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

Percentage Accurate: 77.3% → 99.5%

Time: 10.0s

Alternatives: 21

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: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x\_m \cdot 0.5\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0005:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001953125\right), x\_m \cdot x\_m, -0.09375\right), x\_m \cdot x\_m, 0.75\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{t\_0}^{2}}{\sin x\_m}}{0.375}\\ \end{array} \end{array} \end{array} \]

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
    (if (<= x_m 0.0005)
      (/
       t_0
       (fma
        (fma
         (fma -1.6276041666666666e-5 (* x_m x_m) 0.001953125)
         (* x_m x_m)
         -0.09375)
        (* x_m x_m)
        0.75))
      (/ (/ (pow t_0 2.0) (sin x_m)) 0.375)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((x_m * 0.5));
	double tmp;
	if (x_m <= 0.0005) {
		tmp = t_0 / fma(fma(fma(-1.6276041666666666e-5, (x_m * x_m), 0.001953125), (x_m * x_m), -0.09375), (x_m * x_m), 0.75);
	} else {
		tmp = (pow(t_0, 2.0) / sin(x_m)) / 0.375;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = sin(Float64(x_m * 0.5))
	tmp = 0.0
	if (x_m <= 0.0005)
		tmp = Float64(t_0 / fma(fma(fma(-1.6276041666666666e-5, Float64(x_m * x_m), 0.001953125), Float64(x_m * x_m), -0.09375), Float64(x_m * x_m), 0.75));
	else
		tmp = Float64(Float64((t_0 ^ 2.0) / sin(x_m)) / 0.375);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 0.0005], N[(t$95$0 / N[(N[(N[(-1.6276041666666666e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001953125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.09375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.75), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-4

   1. Initial program 69.2%

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

      1. 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. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{\frac{3}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}\right) + \frac{3}{4}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}\right) \cdot {x}^{2}} + \frac{3}{4}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}, {x}^{2}, \frac{3}{4}\right)}} \]

      4. sub-neg N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{32}\right)\right)}, {x}^{2}, \frac{3}{4}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{3}{32}\right)\right), {x}^{2}, \frac{3}{4}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-3}{32}}, {x}^{2}, \frac{3}{4}\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}, {x}^{2}, \frac{-3}{32}\right)}, {x}^{2}, \frac{3}{4}\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{61440} \cdot {x}^{2} + \frac{1}{512}}, {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{61440}, {x}^{2}, \frac{1}{512}\right)}, {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, \color{blue}{x \cdot x}, \frac{1}{512}\right), {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, \color{blue}{x \cdot x}, \frac{1}{512}\right), {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, x \cdot x, \frac{1}{512}\right), \color{blue}{x \cdot x}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, x \cdot x, \frac{1}{512}\right), \color{blue}{x \cdot x}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, x \cdot x, \frac{1}{512}\right), x \cdot x, \frac{-3}{32}\right), \color{blue}{x \cdot x}, \frac{3}{4}\right)} \]

      15. lower-*.f64 65.2

          \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x \cdot x, 0.001953125\right), x \cdot x, -0.09375\right), \color{blue}{x \cdot x}, 0.75\right)} \]

   7. Applied rewrites 65.2%

      \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x \cdot x, 0.001953125\right), x \cdot x, -0.09375\right), x \cdot x, 0.75\right)}} \]

   if 5.0000000000000001e-4 < x

   1. Initial program 99.2%

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

      1. 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 98.8

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

      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 98.8

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

      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 98.8

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

      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 98.8

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

      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 98.8

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

   4. Applied rewrites 98.8%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. lower-/.f64 99.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.1

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

   6. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(0.5 \cdot x\right)}^{2}}{\sin x}}{0.375}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x \cdot x, 0.001953125\right), x \cdot x, -0.09375\right), x \cdot x, 0.75\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x\_m \cdot 0.5\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{8}{3} \cdot t\_0\right) \cdot t\_0}{\sin x\_m} \leq 10^{-5}:\\ \;\;\;\;\frac{t\_0}{0.75}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\cos x\_m, -0.5, 0.5\right)} \cdot 2.6666666666666665\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

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
    (if (<= (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x_m)) 1e-5)
      (/ t_0 0.75)
      (* (* (sqrt (fma (cos x_m) -0.5 0.5)) 2.6666666666666665) 0.5)))))

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));
	double tmp;
	if (((((8.0 / 3.0) * t_0) * t_0) / sin(x_m)) <= 1e-5) {
		tmp = t_0 / 0.75;
	} else {
		tmp = (sqrt(fma(cos(x_m), -0.5, 0.5)) * 2.6666666666666665) * 0.5;
	}
	return x_s * tmp;
}

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))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x_m)) <= 1e-5)
		tmp = Float64(t_0 / 0.75);
	else
		tmp = Float64(Float64(sqrt(fma(cos(x_m), -0.5, 0.5)) * 2.6666666666666665) * 0.5);
	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_] := Block[{t$95$0 = N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], 1e-5], N[(t$95$0 / 0.75), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Cos[x$95$m], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision] * 2.6666666666666665), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 (/.f64 #s(literal 8 binary64) #s(literal 3 binary64)) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 x)) < 1.00000000000000008e-5

   1. Initial program 67.1%

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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 70.6%

      \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\color{blue}{0.75}} \]

   if 1.00000000000000008e-5 < (/.f64 (*.f64 (*.f64 (/.f64 #s(literal 8 binary64) #s(literal 3 binary64)) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 x))

   1. Initial program 99.2%

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

      1. 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 98.9

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

      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 98.9

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

      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 98.9

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

      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 98.9

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

      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 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 11.6%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. fabs-sqr-rev N/A

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

      4. sqr-pow N/A

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

      5. unpow1 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sqr-sin-a N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

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

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

      15. neg-mul-1 N/A

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

      16. remove-double-neg N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. cancel-sub-sign-inv N/A

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

   9. Applied rewrites 22.2%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \leq 10^{-5}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\cos x, -0.5, 0.5\right)} \cdot 2.6666666666666665\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x\_m \cdot 0.5\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{8}{3} \cdot t\_0\right) \cdot t\_0}{\sin x\_m} \leq 0.0004:\\ \;\;\;\;\frac{t\_0}{0.75}\\ \mathbf{else}:\\ \;\;\;\;\left(\left|t\_0\right| \cdot 2.6666666666666665\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

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
    (if (<= (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x_m)) 0.0004)
      (/ t_0 0.75)
      (* (* (fabs t_0) 2.6666666666666665) 0.5)))))

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));
	double tmp;
	if (((((8.0 / 3.0) * t_0) * t_0) / sin(x_m)) <= 0.0004) {
		tmp = t_0 / 0.75;
	} else {
		tmp = (fabs(t_0) * 2.6666666666666665) * 0.5;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = sin((x_m * 0.5d0))
    if (((((8.0d0 / 3.0d0) * t_0) * t_0) / sin(x_m)) <= 0.0004d0) then
        tmp = t_0 / 0.75d0
    else
        tmp = (abs(t_0) * 2.6666666666666665d0) * 0.5d0
    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 t_0 = Math.sin((x_m * 0.5));
	double tmp;
	if (((((8.0 / 3.0) * t_0) * t_0) / Math.sin(x_m)) <= 0.0004) {
		tmp = t_0 / 0.75;
	} else {
		tmp = (Math.abs(t_0) * 2.6666666666666665) * 0.5;
	}
	return x_s * tmp;
}

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))
	tmp = 0
	if ((((8.0 / 3.0) * t_0) * t_0) / math.sin(x_m)) <= 0.0004:
		tmp = t_0 / 0.75
	else:
		tmp = (math.fabs(t_0) * 2.6666666666666665) * 0.5
	return x_s * tmp

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))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x_m)) <= 0.0004)
		tmp = Float64(t_0 / 0.75);
	else
		tmp = Float64(Float64(abs(t_0) * 2.6666666666666665) * 0.5);
	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)
	t_0 = sin((x_m * 0.5));
	tmp = 0.0;
	if (((((8.0 / 3.0) * t_0) * t_0) / sin(x_m)) <= 0.0004)
		tmp = t_0 / 0.75;
	else
		tmp = (abs(t_0) * 2.6666666666666665) * 0.5;
	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_] := Block[{t$95$0 = N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], 0.0004], N[(t$95$0 / 0.75), $MachinePrecision], N[(N[(N[Abs[t$95$0], $MachinePrecision] * 2.6666666666666665), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 (/.f64 #s(literal 8 binary64) #s(literal 3 binary64)) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 x)) < 4.00000000000000019e-4

   1. Initial program 67.3%

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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 70.5%

      \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\color{blue}{0.75}} \]

   if 4.00000000000000019e-4 < (/.f64 (*.f64 (*.f64 (/.f64 #s(literal 8 binary64) #s(literal 3 binary64)) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 x))

   1. Initial program 99.2%

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

      1. 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 98.9

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

      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 98.9

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

      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 98.9

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

      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 98.9

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

      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 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 10.9%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. fabs-sqr-rev N/A

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

      4. sqr-pow N/A

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

      5. unpow1 N/A

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

      6. lower-fabs.f64 21.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 21.7

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

   9. Applied rewrites 21.7%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \leq 0.0004:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75}\\ \mathbf{else}:\\ \;\;\;\;\left(\left|\sin \left(x \cdot 0.5\right)\right| \cdot 2.6666666666666665\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((x_m * 0.5));
	return x_s * ((t_0 / 0.375) * (t_0 / sin(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
    real(8) :: t_0
    t_0 = sin((x_m * 0.5d0))
    code = x_s * ((t_0 / 0.375d0) * (t_0 / sin(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) {
	double t_0 = Math.sin((x_m * 0.5));
	return x_s * ((t_0 / 0.375) * (t_0 / Math.sin(x_m)));
}

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 / 0.375) * (t_0 / math.sin(x_m)))

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(t_0 / 0.375) * Float64(t_0 / sin(x_m))))
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 / 0.375) * (t_0 / sin(x_m)));
end

Wolfram

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

Derivation

1. Initial program 75.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Add Preprocessing
3. 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 99.1

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

   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 99.1

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

   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 99.1

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

   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 99.1

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

   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 99.1

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

4. Applied rewrites 99.1%

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

   1. metadata-eval N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. lower-/.f64 99.5

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 75.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Add Preprocessing
3. 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 99.1

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

   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 99.1

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

   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 99.1

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

   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 99.1

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

   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 99.1

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

4. Applied rewrites 99.1%

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

   1. metadata-eval N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. lower-/.f64 99.5

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 99.2

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

8. Applied rewrites 99.2%

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

9. Final simplification 99.2%

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

Alternative 6: 99.2% accurate, 1.0× speedup

Math

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

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 (* x_m 0.5))))
   (* x_s (* (* (/ 2.6666666666666665 (sin x_m)) t_0) 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((x_m * 0.5));
	return x_s * (((2.6666666666666665 / sin(x_m)) * t_0) * 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((x_m * 0.5d0))
    code = x_s * (((2.6666666666666665d0 / sin(x_m)) * t_0) * 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((x_m * 0.5));
	return x_s * (((2.6666666666666665 / Math.sin(x_m)) * t_0) * 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((x_m * 0.5))
	return x_s * (((2.6666666666666665 / math.sin(x_m)) * t_0) * t_0)

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

Derivation

1. Initial program 75.6%

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

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 99.2

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

   15. lift-/.f64 N/A

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

   16. metadata-eval 99.2

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.2

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

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. Final simplification 99.2%

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

Alternative 7: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{-19}:\\ \;\;\;\;\frac{0.25 \cdot x\_m}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(x\_m \cdot 0.5\right)}^{2}}{0.375 \cdot \sin x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 4e-19)
    (/ (* 0.25 x_m) 0.375)
    (/ (pow (sin (* x_m 0.5)) 2.0) (* 0.375 (sin x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 4e-19) {
		tmp = (0.25 * x_m) / 0.375;
	} else {
		tmp = pow(sin((x_m * 0.5)), 2.0) / (0.375 * 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 <= 4d-19) then
        tmp = (0.25d0 * x_m) / 0.375d0
    else
        tmp = (sin((x_m * 0.5d0)) ** 2.0d0) / (0.375d0 * 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 <= 4e-19) {
		tmp = (0.25 * x_m) / 0.375;
	} else {
		tmp = Math.pow(Math.sin((x_m * 0.5)), 2.0) / (0.375 * 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 <= 4e-19:
		tmp = (0.25 * x_m) / 0.375
	else:
		tmp = math.pow(math.sin((x_m * 0.5)), 2.0) / (0.375 * 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 <= 4e-19)
		tmp = Float64(Float64(0.25 * x_m) / 0.375);
	else
		tmp = Float64((sin(Float64(x_m * 0.5)) ^ 2.0) / Float64(0.375 * 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 <= 4e-19)
		tmp = (0.25 * x_m) / 0.375;
	else
		tmp = (sin((x_m * 0.5)) ^ 2.0) / (0.375 * 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, 4e-19], N[(N[(0.25 * x$95$m), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[(0.375 * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999999e-19

   1. Initial program 68.7%

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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\frac{\frac{0.5 - 0.5 \cdot \cos x}{\sin x}}{0.375}} \]

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 64.2

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

   9. Applied rewrites 64.2%

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

   if 3.9999999999999999e-19 < x

   1. Initial program 99.2%

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

      1. 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. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. clear-num-rev N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. div-inv N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lift-/.f64 N/A

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

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{{\sin \left(0.5 \cdot x\right)}^{2}}{\sin x \cdot 0.375}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-19}:\\ \;\;\;\;\frac{0.25 \cdot x}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375 \cdot \sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25 \cdot x\_m}{0.375}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x\_m \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x\_m}\\ \end{array} \end{array} \]

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 5e-10)
    (/ (* 0.25 x_m) 0.375)
    (* (pow (sin (* x_m 0.5)) 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 <= 5e-10) {
		tmp = (0.25 * x_m) / 0.375;
	} else {
		tmp = pow(sin((x_m * 0.5)), 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 <= 5d-10) then
        tmp = (0.25d0 * x_m) / 0.375d0
    else
        tmp = (sin((x_m * 0.5d0)) ** 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 <= 5e-10) {
		tmp = (0.25 * x_m) / 0.375;
	} else {
		tmp = Math.pow(Math.sin((x_m * 0.5)), 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 <= 5e-10:
		tmp = (0.25 * x_m) / 0.375
	else:
		tmp = math.pow(math.sin((x_m * 0.5)), 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 <= 5e-10)
		tmp = Float64(Float64(0.25 * x_m) / 0.375);
	else
		tmp = Float64((sin(Float64(x_m * 0.5)) ^ 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 <= 5e-10)
		tmp = (0.25 * x_m) / 0.375;
	else
		tmp = (sin((x_m * 0.5)) ^ 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, 5e-10], N[(N[(0.25 * x$95$m), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[Power[N[Sin[N[(x$95$m * 0.5), $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 < 5.00000000000000031e-10

   1. Initial program 68.9%

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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\frac{\frac{0.5 - 0.5 \cdot \cos x}{\sin x}}{0.375}} \]

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 64.3

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

   9. Applied rewrites 64.3%

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

   if 5.00000000000000031e-10 < x

   1. Initial program 99.2%

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

      1. 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 99.3

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

      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 99.3

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

   4. Applied rewrites 99.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25 \cdot x}{0.375}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{0.25 \cdot x\_m}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(x\_m \cdot 0.5\right)}^{2}}{\sin x\_m} \cdot 2.6666666666666665\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2e-15)
    (/ (* 0.25 x_m) 0.375)
    (* (/ (pow (sin (* x_m 0.5)) 2.0) (sin x_m)) 2.6666666666666665))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2e-15) {
		tmp = (0.25 * x_m) / 0.375;
	} else {
		tmp = (pow(sin((x_m * 0.5)), 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 <= 2d-15) then
        tmp = (0.25d0 * x_m) / 0.375d0
    else
        tmp = ((sin((x_m * 0.5d0)) ** 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 <= 2e-15) {
		tmp = (0.25 * x_m) / 0.375;
	} else {
		tmp = (Math.pow(Math.sin((x_m * 0.5)), 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 <= 2e-15:
		tmp = (0.25 * x_m) / 0.375
	else:
		tmp = (math.pow(math.sin((x_m * 0.5)), 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 <= 2e-15)
		tmp = Float64(Float64(0.25 * x_m) / 0.375);
	else
		tmp = Float64(Float64((sin(Float64(x_m * 0.5)) ^ 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 <= 2e-15)
		tmp = (0.25 * x_m) / 0.375;
	else
		tmp = ((sin((x_m * 0.5)) ^ 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, 2e-15], N[(N[(0.25 * x$95$m), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(N[Power[N[Sin[N[(x$95$m * 0.5), $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 < 2.0000000000000002e-15

   1. Initial program 68.7%

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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\frac{\frac{0.5 - 0.5 \cdot \cos x}{\sin x}}{0.375}} \]

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 64.2

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

   9. Applied rewrites 64.2%

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

   if 2.0000000000000002e-15 < x

   1. Initial program 99.2%

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

      1. 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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 99.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.2

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

      14. lift-/.f64 N/A

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

      15. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{0.25 \cdot x}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x} \cdot 2.6666666666666665\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.124:\\ \;\;\;\;\frac{\sin \left(x\_m \cdot 0.5\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001953125\right), x\_m \cdot x\_m, -0.09375\right), x\_m \cdot x\_m, 0.75\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.375}{\mathsf{fma}\left(-0.5, \cos x\_m, 0.5\right)} \cdot \sin x\_m}\\ \end{array} \end{array} \]

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 0.124)
    (/
     (sin (* x_m 0.5))
     (fma
      (fma
       (fma -1.6276041666666666e-5 (* x_m x_m) 0.001953125)
       (* x_m x_m)
       -0.09375)
      (* x_m x_m)
      0.75))
    (/ 1.0 (* (/ 0.375 (fma -0.5 (cos x_m) 0.5)) (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 <= 0.124) {
		tmp = sin((x_m * 0.5)) / fma(fma(fma(-1.6276041666666666e-5, (x_m * x_m), 0.001953125), (x_m * x_m), -0.09375), (x_m * x_m), 0.75);
	} else {
		tmp = 1.0 / ((0.375 / fma(-0.5, cos(x_m), 0.5)) * 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 <= 0.124)
		tmp = Float64(sin(Float64(x_m * 0.5)) / fma(fma(fma(-1.6276041666666666e-5, Float64(x_m * x_m), 0.001953125), Float64(x_m * x_m), -0.09375), Float64(x_m * x_m), 0.75));
	else
		tmp = Float64(1.0 / Float64(Float64(0.375 / fma(-0.5, cos(x_m), 0.5)) * 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, 0.124], N[(N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(-1.6276041666666666e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001953125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.09375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.75), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.375 / N[(-0.5 * N[Cos[x$95$m], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.124

   1. Initial program 69.2%

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

      1. 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. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{\frac{3}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}\right) + \frac{3}{4}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}\right) \cdot {x}^{2}} + \frac{3}{4}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}, {x}^{2}, \frac{3}{4}\right)}} \]

      4. sub-neg N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{32}\right)\right)}, {x}^{2}, \frac{3}{4}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{3}{32}\right)\right), {x}^{2}, \frac{3}{4}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-3}{32}}, {x}^{2}, \frac{3}{4}\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}, {x}^{2}, \frac{-3}{32}\right)}, {x}^{2}, \frac{3}{4}\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{61440} \cdot {x}^{2} + \frac{1}{512}}, {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{61440}, {x}^{2}, \frac{1}{512}\right)}, {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, \color{blue}{x \cdot x}, \frac{1}{512}\right), {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, \color{blue}{x \cdot x}, \frac{1}{512}\right), {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, x \cdot x, \frac{1}{512}\right), \color{blue}{x \cdot x}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, x \cdot x, \frac{1}{512}\right), \color{blue}{x \cdot x}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, x \cdot x, \frac{1}{512}\right), x \cdot x, \frac{-3}{32}\right), \color{blue}{x \cdot x}, \frac{3}{4}\right)} \]

      15. lower-*.f64 65.2

          \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x \cdot x, 0.001953125\right), x \cdot x, -0.09375\right), \color{blue}{x \cdot x}, 0.75\right)} \]

   7. Applied rewrites 65.2%

      \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x \cdot x, 0.001953125\right), x \cdot x, -0.09375\right), x \cdot x, 0.75\right)}} \]

   if 0.124 < x

   1. Initial program 99.2%

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

      1. 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. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\frac{\frac{0.5 - 0.5 \cdot \cos x}{\sin x}}{0.375}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 98.8

         \[\leadsto \frac{1}{\color{blue}{\frac{0.375}{0.5 - 0.5 \cdot \cos x}} \cdot \sin x} \]

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 98.8

          \[\leadsto \frac{1}{\frac{0.375}{\color{blue}{\mathsf{fma}\left(-0.5, \cos x, 0.5\right)}} \cdot \sin x} \]

   8. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{0.375}{\mathsf{fma}\left(-0.5, \cos x, 0.5\right)} \cdot \sin x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.124:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x \cdot x, 0.001953125\right), x \cdot x, -0.09375\right), x \cdot x, 0.75\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.375}{\mathsf{fma}\left(-0.5, \cos x, 0.5\right)} \cdot \sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.124:\\ \;\;\;\;\frac{\sin \left(x\_m \cdot 0.5\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001953125\right), x\_m \cdot x\_m, -0.09375\right), x\_m \cdot x\_m, 0.75\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin x\_m}{\mathsf{fma}\left(-1.3333333333333333, \cos x\_m, 1.3333333333333333\right)}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.124)
    (/
     (sin (* x_m 0.5))
     (fma
      (fma
       (fma -1.6276041666666666e-5 (* x_m x_m) 0.001953125)
       (* x_m x_m)
       -0.09375)
      (* x_m x_m)
      0.75))
    (/
     1.0
     (/ (sin x_m) (fma -1.3333333333333333 (cos x_m) 1.3333333333333333))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.124) {
		tmp = sin((x_m * 0.5)) / fma(fma(fma(-1.6276041666666666e-5, (x_m * x_m), 0.001953125), (x_m * x_m), -0.09375), (x_m * x_m), 0.75);
	} else {
		tmp = 1.0 / (sin(x_m) / fma(-1.3333333333333333, cos(x_m), 1.3333333333333333));
	}
	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 <= 0.124)
		tmp = Float64(sin(Float64(x_m * 0.5)) / fma(fma(fma(-1.6276041666666666e-5, Float64(x_m * x_m), 0.001953125), Float64(x_m * x_m), -0.09375), Float64(x_m * x_m), 0.75));
	else
		tmp = Float64(1.0 / Float64(sin(x_m) / fma(-1.3333333333333333, cos(x_m), 1.3333333333333333)));
	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, 0.124], N[(N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(-1.6276041666666666e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001953125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.09375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.75), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sin[x$95$m], $MachinePrecision] / N[(-1.3333333333333333 * N[Cos[x$95$m], $MachinePrecision] + 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.124

   1. Initial program 69.2%

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

      1. 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. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{\frac{3}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}\right) + \frac{3}{4}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}\right) \cdot {x}^{2}} + \frac{3}{4}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}, {x}^{2}, \frac{3}{4}\right)}} \]

      4. sub-neg N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{32}\right)\right)}, {x}^{2}, \frac{3}{4}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{3}{32}\right)\right), {x}^{2}, \frac{3}{4}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-3}{32}}, {x}^{2}, \frac{3}{4}\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}, {x}^{2}, \frac{-3}{32}\right)}, {x}^{2}, \frac{3}{4}\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{61440} \cdot {x}^{2} + \frac{1}{512}}, {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{61440}, {x}^{2}, \frac{1}{512}\right)}, {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, \color{blue}{x \cdot x}, \frac{1}{512}\right), {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, \color{blue}{x \cdot x}, \frac{1}{512}\right), {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, x \cdot x, \frac{1}{512}\right), \color{blue}{x \cdot x}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, x \cdot x, \frac{1}{512}\right), \color{blue}{x \cdot x}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, x \cdot x, \frac{1}{512}\right), x \cdot x, \frac{-3}{32}\right), \color{blue}{x \cdot x}, \frac{3}{4}\right)} \]

      15. lower-*.f64 65.2

          \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x \cdot x, 0.001953125\right), x \cdot x, -0.09375\right), \color{blue}{x \cdot x}, 0.75\right)} \]

   7. Applied rewrites 65.2%

      \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x \cdot x, 0.001953125\right), x \cdot x, -0.09375\right), x \cdot x, 0.75\right)}} \]

   if 0.124 < x

   1. Initial program 99.2%

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

      1. 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 98.8

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

      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 98.8

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

      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 98.8

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

      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 98.8

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

      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 98.8

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

   4. Applied rewrites 98.8%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. lower-/.f64 99.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.1

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

   6. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 99.1

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

   8. Applied rewrites 99.1%

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

   10. Applied rewrites 98.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\mathsf{fma}\left(-1.3333333333333333, \cos x, 1.3333333333333333\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.124:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x \cdot x, 0.001953125\right), x \cdot x, -0.09375\right), x \cdot x, 0.75\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin x}{\mathsf{fma}\left(-1.3333333333333333, \cos x, 1.3333333333333333\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.124:\\ \;\;\;\;\frac{\sin \left(x\_m \cdot 0.5\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001953125\right), x\_m \cdot x\_m, -0.09375\right), x\_m \cdot x\_m, 0.75\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x\_m, -1.3333333333333333, 1.3333333333333333\right)}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.124)
    (/
     (sin (* x_m 0.5))
     (fma
      (fma
       (fma -1.6276041666666666e-5 (* x_m x_m) 0.001953125)
       (* x_m x_m)
       -0.09375)
      (* x_m x_m)
      0.75))
    (/ (fma (cos x_m) -1.3333333333333333 1.3333333333333333) (sin x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.124) {
		tmp = sin((x_m * 0.5)) / fma(fma(fma(-1.6276041666666666e-5, (x_m * x_m), 0.001953125), (x_m * x_m), -0.09375), (x_m * x_m), 0.75);
	} else {
		tmp = fma(cos(x_m), -1.3333333333333333, 1.3333333333333333) / sin(x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.124)
		tmp = Float64(sin(Float64(x_m * 0.5)) / fma(fma(fma(-1.6276041666666666e-5, Float64(x_m * x_m), 0.001953125), Float64(x_m * x_m), -0.09375), Float64(x_m * x_m), 0.75));
	else
		tmp = Float64(fma(cos(x_m), -1.3333333333333333, 1.3333333333333333) / sin(x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.124], N[(N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(-1.6276041666666666e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001953125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.09375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.75), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x$95$m], $MachinePrecision] * -1.3333333333333333 + 1.3333333333333333), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.124

   1. Initial program 69.2%

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

      1. 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. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{\frac{3}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}\right) + \frac{3}{4}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}\right) \cdot {x}^{2}} + \frac{3}{4}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) - \frac{3}{32}, {x}^{2}, \frac{3}{4}\right)}} \]

      4. sub-neg N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{32}\right)\right)}, {x}^{2}, \frac{3}{4}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{3}{32}\right)\right), {x}^{2}, \frac{3}{4}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-3}{32}}, {x}^{2}, \frac{3}{4}\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{512} + \frac{-1}{61440} \cdot {x}^{2}, {x}^{2}, \frac{-3}{32}\right)}, {x}^{2}, \frac{3}{4}\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{61440} \cdot {x}^{2} + \frac{1}{512}}, {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{61440}, {x}^{2}, \frac{1}{512}\right)}, {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, \color{blue}{x \cdot x}, \frac{1}{512}\right), {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, \color{blue}{x \cdot x}, \frac{1}{512}\right), {x}^{2}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, x \cdot x, \frac{1}{512}\right), \color{blue}{x \cdot x}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, x \cdot x, \frac{1}{512}\right), \color{blue}{x \cdot x}, \frac{-3}{32}\right), {x}^{2}, \frac{3}{4}\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{61440}, x \cdot x, \frac{1}{512}\right), x \cdot x, \frac{-3}{32}\right), \color{blue}{x \cdot x}, \frac{3}{4}\right)} \]

      15. lower-*.f64 65.2

          \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x \cdot x, 0.001953125\right), x \cdot x, -0.09375\right), \color{blue}{x \cdot x}, 0.75\right)} \]

   7. Applied rewrites 65.2%

      \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x \cdot x, 0.001953125\right), x \cdot x, -0.09375\right), x \cdot x, 0.75\right)}} \]

   if 0.124 < x

   1. Initial program 99.2%

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

      1. 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. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\frac{\frac{0.5 - 0.5 \cdot \cos x}{\sin x}}{0.375}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. metadata-eval 98.6

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

   8. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip3-+ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. flip3-+ N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   10. Applied rewrites 98.7%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.124:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.6276041666666666 \cdot 10^{-5}, x \cdot x, 0.001953125\right), x \cdot x, -0.09375\right), x \cdot x, 0.75\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, -1.3333333333333333, 1.3333333333333333\right)}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 9.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00021081349206349207, x\_m \cdot x\_m, 0.0020833333333333333\right), x\_m \cdot x\_m, 0.020833333333333332\right), x\_m \cdot x\_m, 0.25\right) \cdot x\_m}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\mathsf{fma}\left(-0.5, x\_m, \mathsf{PI}\left(\right)\right)\right) \cdot 2.6666666666666665\right) \cdot 0.5\\ \end{array} \end{array} \]

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 9.5)
    (/
     (*
      (fma
       (fma
        (fma 0.00021081349206349207 (* x_m x_m) 0.0020833333333333333)
        (* x_m x_m)
        0.020833333333333332)
       (* x_m x_m)
       0.25)
      x_m)
     0.375)
    (* (* (sin (fma -0.5 x_m (PI))) 2.6666666666666665) 0.5))))

Derivation

1. Split input into 2 regimes

2. if x < 9.5

   1. Initial program 69.5%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Add Preprocessing
   3. 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. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\frac{0.5 - 0.5 \cdot \cos x}{\sin x}}{0.375}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{48} + {x}^{2} \cdot \left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right)\right)\right)}}{\frac{3}{8}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{48} + {x}^{2} \cdot \left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right)\right)\right) \cdot x}}{\frac{3}{8}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{48} + {x}^{2} \cdot \left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right)\right)\right) \cdot x}}{\frac{3}{8}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{48} + {x}^{2} \cdot \left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right)\right) + \frac{1}{4}\right)} \cdot x}{\frac{3}{8}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{48} + {x}^{2} \cdot \left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{48} + {x}^{2} \cdot \left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right), {x}^{2}, \frac{1}{4}\right)} \cdot x}{\frac{3}{8}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right) + \frac{1}{48}}, {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{48}, {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}, {x}^{2}, \frac{1}{48}\right)}, {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{17}{80640} \cdot {x}^{2} + \frac{1}{480}}, {x}^{2}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{17}{80640}, {x}^{2}, \frac{1}{480}\right)}, {x}^{2}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{80640}, \color{blue}{x \cdot x}, \frac{1}{480}\right), {x}^{2}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{80640}, \color{blue}{x \cdot x}, \frac{1}{480}\right), {x}^{2}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{80640}, x \cdot x, \frac{1}{480}\right), \color{blue}{x \cdot x}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{80640}, x \cdot x, \frac{1}{480}\right), \color{blue}{x \cdot x}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{80640}, x \cdot x, \frac{1}{480}\right), x \cdot x, \frac{1}{48}\right), \color{blue}{x \cdot x}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

      16. lower-*.f64 64.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00021081349206349207, x \cdot x, 0.0020833333333333333\right), x \cdot x, 0.020833333333333332\right), \color{blue}{x \cdot x}, 0.25\right) \cdot x}{0.375} \]

   9. Applied rewrites 64.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00021081349206349207, x \cdot x, 0.0020833333333333333\right), x \cdot x, 0.020833333333333332\right), x \cdot x, 0.25\right) \cdot x}}{0.375} \]

   if 9.5 < x

   1. Initial program 99.2%

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

      1. 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 98.8

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

      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 98.8

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

      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 98.8

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

      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 98.8

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

      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 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 11.0%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f64 N/A

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

      3. sin-neg-rev N/A

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

      4. sin-+PI-rev N/A

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

      5. lower-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-PI.f64 11.3

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

   9. Applied rewrites 11.3%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00021081349206349207, x \cdot x, 0.0020833333333333333\right), x \cdot x, 0.020833333333333332\right), x \cdot x, 0.25\right) \cdot x}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\mathsf{fma}\left(-0.5, x, \mathsf{PI}\left(\right)\right)\right) \cdot 2.6666666666666665\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{\sin \left(x\_m \cdot 0.5\right)}{0.75} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ (sin (* x_m 0.5)) 0.75)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (sin((x_m * 0.5)) / 0.75);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (sin((x_m * 0.5d0)) / 0.75d0)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (Math.sin((x_m * 0.5)) / 0.75);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (math.sin((x_m * 0.5)) / 0.75)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(sin(Float64(x_m * 0.5)) / 0.75))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (sin((x_m * 0.5)) / 0.75);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Add Preprocessing
3. 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. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. clear-num-rev N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 54.9%

   \[\leadsto \frac{\sin \left(0.5 \cdot x\right)}{\color{blue}{0.75}} \]

8. Final simplification 54.9%

   \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{0.75} \]
9. Add Preprocessing

Alternative 15: 54.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Add Preprocessing
3. 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 99.1

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

   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 99.1

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

   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 99.1

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

   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 99.1

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

   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 99.1

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

4. Applied rewrites 99.1%

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

   1. metadata-eval N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. lower-/.f64 99.5

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 99.2

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 54.7%

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

12. Final simplification 54.7%

    \[\leadsto 1.3333333333333333 \cdot \sin \left(x \cdot 0.5\right) \]
13. Add Preprocessing

Alternative 16: 50.4% accurate, 6.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00021081349206349207, x\_m \cdot x\_m, 0.0020833333333333333\right), x\_m \cdot x\_m, 0.020833333333333332\right), x\_m \cdot x\_m, 0.25\right) \cdot x\_m}{0.375} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/
   (*
    (fma
     (fma
      (fma 0.00021081349206349207 (* x_m x_m) 0.0020833333333333333)
      (* x_m x_m)
      0.020833333333333332)
     (* x_m x_m)
     0.25)
    x_m)
   0.375)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((fma(fma(fma(0.00021081349206349207, (x_m * x_m), 0.0020833333333333333), (x_m * x_m), 0.020833333333333332), (x_m * x_m), 0.25) * x_m) / 0.375);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(fma(fma(fma(0.00021081349206349207, Float64(x_m * x_m), 0.0020833333333333333), Float64(x_m * x_m), 0.020833333333333332), Float64(x_m * x_m), 0.25) * x_m) / 0.375))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(N[(N[(N[(0.00021081349206349207 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0020833333333333333), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.25), $MachinePrecision] * x$95$m), $MachinePrecision] / 0.375), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Add Preprocessing
3. 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. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. clear-num-rev N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 53.2%

   \[\leadsto \color{blue}{\frac{\frac{0.5 - 0.5 \cdot \cos x}{\sin x}}{0.375}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{48} + {x}^{2} \cdot \left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right)\right)\right)}}{\frac{3}{8}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{48} + {x}^{2} \cdot \left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right)\right)\right) \cdot x}}{\frac{3}{8}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{48} + {x}^{2} \cdot \left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right)\right)\right) \cdot x}}{\frac{3}{8}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{48} + {x}^{2} \cdot \left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right)\right) + \frac{1}{4}\right)} \cdot x}{\frac{3}{8}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{48} + {x}^{2} \cdot \left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{48} + {x}^{2} \cdot \left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right), {x}^{2}, \frac{1}{4}\right)} \cdot x}{\frac{3}{8}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right) + \frac{1}{48}}, {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{48}, {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{480} + \frac{17}{80640} \cdot {x}^{2}, {x}^{2}, \frac{1}{48}\right)}, {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{17}{80640} \cdot {x}^{2} + \frac{1}{480}}, {x}^{2}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{17}{80640}, {x}^{2}, \frac{1}{480}\right)}, {x}^{2}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{80640}, \color{blue}{x \cdot x}, \frac{1}{480}\right), {x}^{2}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{80640}, \color{blue}{x \cdot x}, \frac{1}{480}\right), {x}^{2}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{80640}, x \cdot x, \frac{1}{480}\right), \color{blue}{x \cdot x}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{80640}, x \cdot x, \frac{1}{480}\right), \color{blue}{x \cdot x}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{80640}, x \cdot x, \frac{1}{480}\right), x \cdot x, \frac{1}{48}\right), \color{blue}{x \cdot x}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   16. lower-*.f64 51.8

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00021081349206349207, x \cdot x, 0.0020833333333333333\right), x \cdot x, 0.020833333333333332\right), \color{blue}{x \cdot x}, 0.25\right) \cdot x}{0.375} \]

9. Applied rewrites 51.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00021081349206349207, x \cdot x, 0.0020833333333333333\right), x \cdot x, 0.020833333333333332\right), x \cdot x, 0.25\right) \cdot x}}{0.375} \]
10. Add Preprocessing

Alternative 17: 50.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0020833333333333333, x\_m \cdot x\_m, 0.020833333333333332\right), x\_m \cdot x\_m, 0.25\right) \cdot x\_m}{0.375} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/
   (*
    (fma
     (fma 0.0020833333333333333 (* x_m x_m) 0.020833333333333332)
     (* x_m x_m)
     0.25)
    x_m)
   0.375)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((fma(fma(0.0020833333333333333, (x_m * x_m), 0.020833333333333332), (x_m * x_m), 0.25) * x_m) / 0.375);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(fma(fma(0.0020833333333333333, Float64(x_m * x_m), 0.020833333333333332), Float64(x_m * x_m), 0.25) * x_m) / 0.375))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(N[(N[(0.0020833333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.25), $MachinePrecision] * x$95$m), $MachinePrecision] / 0.375), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Add Preprocessing
3. 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. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. clear-num-rev N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 53.2%

   \[\leadsto \color{blue}{\frac{\frac{0.5 - 0.5 \cdot \cos x}{\sin x}}{0.375}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{48} + \frac{1}{480} \cdot {x}^{2}\right)\right)}}{\frac{3}{8}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{48} + \frac{1}{480} \cdot {x}^{2}\right)\right) \cdot x}}{\frac{3}{8}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{48} + \frac{1}{480} \cdot {x}^{2}\right)\right) \cdot x}}{\frac{3}{8}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{48} + \frac{1}{480} \cdot {x}^{2}\right) + \frac{1}{4}\right)} \cdot x}{\frac{3}{8}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{48} + \frac{1}{480} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{48} + \frac{1}{480} \cdot {x}^{2}, {x}^{2}, \frac{1}{4}\right)} \cdot x}{\frac{3}{8}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{480} \cdot {x}^{2} + \frac{1}{48}}, {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{480}, {x}^{2}, \frac{1}{48}\right)}, {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{480}, \color{blue}{x \cdot x}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{480}, \color{blue}{x \cdot x}, \frac{1}{48}\right), {x}^{2}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   10. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{480}, x \cdot x, \frac{1}{48}\right), \color{blue}{x \cdot x}, \frac{1}{4}\right) \cdot x}{\frac{3}{8}} \]

   11. lower-*.f64 51.7

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0020833333333333333, x \cdot x, 0.020833333333333332\right), \color{blue}{x \cdot x}, 0.25\right) \cdot x}{0.375} \]

9. Applied rewrites 51.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0020833333333333333, x \cdot x, 0.020833333333333332\right), x \cdot x, 0.25\right) \cdot x}}{0.375} \]
10. Add Preprocessing

Alternative 18: 50.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0005621693121693122, x\_m \cdot x\_m, 0.005555555555555556\right), x\_m \cdot x\_m, 0.05555555555555555\right), x\_m \cdot x\_m, 0.6666666666666666\right) \cdot x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   (fma
    (fma
     (fma 0.0005621693121693122 (* x_m x_m) 0.005555555555555556)
     (* x_m x_m)
     0.05555555555555555)
    (* x_m x_m)
    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 * (fma(fma(fma(0.0005621693121693122, (x_m * x_m), 0.005555555555555556), (x_m * x_m), 0.05555555555555555), (x_m * x_m), 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(fma(fma(fma(0.0005621693121693122, Float64(x_m * x_m), 0.005555555555555556), Float64(x_m * x_m), 0.05555555555555555), Float64(x_m * x_m), 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[(N[(N[(N[(0.0005621693121693122 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.005555555555555556), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.05555555555555555), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \cdot x} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right) + \frac{2}{3}\right)} \cdot x \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + \frac{2}{3}\right) \cdot x \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right), {x}^{2}, \frac{2}{3}\right)} \cdot x \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right) + \frac{1}{18}}, {x}^{2}, \frac{2}{3}\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{18}, {x}^{2}, \frac{2}{3}\right) \cdot x \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}, {x}^{2}, \frac{1}{18}\right)}, {x}^{2}, \frac{2}{3}\right) \cdot x \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{17}{30240} \cdot {x}^{2} + \frac{1}{180}}, {x}^{2}, \frac{1}{18}\right), {x}^{2}, \frac{2}{3}\right) \cdot x \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{17}{30240}, {x}^{2}, \frac{1}{180}\right)}, {x}^{2}, \frac{1}{18}\right), {x}^{2}, \frac{2}{3}\right) \cdot x \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{30240}, \color{blue}{x \cdot x}, \frac{1}{180}\right), {x}^{2}, \frac{1}{18}\right), {x}^{2}, \frac{2}{3}\right) \cdot x \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{30240}, \color{blue}{x \cdot x}, \frac{1}{180}\right), {x}^{2}, \frac{1}{18}\right), {x}^{2}, \frac{2}{3}\right) \cdot x \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{30240}, x \cdot x, \frac{1}{180}\right), \color{blue}{x \cdot x}, \frac{1}{18}\right), {x}^{2}, \frac{2}{3}\right) \cdot x \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{30240}, x \cdot x, \frac{1}{180}\right), \color{blue}{x \cdot x}, \frac{1}{18}\right), {x}^{2}, \frac{2}{3}\right) \cdot x \]

   15. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{30240}, x \cdot x, \frac{1}{180}\right), x \cdot x, \frac{1}{18}\right), \color{blue}{x \cdot x}, \frac{2}{3}\right) \cdot x \]

   16. lower-*.f64 51.5

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0005621693121693122, x \cdot x, 0.005555555555555556\right), x \cdot x, 0.05555555555555555\right), \color{blue}{x \cdot x}, 0.6666666666666666\right) \cdot x \]

5. Applied rewrites 51.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0005621693121693122, x \cdot x, 0.005555555555555556\right), x \cdot x, 0.05555555555555555\right), x \cdot x, 0.6666666666666666\right) \cdot x} \]
6. Add Preprocessing

Alternative 19: 50.7% accurate, 20.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Add Preprocessing
3. 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. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. clear-num-rev N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f64 N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 53.2%

   \[\leadsto \color{blue}{\frac{\frac{0.5 - 0.5 \cdot \cos x}{\sin x}}{0.375}} \]

7. Taylor expanded in x around 0

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

   1. lower-*.f64 51.5

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

9. Applied rewrites 51.5%

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

Alternative 20: 50.2% accurate, 20.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 75.6%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 51.3

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

5. Applied rewrites 51.3%

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

Alternative 21: 50.4% accurate, 57.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 51.2

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

5. Applied rewrites 51.2%

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

Developer Target 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024298 
(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)))