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

Percentage Accurate: 77.2% → 99.5%

Time: 14.8s

Alternatives: 18

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: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0002:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x\_m}^{3} + x\_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x\_m \cdot \frac{0.375}{{\sin \left(x\_m \cdot 0.5\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0002)
    (/ (+ (* 0.020833333333333332 (pow x_m 3.0)) (* x_m 0.25)) 0.375)
    (/ 1.0 (* (sin x_m) (/ 0.375 (pow (sin (* x_m 0.5)) 2.0)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0002) {
		tmp = ((0.020833333333333332 * pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = 1.0 / (sin(x_m) * (0.375 / pow(sin((x_m * 0.5)), 2.0)));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0002d0) then
        tmp = ((0.020833333333333332d0 * (x_m ** 3.0d0)) + (x_m * 0.25d0)) / 0.375d0
    else
        tmp = 1.0d0 / (sin(x_m) * (0.375d0 / (sin((x_m * 0.5d0)) ** 2.0d0)))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0002) {
		tmp = ((0.020833333333333332 * Math.pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = 1.0 / (Math.sin(x_m) * (0.375 / Math.pow(Math.sin((x_m * 0.5)), 2.0)));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.0002:
		tmp = ((0.020833333333333332 * math.pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375
	else:
		tmp = 1.0 / (math.sin(x_m) * (0.375 / math.pow(math.sin((x_m * 0.5)), 2.0)))
	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 <= 0.0002)
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x_m ^ 3.0)) + Float64(x_m * 0.25)) / 0.375);
	else
		tmp = Float64(1.0 / Float64(sin(x_m) * Float64(0.375 / (sin(Float64(x_m * 0.5)) ^ 2.0))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.0002)
		tmp = ((0.020833333333333332 * (x_m ^ 3.0)) + (x_m * 0.25)) / 0.375;
	else
		tmp = 1.0 / (sin(x_m) * (0.375 / (sin((x_m * 0.5)) ^ 2.0)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0002], N[(N[(N[(0.020833333333333332 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(1.0 / N[(N[Sin[x$95$m], $MachinePrecision] * N[(0.375 / N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-4

   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. Step-by-step derivation

      1. *-commutative 68.9%

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

      2. remove-double-neg 68.9%

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

      3. sin-neg 68.9%

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

      4. distribute-lft-neg-out 68.9%

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

      5. distribute-rgt-neg-in 68.9%

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

      6. associate-*l/ 99.2%

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

      7. *-commutative 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. sin-neg 99.2%

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

      11. remove-double-neg 99.2%

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

      12. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r/ 69.0%

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

      3. associate-/r/ 69.0%

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

      4. div-inv 69.0%

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

      5. associate-/r* 69.2%

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

      6. pow2 69.2%

         \[\leadsto \frac{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}{\frac{1}{2.6666666666666665}} \]

      7. metadata-eval 69.2%

         \[\leadsto \frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{\color{blue}{0.375}} \]

   6. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Taylor expanded in x around 0 72.8%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]

   if 2.0000000000000001e-4 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l/ 99.1%

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

      4. metadata-eval 99.1%

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

      5. metadata-eval 99.1%

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

      6. metadata-eval 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\frac{8}{3}}}{1} \]

      7. metadata-eval 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\frac{8}{3}}{\color{blue}{\frac{-1}{-1}}} \]

      8. times-frac 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}}} \]

      9. *-commutative 99.1%

         \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}} \]

      10. times-frac 98.9%

          \[\leadsto \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}}} \]

      11. associate-/l* 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \]

      12. *-commutative 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \]

      13. neg-mul-1 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \]

      14. sin-neg 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \]

      15. distribute-lft-neg-out 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \]

      16. associate-*l/ 99.1%

          \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.1%

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

      1. associate-/r/ 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l/ 99.2%

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

      4. associate-/r/ 98.9%

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

      5. associate-*l/ 99.0%

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

      6. div-inv 99.2%

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

      7. times-frac 99.2%

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

      8. metadata-eval 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. associate-*r/ 99.1%

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

      2. clear-num 99.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{0.375}{\frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)}}} \]

      3. associate-*l/ 99.2%

         \[\leadsto \frac{1}{\frac{0.375}{\color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}}}} \]

      4. unpow2 99.2%

         \[\leadsto \frac{1}{\frac{0.375}{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}} \]

      5. div-inv 99.1%

         \[\leadsto \frac{1}{\color{blue}{0.375 \cdot \frac{1}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}}} \]

      6. clear-num 99.1%

         \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

      7. div-inv 99.0%

         \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\left(\sin x \cdot \frac{1}{{\sin \left(x \cdot 0.5\right)}^{2}}\right)}} \]

      8. unpow2 99.0%

         \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot \frac{1}{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}\right)} \]

      9. associate-/l/ 99.1%

         \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot \color{blue}{\frac{\frac{1}{\sin \left(x \cdot 0.5\right)}}{\sin \left(x \cdot 0.5\right)}}\right)} \]

      10. inv-pow 99.1%

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

      11. pow1 99.1%

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

      12. pow-div 99.0%

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

      13. metadata-eval 99.0%

          \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{\color{blue}{-2}}\right)} \]

   8. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{1}{0.375 \cdot \left(\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)}} \]

   9. Taylor expanded in x around inf 99.1%

      \[\leadsto \frac{1}{\color{blue}{0.375 \cdot \frac{\sin x}{{\sin \left(0.5 \cdot x\right)}^{2}}}} \]
   10. Step-by-step derivation

       1. *-commutative 99.1%

          \[\leadsto \frac{1}{0.375 \cdot \frac{\sin x}{{\sin \color{blue}{\left(x \cdot 0.5\right)}}^{2}}} \]

       2. associate-*r/ 99.3%

          \[\leadsto \frac{1}{\color{blue}{\frac{0.375 \cdot \sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

       3. associate-*l/ 99.1%

          \[\leadsto \frac{1}{\color{blue}{\frac{0.375}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \sin x}} \]

       4. *-commutative 99.1%

          \[\leadsto \frac{1}{\color{blue}{\sin x \cdot \frac{0.375}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   11. Simplified 99.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0002:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x \cdot \frac{0.375}{{\sin \left(x \cdot 0.5\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0004:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x\_m}^{3} + x\_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x\_m \cdot 0.5\right)}^{2}}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0004)
    (/ (+ (* 0.020833333333333332 (pow x_m 3.0)) (* x_m 0.25)) 0.375)
    (* 2.6666666666666665 (/ (pow (sin (* x_m 0.5)) 2.0) (sin x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0004) {
		tmp = ((0.020833333333333332 * pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 * (pow(sin((x_m * 0.5)), 2.0) / sin(x_m));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0004d0) then
        tmp = ((0.020833333333333332d0 * (x_m ** 3.0d0)) + (x_m * 0.25d0)) / 0.375d0
    else
        tmp = 2.6666666666666665d0 * ((sin((x_m * 0.5d0)) ** 2.0d0) / sin(x_m))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0004) {
		tmp = ((0.020833333333333332 * Math.pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 * (Math.pow(Math.sin((x_m * 0.5)), 2.0) / Math.sin(x_m));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.0004:
		tmp = ((0.020833333333333332 * math.pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375
	else:
		tmp = 2.6666666666666665 * (math.pow(math.sin((x_m * 0.5)), 2.0) / math.sin(x_m))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0004)
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x_m ^ 3.0)) + Float64(x_m * 0.25)) / 0.375);
	else
		tmp = Float64(2.6666666666666665 * Float64((sin(Float64(x_m * 0.5)) ^ 2.0) / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.0004)
		tmp = ((0.020833333333333332 * (x_m ^ 3.0)) + (x_m * 0.25)) / 0.375;
	else
		tmp = 2.6666666666666665 * ((sin((x_m * 0.5)) ^ 2.0) / sin(x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0004], N[(N[(N[(0.020833333333333332 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000019e-4

   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. Step-by-step derivation

      1. *-commutative 68.9%

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

      2. remove-double-neg 68.9%

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

      3. sin-neg 68.9%

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

      4. distribute-lft-neg-out 68.9%

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

      5. distribute-rgt-neg-in 68.9%

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

      6. associate-*l/ 99.2%

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

      7. *-commutative 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. sin-neg 99.2%

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

      11. remove-double-neg 99.2%

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

      12. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r/ 69.0%

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

      3. associate-/r/ 69.0%

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

      4. div-inv 69.0%

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

      5. associate-/r* 69.2%

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

      6. pow2 69.2%

         \[\leadsto \frac{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}{\frac{1}{2.6666666666666665}} \]

      7. metadata-eval 69.2%

         \[\leadsto \frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{\color{blue}{0.375}} \]

   6. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Taylor expanded in x around 0 72.8%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]

   if 4.00000000000000019e-4 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l/ 99.1%

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

      4. metadata-eval 99.1%

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

      5. metadata-eval 99.1%

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

      6. metadata-eval 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\frac{8}{3}}}{1} \]

      7. metadata-eval 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\frac{8}{3}}{\color{blue}{\frac{-1}{-1}}} \]

      8. times-frac 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}}} \]

      9. *-commutative 99.1%

         \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}} \]

      10. times-frac 98.9%

          \[\leadsto \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}}} \]

      11. associate-/l* 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \]

      12. *-commutative 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \]

      13. neg-mul-1 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \]

      14. sin-neg 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \]

      15. distribute-lft-neg-out 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \]

      16. associate-*l/ 99.1%

          \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.1%

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

      1. div-inv 99.0%

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

      2. clear-num 99.2%

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

      3. associate-*r* 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-*r/ 99.1%

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

      6. pow2 99.1%

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0004:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00015:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x\_m}^{3} + x\_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x\_m \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.00015)
    (/ (+ (* 0.020833333333333332 (pow x_m 3.0)) (* x_m 0.25)) 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 <= 0.00015) {
		tmp = ((0.020833333333333332 * pow(x_m, 3.0)) + (x_m * 0.25)) / 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 <= 0.00015d0) then
        tmp = ((0.020833333333333332d0 * (x_m ** 3.0d0)) + (x_m * 0.25d0)) / 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 <= 0.00015) {
		tmp = ((0.020833333333333332 * Math.pow(x_m, 3.0)) + (x_m * 0.25)) / 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 <= 0.00015:
		tmp = ((0.020833333333333332 * math.pow(x_m, 3.0)) + (x_m * 0.25)) / 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 <= 0.00015)
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x_m ^ 3.0)) + Float64(x_m * 0.25)) / 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 <= 0.00015)
		tmp = ((0.020833333333333332 * (x_m ^ 3.0)) + (x_m * 0.25)) / 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, 0.00015], N[(N[(N[(0.020833333333333332 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 0.25), $MachinePrecision]), $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 < 1.49999999999999987e-4

   1. Initial program 68.6%

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

      1. *-commutative 68.6%

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

      2. remove-double-neg 68.6%

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

      3. sin-neg 68.6%

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

      4. distribute-lft-neg-out 68.6%

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

      5. distribute-rgt-neg-in 68.6%

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

      6. associate-*l/ 99.2%

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

      7. *-commutative 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. sin-neg 99.2%

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

      11. remove-double-neg 99.2%

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

      12. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r/ 68.7%

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

      3. associate-/r/ 68.7%

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

      4. div-inv 68.7%

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

      5. associate-/r* 68.9%

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

      6. pow2 68.9%

         \[\leadsto \frac{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}{\frac{1}{2.6666666666666665}} \]

      7. metadata-eval 68.9%

         \[\leadsto \frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{\color{blue}{0.375}} \]

   6. Applied egg-rr 68.9%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Taylor expanded in x around 0 72.5%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]

   if 1.49999999999999987e-4 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l/ 99.1%

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

      4. metadata-eval 99.1%

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

      5. metadata-eval 99.1%

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

      6. metadata-eval 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\frac{8}{3}}}{1} \]

      7. metadata-eval 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\frac{8}{3}}{\color{blue}{\frac{-1}{-1}}} \]

      8. times-frac 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}}} \]

      9. *-commutative 99.1%

         \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}} \]

      10. times-frac 98.9%

          \[\leadsto \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}}} \]

      11. associate-/l* 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \]

      12. *-commutative 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \]

      13. neg-mul-1 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \]

      14. sin-neg 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \]

      15. distribute-lft-neg-out 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \]

      16. associate-*l/ 99.1%

          \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.1%

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

      1. associate-/r/ 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l/ 99.2%

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

      4. associate-/r/ 98.9%

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

      5. associate-*l/ 99.0%

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

      6. div-inv 99.0%

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

      7. clear-num 99.2%

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

      8. pow2 99.2%

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

   6. Applied egg-rr 99.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00015:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00018:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x\_m}^{3} + x\_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x\_m}{{\sin \left(x\_m \cdot 0.5\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.00018)
    (/ (+ (* 0.020833333333333332 (pow x_m 3.0)) (* x_m 0.25)) 0.375)
    (/ 2.6666666666666665 (/ (sin x_m) (pow (sin (* x_m 0.5)) 2.0))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.00018) {
		tmp = ((0.020833333333333332 * pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 / (sin(x_m) / pow(sin((x_m * 0.5)), 2.0));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.00018d0) then
        tmp = ((0.020833333333333332d0 * (x_m ** 3.0d0)) + (x_m * 0.25d0)) / 0.375d0
    else
        tmp = 2.6666666666666665d0 / (sin(x_m) / (sin((x_m * 0.5d0)) ** 2.0d0))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.00018) {
		tmp = ((0.020833333333333332 * Math.pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 / (Math.sin(x_m) / Math.pow(Math.sin((x_m * 0.5)), 2.0));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.00018:
		tmp = ((0.020833333333333332 * math.pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375
	else:
		tmp = 2.6666666666666665 / (math.sin(x_m) / math.pow(math.sin((x_m * 0.5)), 2.0))
	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 <= 0.00018)
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x_m ^ 3.0)) + Float64(x_m * 0.25)) / 0.375);
	else
		tmp = Float64(2.6666666666666665 / Float64(sin(x_m) / (sin(Float64(x_m * 0.5)) ^ 2.0)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.00018)
		tmp = ((0.020833333333333332 * (x_m ^ 3.0)) + (x_m * 0.25)) / 0.375;
	else
		tmp = 2.6666666666666665 / (sin(x_m) / (sin((x_m * 0.5)) ^ 2.0));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.00018], N[(N[(N[(0.020833333333333332 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(2.6666666666666665 / N[(N[Sin[x$95$m], $MachinePrecision] / N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000011e-4

   1. Initial program 68.8%

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

      1. *-commutative 68.8%

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

      2. remove-double-neg 68.8%

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

      3. sin-neg 68.8%

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

      4. distribute-lft-neg-out 68.8%

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

      5. distribute-rgt-neg-in 68.8%

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

      6. associate-*l/ 99.2%

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

      7. *-commutative 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. sin-neg 99.2%

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

      11. remove-double-neg 99.2%

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

      12. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r/ 68.9%

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

      3. associate-/r/ 68.8%

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

      4. div-inv 68.8%

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

      5. associate-/r* 69.1%

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

      6. pow2 69.1%

         \[\leadsto \frac{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}{\frac{1}{2.6666666666666665}} \]

      7. metadata-eval 69.1%

         \[\leadsto \frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{\color{blue}{0.375}} \]

   6. Applied egg-rr 69.1%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Taylor expanded in x around 0 72.6%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]

   if 1.80000000000000011e-4 < x

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. remove-double-neg 99.1%

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

      3. sin-neg 99.1%

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

      4. distribute-lft-neg-out 99.1%

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

      5. distribute-rgt-neg-in 99.1%

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

      6. associate-*l/ 99.2%

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

      7. *-commutative 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. sin-neg 99.2%

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

      11. remove-double-neg 99.2%

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

      12. associate-*l* 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*r/ 99.1%

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

      2. associate-*r/ 99.1%

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

      3. associate-/l* 99.2%

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

      4. pow2 99.2%

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

   6. Applied egg-rr 99.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00018:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5)))) (* x_s (* (/ t_0 (sin x_m)) (/ t_0 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));
	return x_s * ((t_0 / sin(x_m)) * (t_0 / 0.375));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.2%

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

   4. metadata-eval 99.2%

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

   5. metadata-eval 99.2%

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

   6. metadata-eval 99.2%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\frac{8}{3}}}{1} \]

   7. metadata-eval 99.2%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\frac{8}{3}}{\color{blue}{\frac{-1}{-1}}} \]

   8. times-frac 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}}} \]

   9. *-commutative 99.2%

      \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}} \]

   10. times-frac 99.1%

       \[\leadsto \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}}} \]

   11. associate-/l* 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \]

   12. *-commutative 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \]

   13. neg-mul-1 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \]

   14. sin-neg 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \]

   15. distribute-lft-neg-out 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \]

   16. associate-*l/ 99.2%

       \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.2%

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

   1. associate-/r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.2%

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

   4. associate-/r/ 99.0%

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

   5. associate-*l/ 77.4%

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

   6. div-inv 77.5%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 6: 99.2% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5))))
   (* x_s (* 2.6666666666666665 (* t_0 (/ 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 * (2.6666666666666665 * (t_0 * (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 * (2.6666666666666665d0 * (t_0 * (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 * (2.6666666666666665 * (t_0 * (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 * (2.6666666666666665 * (t_0 * (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(2.6666666666666665 * Float64(t_0 * 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 * (2.6666666666666665 * (t_0 * (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[(2.6666666666666665 * N[(t$95$0 * N[(t$95$0 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 77.4%

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

   1. *-commutative 77.4%

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

   2. remove-double-neg 77.4%

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

   3. sin-neg 77.4%

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

   4. distribute-lft-neg-out 77.4%

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

   5. distribute-rgt-neg-in 77.4%

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

   6. associate-*l/ 99.2%

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

   7. *-commutative 99.2%

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

   8. distribute-rgt-neg-in 99.2%

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

   9. distribute-lft-neg-out 99.2%

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

   10. sin-neg 99.2%

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

   11. remove-double-neg 99.2%

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

   12. associate-*l* 99.2%

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

3. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 7: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0038:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x\_m}^{3} + x\_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 - \frac{\cos x\_m}{2}\right) \cdot \frac{1}{\sin x\_m}}{0.375}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0038)
    (/ (+ (* 0.020833333333333332 (pow x_m 3.0)) (* x_m 0.25)) 0.375)
    (/ (* (- 0.5 (/ (cos x_m) 2.0)) (/ 1.0 (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 tmp;
	if (x_m <= 0.0038) {
		tmp = ((0.020833333333333332 * pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = ((0.5 - (cos(x_m) / 2.0)) * (1.0 / sin(x_m))) / 0.375;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0038d0) then
        tmp = ((0.020833333333333332d0 * (x_m ** 3.0d0)) + (x_m * 0.25d0)) / 0.375d0
    else
        tmp = ((0.5d0 - (cos(x_m) / 2.0d0)) * (1.0d0 / sin(x_m))) / 0.375d0
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0038) {
		tmp = ((0.020833333333333332 * Math.pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = ((0.5 - (Math.cos(x_m) / 2.0)) * (1.0 / Math.sin(x_m))) / 0.375;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.0038:
		tmp = ((0.020833333333333332 * math.pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375
	else:
		tmp = ((0.5 - (math.cos(x_m) / 2.0)) * (1.0 / math.sin(x_m))) / 0.375
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0038)
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x_m ^ 3.0)) + Float64(x_m * 0.25)) / 0.375);
	else
		tmp = Float64(Float64(Float64(0.5 - Float64(cos(x_m) / 2.0)) * Float64(1.0 / sin(x_m))) / 0.375);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.0038)
		tmp = ((0.020833333333333332 * (x_m ^ 3.0)) + (x_m * 0.25)) / 0.375;
	else
		tmp = ((0.5 - (cos(x_m) / 2.0)) * (1.0 / sin(x_m))) / 0.375;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0038], N[(N[(N[(0.020833333333333332 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(N[(0.5 - N[(N[Cos[x$95$m], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.00379999999999999999

   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. Step-by-step derivation

      1. *-commutative 68.9%

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

      2. remove-double-neg 68.9%

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

      3. sin-neg 68.9%

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

      4. distribute-lft-neg-out 68.9%

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

      5. distribute-rgt-neg-in 68.9%

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

      6. associate-*l/ 99.2%

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

      7. *-commutative 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. sin-neg 99.2%

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

      11. remove-double-neg 99.2%

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

      12. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r/ 69.0%

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

      3. associate-/r/ 69.0%

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

      4. div-inv 69.0%

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

      5. associate-/r* 69.2%

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

      6. pow2 69.2%

         \[\leadsto \frac{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}{\frac{1}{2.6666666666666665}} \]

      7. metadata-eval 69.2%

         \[\leadsto \frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{\color{blue}{0.375}} \]

   6. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Taylor expanded in x around 0 72.8%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]

   if 0.00379999999999999999 < x

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. remove-double-neg 99.1%

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

      3. sin-neg 99.1%

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

      4. distribute-lft-neg-out 99.1%

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

      5. distribute-rgt-neg-in 99.1%

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

      6. associate-*l/ 99.2%

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

      7. *-commutative 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. sin-neg 99.2%

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

      11. remove-double-neg 99.2%

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

      12. associate-*l* 99.1%

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

   3. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*r/ 99.1%

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

      3. associate-/r/ 99.0%

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

      4. div-inv 99.2%

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

      5. associate-/r* 99.2%

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

      6. pow2 99.2%

         \[\leadsto \frac{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}{\frac{1}{2.6666666666666665}} \]

      7. metadata-eval 99.2%

         \[\leadsto \frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{\color{blue}{0.375}} \]

   6. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]
   7. Step-by-step derivation

      1. div-inv 99.1%

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{1}{\sin x}}}{0.375} \]

   8. Applied egg-rr 99.1%

      \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{1}{\sin x}}}{0.375} \]
   9. Step-by-step derivation

      1. unpow2 99.1%

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

      2. sin-mult 98.3%

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

   10. Applied egg-rr 98.5%

       \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}} \cdot \frac{1}{\sin x}}{0.375} \]
   11. Step-by-step derivation

       1. div-sub 98.3%

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

       2. +-inverses 98.3%

          \[\leadsto \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x} \cdot 2.6666666666666665 \]

       3. cos-0 98.3%

          \[\leadsto \frac{\frac{\color{blue}{1}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x} \cdot 2.6666666666666665 \]

       4. metadata-eval 98.3%

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

       5. distribute-lft-out 98.3%

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

       6. metadata-eval 98.3%

          \[\leadsto \frac{0.5 - \frac{\cos \left(x \cdot \color{blue}{1}\right)}{2}}{\sin x} \cdot 2.6666666666666665 \]

       7. *-rgt-identity 98.3%

          \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x} \cdot 2.6666666666666665 \]

   12. Simplified 98.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 - \frac{\cos x}{2}\right) \cdot \frac{1}{\sin x}}{0.375}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0038:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x\_m}^{3} + x\_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sin x\_m}{0.5 - \frac{\cos x\_m}{2}}}}{0.375}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0038)
    (/ (+ (* 0.020833333333333332 (pow x_m 3.0)) (* x_m 0.25)) 0.375)
    (/ (/ 1.0 (/ (sin x_m) (- 0.5 (/ (cos x_m) 2.0)))) 0.375))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0038) {
		tmp = ((0.020833333333333332 * pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = (1.0 / (sin(x_m) / (0.5 - (cos(x_m) / 2.0)))) / 0.375;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0038d0) then
        tmp = ((0.020833333333333332d0 * (x_m ** 3.0d0)) + (x_m * 0.25d0)) / 0.375d0
    else
        tmp = (1.0d0 / (sin(x_m) / (0.5d0 - (cos(x_m) / 2.0d0)))) / 0.375d0
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0038) {
		tmp = ((0.020833333333333332 * Math.pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = (1.0 / (Math.sin(x_m) / (0.5 - (Math.cos(x_m) / 2.0)))) / 0.375;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.0038:
		tmp = ((0.020833333333333332 * math.pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375
	else:
		tmp = (1.0 / (math.sin(x_m) / (0.5 - (math.cos(x_m) / 2.0)))) / 0.375
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0038)
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x_m ^ 3.0)) + Float64(x_m * 0.25)) / 0.375);
	else
		tmp = Float64(Float64(1.0 / Float64(sin(x_m) / Float64(0.5 - Float64(cos(x_m) / 2.0)))) / 0.375);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.0038)
		tmp = ((0.020833333333333332 * (x_m ^ 3.0)) + (x_m * 0.25)) / 0.375;
	else
		tmp = (1.0 / (sin(x_m) / (0.5 - (cos(x_m) / 2.0)))) / 0.375;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0038], N[(N[(N[(0.020833333333333332 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(1.0 / N[(N[Sin[x$95$m], $MachinePrecision] / N[(0.5 - N[(N[Cos[x$95$m], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.00379999999999999999

   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. Step-by-step derivation

      1. *-commutative 68.9%

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

      2. remove-double-neg 68.9%

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

      3. sin-neg 68.9%

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

      4. distribute-lft-neg-out 68.9%

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

      5. distribute-rgt-neg-in 68.9%

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

      6. associate-*l/ 99.2%

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

      7. *-commutative 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. sin-neg 99.2%

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

      11. remove-double-neg 99.2%

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

      12. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r/ 69.0%

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

      3. associate-/r/ 69.0%

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

      4. div-inv 69.0%

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

      5. associate-/r* 69.2%

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

      6. pow2 69.2%

         \[\leadsto \frac{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}{\frac{1}{2.6666666666666665}} \]

      7. metadata-eval 69.2%

         \[\leadsto \frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{\color{blue}{0.375}} \]

   6. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Taylor expanded in x around 0 72.8%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]

   if 0.00379999999999999999 < x

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. remove-double-neg 99.1%

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

      3. sin-neg 99.1%

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

      4. distribute-lft-neg-out 99.1%

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

      5. distribute-rgt-neg-in 99.1%

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

      6. associate-*l/ 99.2%

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

      7. *-commutative 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. sin-neg 99.2%

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

      11. remove-double-neg 99.2%

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

      12. associate-*l* 99.1%

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

   3. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*r/ 99.1%

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

      3. associate-/r/ 99.0%

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

      4. div-inv 99.2%

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

      5. associate-/r* 99.2%

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

      6. pow2 99.2%

         \[\leadsto \frac{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}{\frac{1}{2.6666666666666665}} \]

      7. metadata-eval 99.2%

         \[\leadsto \frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{\color{blue}{0.375}} \]

   6. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]
   7. Step-by-step derivation

      1. div-inv 99.1%

         \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{1}{\sin x}}}{0.375} \]

   8. Applied egg-rr 99.1%

      \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{1}{\sin x}}}{0.375} \]
   9. Step-by-step derivation

      1. div-inv 99.2%

         \[\leadsto \frac{\color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}}{0.375} \]

      2. clear-num 99.2%

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}}}{0.375} \]

   10. Applied egg-rr 99.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}}}{0.375} \]
   11. Step-by-step derivation

       1. unpow2 99.1%

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

       2. sin-mult 98.3%

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

   12. Applied egg-rr 98.4%

       \[\leadsto \frac{\frac{1}{\frac{\sin x}{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}}}{0.375} \]
   13. Step-by-step derivation

       1. div-sub 98.3%

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

       2. +-inverses 98.3%

          \[\leadsto \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x} \cdot 2.6666666666666665 \]

       3. cos-0 98.3%

          \[\leadsto \frac{\frac{\color{blue}{1}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x} \cdot 2.6666666666666665 \]

       4. metadata-eval 98.3%

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

       5. distribute-lft-out 98.3%

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

       6. metadata-eval 98.3%

          \[\leadsto \frac{0.5 - \frac{\cos \left(x \cdot \color{blue}{1}\right)}{2}}{\sin x} \cdot 2.6666666666666665 \]

       7. *-rgt-identity 98.3%

          \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x} \cdot 2.6666666666666665 \]

   14. Simplified 98.4%

       \[\leadsto \frac{\frac{1}{\frac{\sin x}{\color{blue}{0.5 - \frac{\cos x}{2}}}}}{0.375} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sin x}{0.5 - \frac{\cos x}{2}}}}{0.375}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0038:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x\_m}^{3} + x\_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - \frac{\cos x\_m}{2}}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0038)
    (/ (+ (* 0.020833333333333332 (pow x_m 3.0)) (* x_m 0.25)) 0.375)
    (* 2.6666666666666665 (/ (- 0.5 (/ (cos x_m) 2.0)) (sin x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0038) {
		tmp = ((0.020833333333333332 * pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (cos(x_m) / 2.0)) / sin(x_m));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0038d0) then
        tmp = ((0.020833333333333332d0 * (x_m ** 3.0d0)) + (x_m * 0.25d0)) / 0.375d0
    else
        tmp = 2.6666666666666665d0 * ((0.5d0 - (cos(x_m) / 2.0d0)) / sin(x_m))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0038) {
		tmp = ((0.020833333333333332 * Math.pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (Math.cos(x_m) / 2.0)) / Math.sin(x_m));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.0038:
		tmp = ((0.020833333333333332 * math.pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375
	else:
		tmp = 2.6666666666666665 * ((0.5 - (math.cos(x_m) / 2.0)) / math.sin(x_m))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0038)
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x_m ^ 3.0)) + Float64(x_m * 0.25)) / 0.375);
	else
		tmp = Float64(2.6666666666666665 * Float64(Float64(0.5 - Float64(cos(x_m) / 2.0)) / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.0038)
		tmp = ((0.020833333333333332 * (x_m ^ 3.0)) + (x_m * 0.25)) / 0.375;
	else
		tmp = 2.6666666666666665 * ((0.5 - (cos(x_m) / 2.0)) / sin(x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0038], N[(N[(N[(0.020833333333333332 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(2.6666666666666665 * N[(N[(0.5 - N[(N[Cos[x$95$m], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.00379999999999999999

   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. Step-by-step derivation

      1. *-commutative 68.9%

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

      2. remove-double-neg 68.9%

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

      3. sin-neg 68.9%

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

      4. distribute-lft-neg-out 68.9%

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

      5. distribute-rgt-neg-in 68.9%

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

      6. associate-*l/ 99.2%

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

      7. *-commutative 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. sin-neg 99.2%

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

      11. remove-double-neg 99.2%

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

      12. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r/ 69.0%

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

      3. associate-/r/ 69.0%

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

      4. div-inv 69.0%

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

      5. associate-/r* 69.2%

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

      6. pow2 69.2%

         \[\leadsto \frac{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}{\frac{1}{2.6666666666666665}} \]

      7. metadata-eval 69.2%

         \[\leadsto \frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{\color{blue}{0.375}} \]

   6. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Taylor expanded in x around 0 72.8%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]

   if 0.00379999999999999999 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l/ 99.1%

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

      4. metadata-eval 99.1%

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

      5. metadata-eval 99.1%

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

      6. metadata-eval 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\frac{8}{3}}}{1} \]

      7. metadata-eval 99.1%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\frac{8}{3}}{\color{blue}{\frac{-1}{-1}}} \]

      8. times-frac 99.1%

         \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}}} \]

      9. *-commutative 99.1%

         \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}} \]

      10. times-frac 98.9%

          \[\leadsto \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}}} \]

      11. associate-/l* 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \]

      12. *-commutative 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \]

      13. neg-mul-1 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \]

      14. sin-neg 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \]

      15. distribute-lft-neg-out 98.9%

          \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \]

      16. associate-*l/ 99.1%

          \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

   3. Simplified 99.1%

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

      1. div-inv 99.0%

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

      2. clear-num 99.2%

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

      3. associate-*r* 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-*r/ 99.1%

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

      6. pow2 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. unpow2 99.1%

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

      2. sin-mult 98.3%

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

   8. Applied egg-rr 98.3%

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

      1. div-sub 98.3%

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

      2. +-inverses 98.3%

         \[\leadsto \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x} \cdot 2.6666666666666665 \]

      3. cos-0 98.3%

         \[\leadsto \frac{\frac{\color{blue}{1}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x} \cdot 2.6666666666666665 \]

      4. metadata-eval 98.3%

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

      5. distribute-lft-out 98.3%

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

      6. metadata-eval 98.3%

         \[\leadsto \frac{0.5 - \frac{\cos \left(x \cdot \color{blue}{1}\right)}{2}}{\sin x} \cdot 2.6666666666666665 \]

      7. *-rgt-identity 98.3%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x} \cdot 2.6666666666666665 \]

   10. Simplified 98.3%

       \[\leadsto \frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{\sin x} \cdot 2.6666666666666665 \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - \frac{\cos x}{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0038:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x\_m}^{3} + x\_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 - \frac{\cos x\_m}{2}}{\sin x\_m}}{0.375}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0038)
    (/ (+ (* 0.020833333333333332 (pow x_m 3.0)) (* x_m 0.25)) 0.375)
    (/ (/ (- 0.5 (/ (cos x_m) 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 tmp;
	if (x_m <= 0.0038) {
		tmp = ((0.020833333333333332 * pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = ((0.5 - (cos(x_m) / 2.0)) / sin(x_m)) / 0.375;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0038d0) then
        tmp = ((0.020833333333333332d0 * (x_m ** 3.0d0)) + (x_m * 0.25d0)) / 0.375d0
    else
        tmp = ((0.5d0 - (cos(x_m) / 2.0d0)) / sin(x_m)) / 0.375d0
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0038) {
		tmp = ((0.020833333333333332 * Math.pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = ((0.5 - (Math.cos(x_m) / 2.0)) / Math.sin(x_m)) / 0.375;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.0038:
		tmp = ((0.020833333333333332 * math.pow(x_m, 3.0)) + (x_m * 0.25)) / 0.375
	else:
		tmp = ((0.5 - (math.cos(x_m) / 2.0)) / math.sin(x_m)) / 0.375
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0038)
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x_m ^ 3.0)) + Float64(x_m * 0.25)) / 0.375);
	else
		tmp = Float64(Float64(Float64(0.5 - Float64(cos(x_m) / 2.0)) / sin(x_m)) / 0.375);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.0038)
		tmp = ((0.020833333333333332 * (x_m ^ 3.0)) + (x_m * 0.25)) / 0.375;
	else
		tmp = ((0.5 - (cos(x_m) / 2.0)) / sin(x_m)) / 0.375;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0038], N[(N[(N[(0.020833333333333332 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(N[(0.5 - N[(N[Cos[x$95$m], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.00379999999999999999

   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. Step-by-step derivation

      1. *-commutative 68.9%

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

      2. remove-double-neg 68.9%

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

      3. sin-neg 68.9%

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

      4. distribute-lft-neg-out 68.9%

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

      5. distribute-rgt-neg-in 68.9%

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

      6. associate-*l/ 99.2%

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

      7. *-commutative 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. sin-neg 99.2%

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

      11. remove-double-neg 99.2%

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

      12. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r/ 69.0%

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

      3. associate-/r/ 69.0%

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

      4. div-inv 69.0%

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

      5. associate-/r* 69.2%

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

      6. pow2 69.2%

         \[\leadsto \frac{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}{\frac{1}{2.6666666666666665}} \]

      7. metadata-eval 69.2%

         \[\leadsto \frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{\color{blue}{0.375}} \]

   6. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

   7. Taylor expanded in x around 0 72.8%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]

   if 0.00379999999999999999 < x

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. remove-double-neg 99.1%

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

      3. sin-neg 99.1%

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

      4. distribute-lft-neg-out 99.1%

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

      5. distribute-rgt-neg-in 99.1%

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

      6. associate-*l/ 99.2%

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

      7. *-commutative 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. distribute-lft-neg-out 99.2%

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

      10. sin-neg 99.2%

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

      11. remove-double-neg 99.2%

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

      12. associate-*l* 99.1%

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

   3. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*r/ 99.1%

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

      3. associate-/r/ 99.0%

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

      4. div-inv 99.2%

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

      5. associate-/r* 99.2%

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

      6. pow2 99.2%

         \[\leadsto \frac{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}{\frac{1}{2.6666666666666665}} \]

      7. metadata-eval 99.2%

         \[\leadsto \frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{\color{blue}{0.375}} \]

   6. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]
   7. Step-by-step derivation

      1. unpow2 99.1%

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

      2. sin-mult 98.3%

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

   8. Applied egg-rr 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\cos \left(x \cdot 0.5 - x \cdot 0.5\right) - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}}{\sin x}}{0.375} \]
   9. Step-by-step derivation

      1. div-sub 98.3%

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

      2. +-inverses 98.3%

         \[\leadsto \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x} \cdot 2.6666666666666665 \]

      3. cos-0 98.3%

         \[\leadsto \frac{\frac{\color{blue}{1}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}{\sin x} \cdot 2.6666666666666665 \]

      4. metadata-eval 98.3%

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

      5. distribute-lft-out 98.3%

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

      6. metadata-eval 98.3%

         \[\leadsto \frac{0.5 - \frac{\cos \left(x \cdot \color{blue}{1}\right)}{2}}{\sin x} \cdot 2.6666666666666665 \]

      7. *-rgt-identity 98.3%

         \[\leadsto \frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x} \cdot 2.6666666666666665 \]

   10. Simplified 98.5%

       \[\leadsto \frac{\frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{\sin x}}{0.375} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 - \frac{\cos x}{2}}{\sin x}}{0.375}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.0% accurate, 2.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* 0.5 (/ (sin (* x_m 0.5)) 0.375))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.2%

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

   4. metadata-eval 99.2%

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

   5. metadata-eval 99.2%

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

   6. metadata-eval 99.2%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\frac{8}{3}}}{1} \]

   7. metadata-eval 99.2%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\frac{8}{3}}{\color{blue}{\frac{-1}{-1}}} \]

   8. times-frac 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}}} \]

   9. *-commutative 99.2%

      \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}} \]

   10. times-frac 99.1%

       \[\leadsto \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}}} \]

   11. associate-/l* 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \]

   12. *-commutative 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \]

   13. neg-mul-1 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \]

   14. sin-neg 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \]

   15. distribute-lft-neg-out 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \]

   16. associate-*l/ 99.2%

       \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.2%

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

   1. associate-/r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.2%

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

   4. associate-/r/ 99.0%

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

   5. associate-*l/ 77.4%

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

   6. div-inv 77.5%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in x around 0 56.5%

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

8. Final simplification 56.5%

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

Alternative 12: 54.8% accurate, 3.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* (sin (* x_m 0.5)) 1.3333333333333333)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. *-commutative 77.4%

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

   2. remove-double-neg 77.4%

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

   3. sin-neg 77.4%

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

   4. distribute-lft-neg-out 77.4%

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

   5. distribute-rgt-neg-in 77.4%

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

   6. associate-*r/ 99.2%

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

   7. neg-mul-1 99.2%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{-1 \cdot \left(\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)\right)}}{\sin x} \]

   8. *-commutative 99.2%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)\right) \cdot -1}}{\sin x} \]

   9. associate-/l* 99.2%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\frac{\sin x}{-1}}} \]

   10. associate-/r/ 99.2%

       \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\left(\frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\sin x} \cdot -1\right)} \]

   11. *-commutative 99.2%

       \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{8}{3} \cdot \sin \left(\left(-x\right) \cdot 0.5\right)}{\sin x}\right)} \]

3. Simplified 99.0%

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

5. Taylor expanded in x around 0 56.2%

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

6. Final simplification 56.2%

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

Alternative 13: 51.2% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{1}{0.375 \cdot \left(x\_m \cdot -0.3333333333333333 + 4 \cdot \frac{1}{x\_m}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ 1.0 (* 0.375 (+ (* x_m -0.3333333333333333) (* 4.0 (/ 1.0 x_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (1.0 / (0.375 * ((x_m * -0.3333333333333333) + (4.0 * (1.0 / x_m)))));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (1.0d0 / (0.375d0 * ((x_m * (-0.3333333333333333d0)) + (4.0d0 * (1.0d0 / x_m)))))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (1.0 / (0.375 * ((x_m * -0.3333333333333333) + (4.0 * (1.0 / x_m)))));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (1.0 / (0.375 * ((x_m * -0.3333333333333333) + (4.0 * (1.0 / x_m)))))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(1.0 / Float64(0.375 * Float64(Float64(x_m * -0.3333333333333333) + Float64(4.0 * Float64(1.0 / x_m))))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (1.0 / (0.375 * ((x_m * -0.3333333333333333) + (4.0 * (1.0 / x_m)))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(1.0 / N[(0.375 * N[(N[(x$95$m * -0.3333333333333333), $MachinePrecision] + N[(4.0 * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.2%

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

   4. metadata-eval 99.2%

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

   5. metadata-eval 99.2%

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

   6. metadata-eval 99.2%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\frac{8}{3}}}{1} \]

   7. metadata-eval 99.2%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\frac{8}{3}}{\color{blue}{\frac{-1}{-1}}} \]

   8. times-frac 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}}} \]

   9. *-commutative 99.2%

      \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}} \]

   10. times-frac 99.1%

       \[\leadsto \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}}} \]

   11. associate-/l* 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \]

   12. *-commutative 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \]

   13. neg-mul-1 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \]

   14. sin-neg 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \]

   15. distribute-lft-neg-out 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \]

   16. associate-*l/ 99.2%

       \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.2%

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

   1. associate-/r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.2%

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

   4. associate-/r/ 99.0%

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

   5. associate-*l/ 77.4%

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

   6. div-inv 77.5%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.5%

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

   2. clear-num 99.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{0.375}{\frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)}}} \]

   3. associate-*l/ 77.5%

      \[\leadsto \frac{1}{\frac{0.375}{\color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}}}} \]

   4. unpow2 77.5%

      \[\leadsto \frac{1}{\frac{0.375}{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}} \]

   5. div-inv 77.5%

      \[\leadsto \frac{1}{\color{blue}{0.375 \cdot \frac{1}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}}} \]

   6. clear-num 77.5%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   7. div-inv 76.7%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\left(\sin x \cdot \frac{1}{{\sin \left(x \cdot 0.5\right)}^{2}}\right)}} \]

   8. unpow2 76.7%

      \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot \frac{1}{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}\right)} \]

   9. associate-/l/ 76.8%

      \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot \color{blue}{\frac{\frac{1}{\sin \left(x \cdot 0.5\right)}}{\sin \left(x \cdot 0.5\right)}}\right)} \]

   10. inv-pow 76.8%

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

   11. pow1 76.8%

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

   12. pow-div 76.8%

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

   13. metadata-eval 76.8%

       \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{\color{blue}{-2}}\right)} \]

8. Applied egg-rr 76.8%

   \[\leadsto \color{blue}{\frac{1}{0.375 \cdot \left(\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)}} \]

9. Taylor expanded in x around 0 53.9%

   \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\left(-0.3333333333333333 \cdot x + 4 \cdot \frac{1}{x}\right)}} \]

10. Final simplification 53.9%

    \[\leadsto \frac{1}{0.375 \cdot \left(x \cdot -0.3333333333333333 + 4 \cdot \frac{1}{x}\right)} \]
11. Add Preprocessing

Alternative 14: 51.4% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{\frac{1}{x\_m \cdot -0.3333333333333333 + 4 \cdot \frac{1}{x\_m}}}{0.375} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (/ 1.0 (+ (* x_m -0.3333333333333333) (* 4.0 (/ 1.0 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 * ((1.0 / ((x_m * -0.3333333333333333) + (4.0 * (1.0 / 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 * ((1.0d0 / ((x_m * (-0.3333333333333333d0)) + (4.0d0 * (1.0d0 / 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 * ((1.0 / ((x_m * -0.3333333333333333) + (4.0 * (1.0 / 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 * ((1.0 / ((x_m * -0.3333333333333333) + (4.0 * (1.0 / 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(1.0 / Float64(Float64(x_m * -0.3333333333333333) + Float64(4.0 * Float64(1.0 / 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 * ((1.0 / ((x_m * -0.3333333333333333) + (4.0 * (1.0 / 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[(1.0 / N[(N[(x$95$m * -0.3333333333333333), $MachinePrecision] + N[(4.0 * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

   1. *-commutative 77.4%

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

   2. remove-double-neg 77.4%

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

   3. sin-neg 77.4%

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

   4. distribute-lft-neg-out 77.4%

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

   5. distribute-rgt-neg-in 77.4%

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

   6. associate-*l/ 99.2%

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

   7. *-commutative 99.2%

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

   8. distribute-rgt-neg-in 99.2%

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

   9. distribute-lft-neg-out 99.2%

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

   10. sin-neg 99.2%

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

   11. remove-double-neg 99.2%

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

   12. associate-*l* 99.2%

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

3. Simplified 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-*r/ 77.5%

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

   3. associate-/r/ 77.4%

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

   4. div-inv 77.5%

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

   5. associate-/r* 77.6%

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

   6. pow2 77.6%

      \[\leadsto \frac{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}{\frac{1}{2.6666666666666665}} \]

   7. metadata-eval 77.6%

      \[\leadsto \frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{\color{blue}{0.375}} \]

6. Applied egg-rr 77.6%

   \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]
7. Step-by-step derivation

   1. div-inv 77.6%

      \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{1}{\sin x}}}{0.375} \]

8. Applied egg-rr 77.6%

   \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{1}{\sin x}}}{0.375} \]
9. Step-by-step derivation

   1. div-inv 77.6%

      \[\leadsto \frac{\color{blue}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}}{0.375} \]

   2. clear-num 77.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}}}{0.375} \]

10. Applied egg-rr 77.6%

    \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}}}{0.375} \]

11. Taylor expanded in x around 0 54.1%

    \[\leadsto \frac{\frac{1}{\color{blue}{-0.3333333333333333 \cdot x + 4 \cdot \frac{1}{x}}}}{0.375} \]

12. Final simplification 54.1%

    \[\leadsto \frac{\frac{1}{x \cdot -0.3333333333333333 + 4 \cdot \frac{1}{x}}}{0.375} \]
13. Add Preprocessing

Alternative 15: 51.2% accurate, 28.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{1}{x\_m \cdot -0.125 + \frac{1}{x\_m} \cdot 1.5} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ 1.0 (+ (* x_m -0.125) (* (/ 1.0 x_m) 1.5)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (1.0 / ((x_m * -0.125) + ((1.0 / x_m) * 1.5)));
}

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 * (1.0d0 / ((x_m * (-0.125d0)) + ((1.0d0 / x_m) * 1.5d0)))
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 * (1.0 / ((x_m * -0.125) + ((1.0 / x_m) * 1.5)));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (1.0 / ((x_m * -0.125) + ((1.0 / x_m) * 1.5)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(1.0 / Float64(Float64(x_m * -0.125) + Float64(Float64(1.0 / x_m) * 1.5))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (1.0 / ((x_m * -0.125) + ((1.0 / x_m) * 1.5)));
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[(1.0 / N[(N[(x$95$m * -0.125), $MachinePrecision] + N[(N[(1.0 / x$95$m), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.2%

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

   4. metadata-eval 99.2%

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

   5. metadata-eval 99.2%

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

   6. metadata-eval 99.2%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\frac{8}{3}}}{1} \]

   7. metadata-eval 99.2%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\frac{8}{3}}{\color{blue}{\frac{-1}{-1}}} \]

   8. times-frac 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}}} \]

   9. *-commutative 99.2%

      \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}} \]

   10. times-frac 99.1%

       \[\leadsto \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}}} \]

   11. associate-/l* 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \]

   12. *-commutative 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \]

   13. neg-mul-1 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \]

   14. sin-neg 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \]

   15. distribute-lft-neg-out 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \]

   16. associate-*l/ 99.2%

       \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.2%

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

   1. associate-/r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.2%

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

   4. associate-/r/ 99.0%

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

   5. associate-*l/ 77.4%

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

   6. div-inv 77.5%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.5%

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

   2. clear-num 99.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{0.375}{\frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)}}} \]

   3. associate-*l/ 77.5%

      \[\leadsto \frac{1}{\frac{0.375}{\color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}}}} \]

   4. unpow2 77.5%

      \[\leadsto \frac{1}{\frac{0.375}{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}} \]

   5. div-inv 77.5%

      \[\leadsto \frac{1}{\color{blue}{0.375 \cdot \frac{1}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}}} \]

   6. clear-num 77.5%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   7. div-inv 76.7%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\left(\sin x \cdot \frac{1}{{\sin \left(x \cdot 0.5\right)}^{2}}\right)}} \]

   8. unpow2 76.7%

      \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot \frac{1}{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}\right)} \]

   9. associate-/l/ 76.8%

      \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot \color{blue}{\frac{\frac{1}{\sin \left(x \cdot 0.5\right)}}{\sin \left(x \cdot 0.5\right)}}\right)} \]

   10. inv-pow 76.8%

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

   11. pow1 76.8%

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

   12. pow-div 76.8%

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

   13. metadata-eval 76.8%

       \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{\color{blue}{-2}}\right)} \]

8. Applied egg-rr 76.8%

   \[\leadsto \color{blue}{\frac{1}{0.375 \cdot \left(\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)}} \]

9. Taylor expanded in x around 0 53.9%

   \[\leadsto \frac{1}{\color{blue}{-0.125 \cdot x + 1.5 \cdot \frac{1}{x}}} \]

10. Final simplification 53.9%

    \[\leadsto \frac{1}{x \cdot -0.125 + \frac{1}{x} \cdot 1.5} \]
11. Add Preprocessing

Alternative 16: 50.7% accurate, 62.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{1}{\frac{1.5}{x\_m}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ 1.0 (/ 1.5 x_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (1.0 / (1.5 / x_m));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (1.0d0 / (1.5d0 / x_m))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (1.0 / (1.5 / x_m));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (1.0 / (1.5 / x_m))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(1.0 / Float64(1.5 / x_m)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (1.0 / (1.5 / x_m));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(1.0 / N[(1.5 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.2%

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

   4. metadata-eval 99.2%

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

   5. metadata-eval 99.2%

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

   6. metadata-eval 99.2%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\frac{8}{3}}}{1} \]

   7. metadata-eval 99.2%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\frac{8}{3}}{\color{blue}{\frac{-1}{-1}}} \]

   8. times-frac 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}}} \]

   9. *-commutative 99.2%

      \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)} \cdot \frac{-1}{-1}} \]

   10. times-frac 99.1%

       \[\leadsto \color{blue}{\frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{-1}{-1}}} \]

   11. associate-/l* 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot -1}{-1}} \]

   12. *-commutative 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-1 \cdot \sin \left(x \cdot 0.5\right)}}{-1} \]

   13. neg-mul-1 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{-\sin \left(x \cdot 0.5\right)}}{-1} \]

   14. sin-neg 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\color{blue}{\sin \left(-x \cdot 0.5\right)}}{-1} \]

   15. distribute-lft-neg-out 99.1%

       \[\leadsto \frac{\frac{8}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}} \cdot \frac{\sin \color{blue}{\left(\left(-x\right) \cdot 0.5\right)}}{-1} \]

   16. associate-*l/ 99.2%

       \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \frac{\sin \left(\left(-x\right) \cdot 0.5\right)}{-1}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]

3. Simplified 99.2%

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

   1. associate-/r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.2%

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

   4. associate-/r/ 99.0%

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

   5. associate-*l/ 77.4%

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

   6. div-inv 77.5%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.5%

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

   2. clear-num 99.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{0.375}{\frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \sin \left(x \cdot 0.5\right)}}} \]

   3. associate-*l/ 77.5%

      \[\leadsto \frac{1}{\frac{0.375}{\color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}}}} \]

   4. unpow2 77.5%

      \[\leadsto \frac{1}{\frac{0.375}{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}} \]

   5. div-inv 77.5%

      \[\leadsto \frac{1}{\color{blue}{0.375 \cdot \frac{1}{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}}} \]

   6. clear-num 77.5%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]

   7. div-inv 76.7%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\left(\sin x \cdot \frac{1}{{\sin \left(x \cdot 0.5\right)}^{2}}\right)}} \]

   8. unpow2 76.7%

      \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot \frac{1}{\color{blue}{\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)}}\right)} \]

   9. associate-/l/ 76.8%

      \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot \color{blue}{\frac{\frac{1}{\sin \left(x \cdot 0.5\right)}}{\sin \left(x \cdot 0.5\right)}}\right)} \]

   10. inv-pow 76.8%

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

   11. pow1 76.8%

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

   12. pow-div 76.8%

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

   13. metadata-eval 76.8%

       \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{\color{blue}{-2}}\right)} \]

8. Applied egg-rr 76.8%

   \[\leadsto \color{blue}{\frac{1}{0.375 \cdot \left(\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)}} \]

9. Taylor expanded in x around 0 52.4%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]

10. Final simplification 52.4%

    \[\leadsto \frac{1}{\frac{1.5}{x}} \]
11. Add Preprocessing

Alternative 17: 50.9% accurate, 62.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ (* x_m 0.25) 0.375)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. *-commutative 77.4%

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

   2. remove-double-neg 77.4%

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

   3. sin-neg 77.4%

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

   4. distribute-lft-neg-out 77.4%

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

   5. distribute-rgt-neg-in 77.4%

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

   6. associate-*l/ 99.2%

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

   7. *-commutative 99.2%

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

   8. distribute-rgt-neg-in 99.2%

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

   9. distribute-lft-neg-out 99.2%

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

   10. sin-neg 99.2%

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

   11. remove-double-neg 99.2%

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

   12. associate-*l* 99.2%

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

3. Simplified 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-*r/ 77.5%

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

   3. associate-/r/ 77.4%

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

   4. div-inv 77.5%

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

   5. associate-/r* 77.6%

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

   6. pow2 77.6%

      \[\leadsto \frac{\frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x}}{\frac{1}{2.6666666666666665}} \]

   7. metadata-eval 77.6%

      \[\leadsto \frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{\color{blue}{0.375}} \]

6. Applied egg-rr 77.6%

   \[\leadsto \color{blue}{\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}} \]

7. Taylor expanded in x around 0 52.6%

   \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]
8. Step-by-step derivation

   1. *-commutative 52.6%

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

9. Simplified 52.6%

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

10. Final simplification 52.6%

    \[\leadsto \frac{x \cdot 0.25}{0.375} \]
11. Add Preprocessing

Alternative 18: 50.7% accurate, 104.3× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* x_m 0.6666666666666666)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. *-commutative 77.4%

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

   2. remove-double-neg 77.4%

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

   3. sin-neg 77.4%

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

   4. distribute-lft-neg-out 77.4%

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

   5. distribute-rgt-neg-in 77.4%

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

   6. associate-*l/ 99.2%

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

   7. *-commutative 99.2%

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

   8. distribute-rgt-neg-in 99.2%

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

   9. distribute-lft-neg-out 99.2%

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

   10. sin-neg 99.2%

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

   11. remove-double-neg 99.2%

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

   12. associate-*l* 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 52.2%

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

6. Final simplification 52.2%

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

Developer target: 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 2024040 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :herbie-target
  (/ (/ (* 8.0 (sin (* x 0.5))) 3.0) (/ (sin x) (sin (* x 0.5))))

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