x, \color{blue}{-1.3333333333333333}, 1.3333333333333333\right)}{\sin x} \]

   6. Applied egg-rr 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x, -1.3333333333333333, 1.3333333333333333\right)}}{\sin x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 55.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(0.5 \cdot \frac{\sin \left(0.5 \cdot x\_m\right)}{0.375}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* 0.5 (/ (sin (* 0.5 x_m)) 0.375))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (0.5 * (sin((0.5 * x_m)) / 0.375));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (0.5d0 * (sin((0.5d0 * x_m)) / 0.375d0))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (0.5 * (Math.sin((0.5 * x_m)) / 0.375));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (0.5 * (math.sin((0.5 * x_m)) / 0.375))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(0.5 * Float64(sin(Float64(0.5 * x_m)) / 0.375)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (0.5 * (sin((0.5 * x_m)) / 0.375));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(0.5 * N[(N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision] / 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   2. associate-*l* N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin x}{\frac{8}{3}}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}}} \]

   5. frac-2neg N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\frac{\mathsf{neg}\left(\sin x\right)}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   6. neg-mul-1 N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\color{blue}{-1 \cdot \sin x}}{\mathsf{neg}\left(\frac{8}{3}\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\color{blue}{\sin x \cdot -1}}{\mathsf{neg}\left(\frac{8}{3}\right)}} \]

   8. associate-/l* N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x \cdot \frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   9. times-frac N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \frac{\sin \left(0.5 \cdot x\right)}{0.375}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\frac{3}{8}} \]
6. Step-by-step derivation

7. Simplified 54.6%

   \[\leadsto \color{blue}{0.5} \cdot \frac{\sin \left(0.5 \cdot x\right)}{0.375} \]
8. Add Preprocessing

Alternative 9: 55.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\sin \left(0.5 \cdot x\_m\right) \cdot 1.3333333333333333\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* (sin (* 0.5 x_m)) 1.3333333333333333)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (sin((0.5 * x_m)) * 1.3333333333333333);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (sin((0.5d0 * x_m)) * 1.3333333333333333d0)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (Math.sin((0.5 * x_m)) * 1.3333333333333333);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (math.sin((0.5 * x_m)) * 1.3333333333333333)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(sin(Float64(0.5 * x_m)) * 1.3333333333333333))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (sin((0.5 * x_m)) * 1.3333333333333333);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision] * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   2. associate-*l* N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin x}{\frac{8}{3}}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}}} \]

   5. frac-2neg N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\frac{\mathsf{neg}\left(\sin x\right)}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   6. neg-mul-1 N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\color{blue}{-1 \cdot \sin x}}{\mathsf{neg}\left(\frac{8}{3}\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\color{blue}{\sin x \cdot -1}}{\mathsf{neg}\left(\frac{8}{3}\right)}} \]

   8. associate-/l* N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x \cdot \frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   9. times-frac N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \frac{\sin \left(0.5 \cdot x\right)}{0.375}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\frac{3}{8}} \]
6. Step-by-step derivation

7. Simplified 54.6%

   \[\leadsto \color{blue}{0.5} \cdot \frac{\sin \left(0.5 \cdot x\right)}{0.375} \]
8. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{1}{\frac{3}{8}}\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\frac{8}{3}}\right) \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}\right) \cdot \frac{1}{2}} \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{8}{3} \cdot \frac{1}{2}\right)} \]

   5. metadata-eval N/A

      \[\leadsto \sin \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\frac{4}{3}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{4}{3}} \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot x\right)} \cdot \frac{4}{3} \]

   8. *-lowering-*.f64 54.4

      \[\leadsto \sin \color{blue}{\left(0.5 \cdot x\right)} \cdot 1.3333333333333333 \]

9. Applied egg-rr 54.4%

   \[\leadsto \color{blue}{\sin \left(0.5 \cdot x\right) \cdot 1.3333333333333333} \]
10. Add Preprocessing

Alternative 10: 51.1% accurate, 20.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m \cdot 0.25}{0.375} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ (* x_m 0.25) 0.375)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((x_m * 0.25) / 0.375);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((x_m * 0.25d0) / 0.375d0)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * ((x_m * 0.25) / 0.375);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((x_m * 0.25) / 0.375)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m * 0.25) / 0.375))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((x_m * 0.25) / 0.375);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(x$95$m * 0.25), $MachinePrecision] / 0.375), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   2. associate-*l* N/A

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin x}{\frac{8}{3}}}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{\frac{8}{3}}}} \]

   5. frac-2neg N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\frac{\mathsf{neg}\left(\sin x\right)}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   6. neg-mul-1 N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\color{blue}{-1 \cdot \sin x}}{\mathsf{neg}\left(\frac{8}{3}\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\frac{\color{blue}{\sin x \cdot -1}}{\mathsf{neg}\left(\frac{8}{3}\right)}} \]

   8. associate-/l* N/A

      \[\leadsto \frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{\sin x \cdot \frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   9. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x}}{\frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x}}{\frac{-1}{\mathsf{neg}\left(\frac{8}{3}\right)}}} \]

4. Applied egg-rr 54.4%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\cos x, -0.5, 0.5\right)}{\sin x}}{0.375}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot x}}{\frac{3}{8}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{1}{4}}}{\frac{3}{8}} \]

   2. *-lowering-*.f64 49.3

      \[\leadsto \frac{\color{blue}{x \cdot 0.25}}{0.375} \]

7. Simplified 49.3%

   \[\leadsto \frac{\color{blue}{x \cdot 0.25}}{0.375} \]
8. Add Preprocessing

Alternative 11: 50.8% accurate, 57.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot 0.6666666666666666\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* x_m 0.6666666666666666)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m * 0.6666666666666666);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (x_m * 0.6666666666666666d0)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (x_m * 0.6666666666666666);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m * 0.6666666666666666)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m * 0.6666666666666666))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (x_m * 0.6666666666666666);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(x$95$m * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{2}{3} \cdot x} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 49.1

      \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]

5. Simplified 49.1%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]

6. Final simplification 49.1%

   \[\leadsto x \cdot 0.6666666666666666 \]
7. Add Preprocessing

Developer Target 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\frac{8 \cdot t\_0}{3}}{\frac{\sin x}{t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ (/ (* 8.0 t_0) 3.0) (/ (sin x) t_0))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return ((8.0 * t_0) / 3.0) / (sin(x) / t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = ((8.0d0 * t_0) / 3.0d0) / (sin(x) / t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return ((8.0 * t_0) / 3.0) / (Math.sin(x) / t_0);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return ((8.0 * t_0) / 3.0) / (math.sin(x) / t_0)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(Float64(8.0 * t_0) / 3.0) / Float64(sin(x) / t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = ((8.0 * t_0) / 3.0) / (sin(x) / t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(8.0 * t$95$0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024195 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (* 8 (sin (* x 1/2))) 3) (/ (sin x) (sin (* x 1/2)))))

  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))