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

Percentage Accurate: 76.8% → 99.2%

Time: 7.6s

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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-38}:\\ \;\;\;\;0.6666666666666666 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(0.5 \cdot x\_m\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x\_m}\\ \end{array} \end{array} \]

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 2e-38)
    (* 0.6666666666666666 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 <= 2e-38) {
		tmp = 0.6666666666666666 * x_m;
	} else {
		tmp = pow(sin((0.5 * x_m)), 2.0) * (2.6666666666666665 / sin(x_m));
	}
	return x_s * tmp;
}

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) :: tmp
    if (x_m <= 2d-38) then
        tmp = 0.6666666666666666d0 * x_m
    else
        tmp = (sin((0.5d0 * x_m)) ** 2.0d0) * (2.6666666666666665d0 / sin(x_m))
    end if
    code = x_s * tmp
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 tmp;
	if (x_m <= 2e-38) {
		tmp = 0.6666666666666666 * x_m;
	} else {
		tmp = Math.pow(Math.sin((0.5 * x_m)), 2.0) * (2.6666666666666665 / Math.sin(x_m));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 2e-38:
		tmp = 0.6666666666666666 * x_m
	else:
		tmp = math.pow(math.sin((0.5 * x_m)), 2.0) * (2.6666666666666665 / math.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 <= 2e-38)
		tmp = Float64(0.6666666666666666 * x_m);
	else
		tmp = Float64((sin(Float64(0.5 * x_m)) ^ 2.0) * Float64(2.6666666666666665 / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 2e-38)
		tmp = 0.6666666666666666 * x_m;
	else
		tmp = (sin((0.5 * x_m)) ^ 2.0) * (2.6666666666666665 / sin(x_m));
	end
	tmp_2 = 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-38], N[(0.6666666666666666 * x$95$m), $MachinePrecision], N[(N[Power[N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(2.6666666666666665 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e-38

   1. Initial program 71.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 63.6

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

   5. Applied rewrites 63.6%

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

   if 1.9999999999999999e-38 < 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. 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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.0

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

      14. lift-/.f64 N/A

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

      15. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{{\sin \left(0.5 \cdot x\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}} \]
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 80.7%

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

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

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 (* x_m 0.5))))
   (* 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((x_m * 0.5));
	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((x_m * 0.5d0))
    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((x_m * 0.5));
	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((x_m * 0.5))
	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(x_m * 0.5))
	return Float64(x_s * Float64(Float64(Float64(t_0 / sin(x_m)) * 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((x_m * 0.5));
	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[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * N[(N[(N[(t$95$0 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * 2.6666666666666665), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 80.7

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

   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 80.7

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

   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 80.7

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

4. Applied rewrites 80.7%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*l/ N/A

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

   5. lift-/.f64 N/A

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

   6. lower-*.f64 99.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 99.2

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 99.2

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

6. Applied rewrites 99.2%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

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

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 80.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-43}:\\ \;\;\;\;0.6666666666666666 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(0.5 \cdot x\_m\right)}^{2}}{\sin x\_m} \cdot 2.6666666666666665\\ \end{array} \end{array} \]

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 5e-43)
    (* 0.6666666666666666 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 <= 5e-43) {
		tmp = 0.6666666666666666 * x_m;
	} else {
		tmp = (pow(sin((0.5 * x_m)), 2.0) / sin(x_m)) * 2.6666666666666665;
	}
	return x_s * tmp;
}

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) :: tmp
    if (x_m <= 5d-43) then
        tmp = 0.6666666666666666d0 * x_m
    else
        tmp = ((sin((0.5d0 * x_m)) ** 2.0d0) / sin(x_m)) * 2.6666666666666665d0
    end if
    code = x_s * tmp
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 tmp;
	if (x_m <= 5e-43) {
		tmp = 0.6666666666666666 * x_m;
	} else {
		tmp = (Math.pow(Math.sin((0.5 * x_m)), 2.0) / Math.sin(x_m)) * 2.6666666666666665;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 5e-43:
		tmp = 0.6666666666666666 * x_m
	else:
		tmp = (math.pow(math.sin((0.5 * x_m)), 2.0) / math.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 <= 5e-43)
		tmp = Float64(0.6666666666666666 * 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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 5e-43)
		tmp = 0.6666666666666666 * x_m;
	else
		tmp = ((sin((0.5 * x_m)) ^ 2.0) / sin(x_m)) * 2.6666666666666665;
	end
	tmp_2 = 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, 5e-43], N[(0.6666666666666666 * x$95$m), $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 < 5.00000000000000019e-43

   1. Initial program 71.5%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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 63.4

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

   5. Applied rewrites 63.4%

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

   if 5.00000000000000019e-43 < 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. 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 6: 98.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0001:\\ \;\;\;\;0.6666666666666666 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 - \cos x\_m \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x\_m}\\ \end{array} \end{array} \]

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 0.0001)
    (* 0.6666666666666666 x_m)
    (/ (* (- 0.5 (* (cos x_m) 0.5)) 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 <= 0.0001) {
		tmp = 0.6666666666666666 * x_m;
	} else {
		tmp = ((0.5 - (cos(x_m) * 0.5)) * 2.6666666666666665) / sin(x_m);
	}
	return x_s * tmp;
}

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) :: tmp
    if (x_m <= 0.0001d0) then
        tmp = 0.6666666666666666d0 * x_m
    else
        tmp = ((0.5d0 - (cos(x_m) * 0.5d0)) * 2.6666666666666665d0) / sin(x_m)
    end if
    code = x_s * tmp
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 tmp;
	if (x_m <= 0.0001) {
		tmp = 0.6666666666666666 * x_m;
	} else {
		tmp = ((0.5 - (Math.cos(x_m) * 0.5)) * 2.6666666666666665) / Math.sin(x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.0001:
		tmp = 0.6666666666666666 * x_m
	else:
		tmp = ((0.5 - (math.cos(x_m) * 0.5)) * 2.6666666666666665) / math.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 <= 0.0001)
		tmp = Float64(0.6666666666666666 * x_m);
	else
		tmp = Float64(Float64(Float64(0.5 - Float64(cos(x_m) * 0.5)) * 2.6666666666666665) / sin(x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.0001)
		tmp = 0.6666666666666666 * x_m;
	else
		tmp = ((0.5 - (cos(x_m) * 0.5)) * 2.6666666666666665) / sin(x_m);
	end
	tmp_2 = 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, 0.0001], N[(0.6666666666666666 * x$95$m), $MachinePrecision], N[(N[(N[(0.5 - N[(N[Cos[x$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * 2.6666666666666665), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000005e-4

   1. Initial program 73.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   if 1.00000000000000005e-4 < x

   1. Initial program 98.9%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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 98.9

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

      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 98.9

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

      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 98.9

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

   4. Applied rewrites 98.9%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. sin-neg-rev N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-PI.f64 96.7

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

   6. Applied rewrites 96.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. sin-+PI N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. sin-+PI N/A

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

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

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

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

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

      13. 1-sub-sin-rev N/A

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

      14. lower--.f64 N/A

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

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

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

      16. cos-neg-rev N/A

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

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

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

      18. cos-neg-rev N/A

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

      19. sqr-cos-a-rev N/A

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

   8. Applied rewrites 97.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 97.8

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate--r+ N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 98.2

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

      11. lift-*.f64 N/A

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

      12. *-lft-identity 98.2

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

   10. Applied rewrites 98.2%

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

Alternative 7: 98.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0001:\\ \;\;\;\;0.6666666666666666 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x\_m, -0.5, 0.5\right)}{\sin x\_m} \cdot 2.6666666666666665\\ \end{array} \end{array} \]

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 0.0001)
    (* 0.6666666666666666 x_m)
    (* (/ (fma (cos x_m) -0.5 0.5) (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 <= 0.0001) {
		tmp = 0.6666666666666666 * x_m;
	} else {
		tmp = (fma(cos(x_m), -0.5, 0.5) / 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 <= 0.0001)
		tmp = Float64(0.6666666666666666 * x_m);
	else
		tmp = Float64(Float64(fma(cos(x_m), -0.5, 0.5) / 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, 0.0001], N[(0.6666666666666666 * x$95$m), $MachinePrecision], N[(N[(N[(N[Cos[x$95$m], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] * 2.6666666666666665), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000005e-4

   1. Initial program 73.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   if 1.00000000000000005e-4 < x

   1. Initial program 98.9%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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 98.9

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

      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 98.9

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

      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 98.9

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

   4. Applied rewrites 98.9%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. sin-neg-rev N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-PI.f64 96.7

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

   6. Applied rewrites 96.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. sin-+PI N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. sin-+PI N/A

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

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

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

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

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

      13. 1-sub-sin-rev N/A

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

      14. lower--.f64 N/A

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

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

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

      16. cos-neg-rev N/A

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

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

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

      18. cos-neg-rev N/A

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

      19. sqr-cos-a-rev N/A

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in x around inf

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

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

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

       2. +-commutative N/A

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

       3. metadata-eval N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-cos.f64 98.1

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

   11. Applied rewrites 98.1%

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

Alternative 8: 98.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0001:\\ \;\;\;\;0.6666666666666666 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x\_m, -0.5, 0.5\right) \cdot \frac{2.6666666666666665}{\sin x\_m}\\ \end{array} \end{array} \]

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 0.0001)
    (* 0.6666666666666666 x_m)
    (* (fma (cos x_m) -0.5 0.5) (/ 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 <= 0.0001) {
		tmp = 0.6666666666666666 * x_m;
	} else {
		tmp = fma(cos(x_m), -0.5, 0.5) * (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 <= 0.0001)
		tmp = Float64(0.6666666666666666 * x_m);
	else
		tmp = Float64(fma(cos(x_m), -0.5, 0.5) * Float64(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, 0.0001], N[(0.6666666666666666 * x$95$m), $MachinePrecision], N[(N[(N[Cos[x$95$m], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(2.6666666666666665 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000005e-4

   1. Initial program 73.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   if 1.00000000000000005e-4 < x

   1. Initial program 98.9%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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 98.9

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

      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 98.9

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

      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 98.9

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

   4. Applied rewrites 98.9%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. sin-neg-rev N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-PI.f64 96.7

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

   6. Applied rewrites 96.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. sin-+PI N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. sin-+PI N/A

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

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

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

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

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

      13. 1-sub-sin-rev N/A

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

      14. lower--.f64 N/A

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

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

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

      16. cos-neg-rev N/A

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

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

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

      18. cos-neg-rev N/A

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

      19. sqr-cos-a-rev N/A

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

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

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

       6. +-commutative N/A

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

       7. metadata-eval N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-cos.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-sin.f64 98.1

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

   11. Applied rewrites 98.1%

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

Alternative 9: 55.1% accurate, 3.1× speedup

Math

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

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 (* 0.5 x_m)))))

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

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((0.5 * x_m)))

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(0.5 * 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 * (1.3333333333333333 * sin((0.5 * 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[(1.3333333333333333 * N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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. Taylor expanded in x around 0

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

7. Applied rewrites 53.1%

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

Alternative 10: 50.9% 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 80.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 48.5

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

5. Applied rewrites 48.5%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
6. Add Preprocessing

Alternative 11: 4.3% accurate, 343.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot 0 \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s 0.0))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * 0.0;
}

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
    code = x_s * 0.0d0
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 * 0.0;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 0.0

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 0.0)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 0.0;
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_] := N[(x$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 80.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 80.7

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

   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 80.7

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

   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 80.7

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

4. Applied rewrites 80.7%

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

   1. remove-double-neg N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. sin-neg-rev N/A

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

   7. sin-+PI-rev N/A

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

   8. lower-sin.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-PI.f64 53.4

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

6. Applied rewrites 53.4%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. unpow2 N/A

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

   4. sin-PI N/A

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

   5. sin-PI N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} \cdot \frac{8}{3}}{x} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{x} \]

   8. div0 4.1

      \[\leadsto \color{blue}{0} \]

9. Applied rewrites 4.1%

   \[\leadsto \color{blue}{0} \]
10. 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 2024326 
(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)))