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

Percentage Accurate: 76.7% → 99.3%

Time: 8.4s

Alternatives: 9

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: 76.7% 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x\_m}^{6}, 0.00017146776406035664, 0.2962962962962963\right) \cdot x\_m}{\mathsf{fma}\left({x\_m}^{4}, 0.0030864197530864196, 0.4444444444444444 - 0.037037037037037035 \cdot \left(x\_m \cdot x\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(0.5 \cdot x\_m\right)}^{2} \cdot 2.6666666666666665}{\sin x\_m}\\ \end{array} \end{array} \]

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 1e-14)
    (/
     (* (fma (pow x_m 6.0) 0.00017146776406035664 0.2962962962962963) x_m)
     (fma
      (pow x_m 4.0)
      0.0030864197530864196
      (- 0.4444444444444444 (* 0.037037037037037035 (* x_m x_m)))))
    (/ (* (pow (sin (* 0.5 x_m)) 2.0) 2.6666666666666665) (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 <= 1e-14) {
		tmp = (fma(pow(x_m, 6.0), 0.00017146776406035664, 0.2962962962962963) * x_m) / fma(pow(x_m, 4.0), 0.0030864197530864196, (0.4444444444444444 - (0.037037037037037035 * (x_m * x_m))));
	} else {
		tmp = (pow(sin((0.5 * x_m)), 2.0) * 2.6666666666666665) / 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 <= 1e-14)
		tmp = Float64(Float64(fma((x_m ^ 6.0), 0.00017146776406035664, 0.2962962962962963) * x_m) / fma((x_m ^ 4.0), 0.0030864197530864196, Float64(0.4444444444444444 - Float64(0.037037037037037035 * Float64(x_m * x_m)))));
	else
		tmp = Float64(Float64((sin(Float64(0.5 * x_m)) ^ 2.0) * 2.6666666666666665) / 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, 1e-14], N[(N[(N[(N[Power[x$95$m, 6.0], $MachinePrecision] * 0.00017146776406035664 + 0.2962962962962963), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(N[Power[x$95$m, 4.0], $MachinePrecision] * 0.0030864197530864196 + N[(0.4444444444444444 - N[(0.037037037037037035 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * 2.6666666666666665), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999999e-15

   1. Initial program 66.8%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   7. Applied rewrites 68.4%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{6}, 0.00017146776406035664, 0.2962962962962963\right) \cdot x}{\color{blue}{\mathsf{fma}\left({x}^{4}, 0.0030864197530864196, 0.4444444444444444 - 0.037037037037037035 \cdot \left(x \cdot x\right)\right)}} \]

   if 9.99999999999999999e-15 < x

   1. Initial program 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 99.0

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \frac{8}{3}}{\sin x} \]

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.0

          \[\leadsto \frac{{\sin \color{blue}{\left(0.5 \cdot x\right)}}^{2} \cdot \frac{8}{3}}{\sin x} \]

      11. lift-/.f64 N/A

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

      12. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

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

Alternative 2: 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 75.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.2

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\sin x}} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \]

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.2

      \[\leadsto \frac{\sin \color{blue}{\left(0.5 \cdot x\right)}}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \]

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.2

       \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}\right)} \]

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.2

       \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \left(\sin \color{blue}{\left(0.5 \cdot x\right)} \cdot \frac{8}{3}\right) \]

   16. lift-/.f64 N/A

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

   17. metadata-eval 99.2

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

4. Applied rewrites 99.2%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

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 (/ 2.6666666666666665 (sin x_m))) t_0))))

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 * (2.6666666666666665 / sin(x_m))) * t_0);
}

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 * (2.6666666666666665d0 / sin(x_m))) * t_0)
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 * (2.6666666666666665 / Math.sin(x_m))) * t_0);
}

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 * (2.6666666666666665 / math.sin(x_m))) * t_0)

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 * Float64(2.6666666666666665 / sin(x_m))) * t_0))
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 * (2.6666666666666665 / sin(x_m))) * t_0);
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[(2.6666666666666665 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 99.2

       \[\leadsto \left(\sin \left(0.5 \cdot x\right) \cdot \color{blue}{\frac{\frac{8}{3}}{\sin x}}\right) \cdot \sin \left(x \cdot 0.5\right) \]

   17. lift-/.f64 N/A

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

   18. metadata-eval 99.2

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

4. Applied rewrites 99.2%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x\_m}^{6}, 0.00017146776406035664, 0.2962962962962963\right) \cdot x\_m}{\mathsf{fma}\left({x\_m}^{4}, 0.0030864197530864196, 0.4444444444444444 - 0.037037037037037035 \cdot \left(x\_m \cdot x\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(0.5 \cdot x\_m\right)}^{2}}{\sin x\_m} \cdot 2.6666666666666665\\ \end{array} \end{array} \]

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 2e-11)
    (/
     (* (fma (pow x_m 6.0) 0.00017146776406035664 0.2962962962962963) x_m)
     (fma
      (pow x_m 4.0)
      0.0030864197530864196
      (- 0.4444444444444444 (* 0.037037037037037035 (* x_m x_m)))))
    (* (/ (pow (sin (* 0.5 x_m)) 2.0) (sin x_m)) 2.6666666666666665))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2e-11) {
		tmp = (fma(pow(x_m, 6.0), 0.00017146776406035664, 0.2962962962962963) * x_m) / fma(pow(x_m, 4.0), 0.0030864197530864196, (0.4444444444444444 - (0.037037037037037035 * (x_m * x_m))));
	} else {
		tmp = (pow(sin((0.5 * x_m)), 2.0) / sin(x_m)) * 2.6666666666666665;
	}
	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 <= 2e-11)
		tmp = Float64(Float64(fma((x_m ^ 6.0), 0.00017146776406035664, 0.2962962962962963) * x_m) / fma((x_m ^ 4.0), 0.0030864197530864196, Float64(0.4444444444444444 - Float64(0.037037037037037035 * Float64(x_m * x_m)))));
	else
		tmp = Float64(Float64((sin(Float64(0.5 * x_m)) ^ 2.0) / sin(x_m)) * 2.6666666666666665);
	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, 2e-11], N[(N[(N[(N[Power[x$95$m, 6.0], $MachinePrecision] * 0.00017146776406035664 + 0.2962962962962963), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(N[Power[x$95$m, 4.0], $MachinePrecision] * 0.0030864197530864196 + N[(0.4444444444444444 - N[(0.037037037037037035 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] * 2.6666666666666665), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999988e-11

   1. Initial program 66.8%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   7. Applied rewrites 68.4%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{6}, 0.00017146776406035664, 0.2962962962962963\right) \cdot x}{\color{blue}{\mathsf{fma}\left({x}^{4}, 0.0030864197530864196, 0.4444444444444444 - 0.037037037037037035 \cdot \left(x \cdot x\right)\right)}} \]

   if 1.99999999999999988e-11 < x

   1. Initial program 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 99.0

          \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x} \cdot \frac{8}{3} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.0

          \[\leadsto \frac{{\sin \color{blue}{\left(0.5 \cdot x\right)}}^{2}}{\sin x} \cdot \frac{8}{3} \]

      15. lift-/.f64 N/A

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

      16. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

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

Alternative 5: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0027:\\ \;\;\;\;\frac{\mathsf{fma}\left({x\_m}^{6}, 0.00017146776406035664, 0.2962962962962963\right) \cdot x\_m}{\mathsf{fma}\left({x\_m}^{4}, 0.0030864197530864196, 0.4444444444444444 - 0.037037037037037035 \cdot \left(x\_m \cdot x\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.3333333333333333, \cos x\_m, 1.3333333333333333\right)}{\sin x\_m}\\ \end{array} \end{array} \]

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 0.0027)
    (/
     (* (fma (pow x_m 6.0) 0.00017146776406035664 0.2962962962962963) x_m)
     (fma
      (pow x_m 4.0)
      0.0030864197530864196
      (- 0.4444444444444444 (* 0.037037037037037035 (* x_m x_m)))))
    (/ (fma -1.3333333333333333 (cos x_m) 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 <= 0.0027) {
		tmp = (fma(pow(x_m, 6.0), 0.00017146776406035664, 0.2962962962962963) * x_m) / fma(pow(x_m, 4.0), 0.0030864197530864196, (0.4444444444444444 - (0.037037037037037035 * (x_m * x_m))));
	} else {
		tmp = fma(-1.3333333333333333, cos(x_m), 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 <= 0.0027)
		tmp = Float64(Float64(fma((x_m ^ 6.0), 0.00017146776406035664, 0.2962962962962963) * x_m) / fma((x_m ^ 4.0), 0.0030864197530864196, Float64(0.4444444444444444 - Float64(0.037037037037037035 * Float64(x_m * x_m)))));
	else
		tmp = Float64(fma(-1.3333333333333333, cos(x_m), 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, 0.0027], N[(N[(N[(N[Power[x$95$m, 6.0], $MachinePrecision] * 0.00017146776406035664 + 0.2962962962962963), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(N[Power[x$95$m, 4.0], $MachinePrecision] * 0.0030864197530864196 + N[(0.4444444444444444 - N[(0.037037037037037035 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.3333333333333333 * N[Cos[x$95$m], $MachinePrecision] + 1.3333333333333333), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0027000000000000001

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 68.6%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{6}, 0.00017146776406035664, 0.2962962962962963\right) \cdot x}{\color{blue}{\mathsf{fma}\left({x}^{4}, 0.0030864197530864196, 0.4444444444444444 - 0.037037037037037035 \cdot \left(x \cdot x\right)\right)}} \]

   if 0.0027000000000000001 < x

   1. Initial program 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 99.0

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \frac{8}{3}}{\sin x} \]

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.0

          \[\leadsto \frac{{\sin \color{blue}{\left(0.5 \cdot x\right)}}^{2} \cdot \frac{8}{3}}{\sin x} \]

      11. lift-/.f64 N/A

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

      12. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{{\sin \left(0.5 \cdot x\right)}^{2} \cdot 2.6666666666666665}{\sin x}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-neg-rev N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. cos-+PI/2-rev N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. cos-+PI/2-rev N/A

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

      14. 1-sub-sin-rev N/A

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

      15. metadata-eval N/A

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

      16. sin-+PI/2-rev N/A

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

      17. sin-+PI/2-rev N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. pow2 N/A

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

   6. Applied rewrites 98.2%

      \[\leadsto \frac{\color{blue}{\left(1 - {\cos \left(-0.5 \cdot x\right)}^{2}\right)} \cdot 2.6666666666666665}{\sin x} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. sqr-cos-a N/A

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

      6. cos-neg-rev N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. count-2-rev N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. flip-+ N/A

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

   8. Applied rewrites 98.5%

      \[\leadsto \frac{\left(1 - \color{blue}{\left(0.5 - -0.5 \cdot \cos \left(-1 \cdot x\right)\right)}\right) \cdot 2.6666666666666665}{\sin x} \]

   9. Taylor expanded in x around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. lower-fma.f64 N/A

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

       5. metadata-eval N/A

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

       6. mul-1-neg N/A

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

       7. cos-neg N/A

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

       8. lower-cos.f64 N/A

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

       9. metadata-eval 99.0

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

   11. Applied rewrites 99.0%

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

Alternative 6: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0051:\\ \;\;\;\;\frac{-0.4444444444444444 \cdot x\_m}{0.05555555555555555 \cdot \left(x\_m \cdot x\_m\right) - 0.6666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.3333333333333333, \cos x\_m, 1.3333333333333333\right)}{\sin x\_m}\\ \end{array} \end{array} \]

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 0.0051)
    (/
     (* -0.4444444444444444 x_m)
     (- (* 0.05555555555555555 (* x_m x_m)) 0.6666666666666666))
    (/ (fma -1.3333333333333333 (cos x_m) 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 <= 0.0051) {
		tmp = (-0.4444444444444444 * x_m) / ((0.05555555555555555 * (x_m * x_m)) - 0.6666666666666666);
	} else {
		tmp = fma(-1.3333333333333333, cos(x_m), 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 <= 0.0051)
		tmp = Float64(Float64(-0.4444444444444444 * x_m) / Float64(Float64(0.05555555555555555 * Float64(x_m * x_m)) - 0.6666666666666666));
	else
		tmp = Float64(fma(-1.3333333333333333, cos(x_m), 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, 0.0051], N[(N[(-0.4444444444444444 * x$95$m), $MachinePrecision] / N[(N[(0.05555555555555555 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(-1.3333333333333333 * N[Cos[x$95$m], $MachinePrecision] + 1.3333333333333333), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051000000000000004

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 68.7%

      \[\leadsto \frac{\left({x}^{4} \cdot 0.0030864197530864196 - 0.4444444444444444\right) \cdot x}{\color{blue}{0.05555555555555555 \cdot \left(x \cdot x\right) - 0.6666666666666666}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{-4}{9} \cdot x}{\color{blue}{\frac{1}{18}} \cdot \left(x \cdot x\right) - \frac{2}{3}} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.4%

       \[\leadsto \frac{-0.4444444444444444 \cdot x}{\color{blue}{0.05555555555555555} \cdot \left(x \cdot x\right) - 0.6666666666666666} \]

   if 0.0051000000000000004 < x

   1. Initial program 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 99.0

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \frac{8}{3}}{\sin x} \]

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.0

          \[\leadsto \frac{{\sin \color{blue}{\left(0.5 \cdot x\right)}}^{2} \cdot \frac{8}{3}}{\sin x} \]

      11. lift-/.f64 N/A

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

      12. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{{\sin \left(0.5 \cdot x\right)}^{2} \cdot 2.6666666666666665}{\sin x}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-neg-rev N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. cos-+PI/2-rev N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. cos-+PI/2-rev N/A

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

      14. 1-sub-sin-rev N/A

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

      15. metadata-eval N/A

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

      16. sin-+PI/2-rev N/A

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

      17. sin-+PI/2-rev N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. pow2 N/A

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

   6. Applied rewrites 98.2%

      \[\leadsto \frac{\color{blue}{\left(1 - {\cos \left(-0.5 \cdot x\right)}^{2}\right)} \cdot 2.6666666666666665}{\sin x} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. sqr-cos-a N/A

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

      6. cos-neg-rev N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. count-2-rev N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. flip-+ N/A

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

   8. Applied rewrites 98.5%

      \[\leadsto \frac{\left(1 - \color{blue}{\left(0.5 - -0.5 \cdot \cos \left(-1 \cdot x\right)\right)}\right) \cdot 2.6666666666666665}{\sin x} \]

   9. Taylor expanded in x around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. lower-fma.f64 N/A

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

       5. metadata-eval N/A

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

       6. mul-1-neg N/A

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

       7. cos-neg N/A

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

       8. lower-cos.f64 N/A

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

       9. metadata-eval 99.0

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

   11. Applied rewrites 99.0%

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

Alternative 7: 55.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

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 * (1.3333333333333333d0 * sin((x_m * 0.5d0)))
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 * (1.3333333333333333 * Math.sin((x_m * 0.5)));
}

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (1.3333333333333333 * sin((x_m * 0.5)));
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[(1.3333333333333333 * N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.2

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\sin x}} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \]

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.2

      \[\leadsto \frac{\sin \color{blue}{\left(0.5 \cdot x\right)}}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \]

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.2

       \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}\right)} \]

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.2

       \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \left(\sin \color{blue}{\left(0.5 \cdot x\right)} \cdot \frac{8}{3}\right) \]

   16. lift-/.f64 N/A

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

   17. metadata-eval 99.2

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

4. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. associate-*r/ N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 99.2

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

   12. lift-sin.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lift-sin.f64 99.2

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

   17. lift-sin.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lift-*.f64 N/A

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

   21. lift-sin.f64 99.2

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

6. Applied rewrites 99.2%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 56.1%

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

Alternative 8: 52.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{-0.4444444444444444 \cdot x\_m}{0.05555555555555555 \cdot \left(x\_m \cdot x\_m\right) - 0.6666666666666666} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/
   (* -0.4444444444444444 x_m)
   (- (* 0.05555555555555555 (* x_m 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 * ((-0.4444444444444444 * x_m) / ((0.05555555555555555 * (x_m * 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 * (((-0.4444444444444444d0) * x_m) / ((0.05555555555555555d0 * (x_m * 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 * ((-0.4444444444444444 * x_m) / ((0.05555555555555555 * (x_m * 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 * ((-0.4444444444444444 * x_m) / ((0.05555555555555555 * (x_m * 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(Float64(-0.4444444444444444 * x_m) / Float64(Float64(0.05555555555555555 * Float64(x_m * 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 * ((-0.4444444444444444 * x_m) / ((0.05555555555555555 * (x_m * 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[(N[(-0.4444444444444444 * x$95$m), $MachinePrecision] / N[(N[(0.05555555555555555 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.4%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 51.5

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

5. Applied rewrites 51.5%

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

7. Applied rewrites 51.0%

   \[\leadsto \frac{\left({x}^{4} \cdot 0.0030864197530864196 - 0.4444444444444444\right) \cdot x}{\color{blue}{0.05555555555555555 \cdot \left(x \cdot x\right) - 0.6666666666666666}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{-4}{9} \cdot x}{\color{blue}{\frac{1}{18}} \cdot \left(x \cdot x\right) - \frac{2}{3}} \]
9. Step-by-step derivation

10. Applied rewrites 52.3%

    \[\leadsto \frac{-0.4444444444444444 \cdot x}{\color{blue}{0.05555555555555555} \cdot \left(x \cdot x\right) - 0.6666666666666666} \]
11. Add Preprocessing

Alternative 9: 51.4% accurate, 57.2× speedup

Math

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

FPCore

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

C

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

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 * (0.6666666666666666d0 * x_m)
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 * (0.6666666666666666 * x_m);
}

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (0.6666666666666666 * 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_] := N[(x$95$s * N[(0.6666666666666666 * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.4%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 51.8

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

5. Applied rewrites 51.8%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
6. 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 2024342 
(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)))