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

Percentage Accurate: 76.8% → 99.5%

Time: 12.2s

Alternatives: 14

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x\_m}^{3} \cdot 0.020833333333333332 + x\_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x\_m} \cdot {\sin \left(x\_m \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2e-5)
    (/ (+ (* (pow x_m 3.0) 0.020833333333333332) (* 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 <= 2e-5) {
		tmp = ((pow(x_m, 3.0) * 0.020833333333333332) + (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 <= 2d-5) then
        tmp = (((x_m ** 3.0d0) * 0.020833333333333332d0) + (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 <= 2e-5) {
		tmp = ((Math.pow(x_m, 3.0) * 0.020833333333333332) + (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 <= 2e-5:
		tmp = ((math.pow(x_m, 3.0) * 0.020833333333333332) + (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 <= 2e-5)
		tmp = Float64(Float64(Float64((x_m ^ 3.0) * 0.020833333333333332) + Float64(x_m * 0.25)) / 0.375);
	else
		tmp = Float64(Float64(2.6666666666666665 / 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 <= 2e-5)
		tmp = (((x_m ^ 3.0) * 0.020833333333333332) + (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, 2e-5], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 0.020833333333333332), $MachinePrecision] + N[(x$95$m * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(2.6666666666666665 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000016e-5

   1. Initial program 66.8%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r/ 66.6%

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

      2. clear-num 66.6%

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

      3. pow2 66.6%

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

   6. Applied egg-rr 66.6%

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

      1. clear-num 66.6%

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

      2. associate-/l* 66.7%

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

      3. rem-square-sqrt 66.7%

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

      4. pow2 66.7%

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

      5. unpow-prod-down 66.8%

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

      6. clear-num 66.7%

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

      7. *-un-lft-identity 66.7%

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

      8. unpow-prod-down 66.6%

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

      9. pow2 66.6%

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

      10. rem-square-sqrt 66.6%

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

      11. times-frac 66.7%

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

      12. metadata-eval 66.7%

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

      13. unpow2 66.7%

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

      14. associate-/r* 99.3%

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

      15. associate-/l* 99.4%

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

      16. clear-num 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 65.8%

      \[\leadsto \frac{\color{blue}{x \cdot \left(0.25 + 0.020833333333333332 \cdot {x}^{2}\right)}}{0.375} \]
   10. Step-by-step derivation

       1. +-commutative 65.8%

          \[\leadsto \frac{x \cdot \color{blue}{\left(0.020833333333333332 \cdot {x}^{2} + 0.25\right)}}{0.375} \]

       2. distribute-lft-in 65.8%

          \[\leadsto \frac{\color{blue}{x \cdot \left(0.020833333333333332 \cdot {x}^{2}\right) + x \cdot 0.25}}{0.375} \]

       3. *-commutative 65.8%

          \[\leadsto \frac{x \cdot \color{blue}{\left({x}^{2} \cdot 0.020833333333333332\right)} + x \cdot 0.25}{0.375} \]

       4. associate-*r* 65.8%

          \[\leadsto \frac{\color{blue}{\left(x \cdot {x}^{2}\right) \cdot 0.020833333333333332} + x \cdot 0.25}{0.375} \]

       5. unpow2 65.8%

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot 0.020833333333333332 + x \cdot 0.25}{0.375} \]

       6. associate-*l* 65.8%

          \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot 0.020833333333333332 + x \cdot 0.25}{0.375} \]

       7. pow3 65.8%

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

   11. Applied egg-rr 65.8%

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

   if 2.00000000000000016e-5 < x

   1. Initial program 99.0%

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

      1. associate-/l* 98.7%

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

      2. 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. metadata-eval 99.1%

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

   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. Taylor expanded in x around inf 99.1%

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

      1. associate-*r/ 99.1%

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

      2. *-commutative 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-/l* 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{3} \cdot 0.020833333333333332 + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x} \cdot {\sin \left(x \cdot 0.5\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x\_m}^{3} \cdot 0.020833333333333332 + x\_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x\_m \cdot 0.5\right)}^{2}}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5e-5)
    (/ (+ (* (pow x_m 3.0) 0.020833333333333332) (* 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 <= 5e-5) {
		tmp = ((pow(x_m, 3.0) * 0.020833333333333332) + (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 <= 5d-5) then
        tmp = (((x_m ** 3.0d0) * 0.020833333333333332d0) + (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 <= 5e-5) {
		tmp = ((Math.pow(x_m, 3.0) * 0.020833333333333332) + (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 <= 5e-5:
		tmp = ((math.pow(x_m, 3.0) * 0.020833333333333332) + (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 <= 5e-5)
		tmp = Float64(Float64(Float64((x_m ^ 3.0) * 0.020833333333333332) + 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 <= 5e-5)
		tmp = (((x_m ^ 3.0) * 0.020833333333333332) + (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, 5e-5], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 0.020833333333333332), $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 < 5.00000000000000024e-5

   1. Initial program 66.8%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r/ 66.6%

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

      2. clear-num 66.6%

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

      3. pow2 66.6%

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

   6. Applied egg-rr 66.6%

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

      1. clear-num 66.6%

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

      2. associate-/l* 66.7%

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

      3. rem-square-sqrt 66.7%

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

      4. pow2 66.7%

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

      5. unpow-prod-down 66.8%

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

      6. clear-num 66.7%

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

      7. *-un-lft-identity 66.7%

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

      8. unpow-prod-down 66.6%

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

      9. pow2 66.6%

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

      10. rem-square-sqrt 66.6%

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

      11. times-frac 66.7%

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

      12. metadata-eval 66.7%

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

      13. unpow2 66.7%

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

      14. associate-/r* 99.3%

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

      15. associate-/l* 99.4%

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

      16. clear-num 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 65.8%

      \[\leadsto \frac{\color{blue}{x \cdot \left(0.25 + 0.020833333333333332 \cdot {x}^{2}\right)}}{0.375} \]
   10. Step-by-step derivation

       1. +-commutative 65.8%

          \[\leadsto \frac{x \cdot \color{blue}{\left(0.020833333333333332 \cdot {x}^{2} + 0.25\right)}}{0.375} \]

       2. distribute-lft-in 65.8%

          \[\leadsto \frac{\color{blue}{x \cdot \left(0.020833333333333332 \cdot {x}^{2}\right) + x \cdot 0.25}}{0.375} \]

       3. *-commutative 65.8%

          \[\leadsto \frac{x \cdot \color{blue}{\left({x}^{2} \cdot 0.020833333333333332\right)} + x \cdot 0.25}{0.375} \]

       4. associate-*r* 65.8%

          \[\leadsto \frac{\color{blue}{\left(x \cdot {x}^{2}\right) \cdot 0.020833333333333332} + x \cdot 0.25}{0.375} \]

       5. unpow2 65.8%

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot 0.020833333333333332 + x \cdot 0.25}{0.375} \]

       6. associate-*l* 65.8%

          \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot 0.020833333333333332 + x \cdot 0.25}{0.375} \]

       7. pow3 65.8%

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

   11. Applied egg-rr 65.8%

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

   if 5.00000000000000024e-5 < x

   1. Initial program 99.0%

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

      1. metadata-eval 99.0%

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

      2. associate-*r/ 98.7%

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

      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%

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

   4. 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 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{3} \cdot 0.020833333333333332 + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (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 / Float64(sin(x_m) / 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[(N[Sin[x$95$m], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.5%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. associate-*r/ 74.4%

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

   2. clear-num 74.4%

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

   3. pow2 74.4%

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

6. Applied egg-rr 74.4%

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

   1. clear-num 74.4%

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

   2. associate-/l* 74.4%

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

   3. rem-square-sqrt 74.4%

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

   4. pow2 74.4%

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

   5. unpow-prod-down 74.5%

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

   6. clear-num 74.4%

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

   7. *-un-lft-identity 74.4%

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

   8. unpow-prod-down 74.4%

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

   9. pow2 74.4%

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

   10. rem-square-sqrt 74.4%

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

   11. times-frac 74.4%

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

   12. metadata-eval 74.4%

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

   13. unpow2 74.4%

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

   14. associate-/r* 99.2%

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

   15. associate-/l* 99.3%

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

   16. clear-num 99.5%

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

8. Applied egg-rr 99.5%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (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(t_0 / Float64(Float64(sin(x_m) / 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[(t$95$0 / N[(N[(N[Sin[x$95$m], $MachinePrecision] / t$95$0), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.5%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. associate-*r* 99.2%

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

   2. *-commutative 99.2%

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

   3. div-inv 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-/r/ 99.2%

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

   6. un-div-inv 99.2%

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

   7. *-un-lft-identity 99.2%

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

   8. times-frac 99.5%

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

   9. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin((x_m * 0.5))
	return x_s * (t_0 * (t_0 * (2.6666666666666665 / 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(t_0 * Float64(t_0 * Float64(2.6666666666666665 / sin(x_m)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	t_0 = sin((x_m * 0.5));
	tmp = x_s * (t_0 * (t_0 * (2.6666666666666665 / 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[(t$95$0 * N[(t$95$0 * N[(2.6666666666666665 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.5%

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

   1. *-commutative 74.5%

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

   2. associate-/l* 99.3%

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

   3. remove-double-neg 99.3%

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

   4. sin-neg 99.3%

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

   5. distribute-lft-neg-out 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

   7. distribute-frac-neg 99.3%

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

   8. distribute-frac-neg2 99.3%

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

   9. neg-mul-1 99.3%

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

   10. associate-/r* 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around inf 99.3%

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

   1. associate-*r/ 99.3%

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

   2. *-commutative 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*r/ 99.3%

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

7. Simplified 99.3%

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

Alternative 6: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. clear-num 99.2%

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

   2. un-div-inv 99.3%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

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

Alternative 7: 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 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5))))
   (* x_s (* 2.6666666666666665 (* t_0 (/ 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 74.5%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

Alternative 8: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0037:\\ \;\;\;\;\frac{{x\_m}^{3} \cdot 0.020833333333333332 + x\_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\left(2.6666666666666665 \cdot \frac{1}{\sin x\_m}\right) \cdot \left(0.5 - \frac{\cos x\_m}{2}\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0037)
    (/ (+ (* (pow x_m 3.0) 0.020833333333333332) (* x_m 0.25)) 0.375)
    (* (* 2.6666666666666665 (/ 1.0 (sin x_m))) (- 0.5 (/ (cos x_m) 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 <= 0.0037) {
		tmp = ((pow(x_m, 3.0) * 0.020833333333333332) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = (2.6666666666666665 * (1.0 / sin(x_m))) * (0.5 - (cos(x_m) / 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 <= 0.0037d0) then
        tmp = (((x_m ** 3.0d0) * 0.020833333333333332d0) + (x_m * 0.25d0)) / 0.375d0
    else
        tmp = (2.6666666666666665d0 * (1.0d0 / sin(x_m))) * (0.5d0 - (cos(x_m) / 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 <= 0.0037) {
		tmp = ((Math.pow(x_m, 3.0) * 0.020833333333333332) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = (2.6666666666666665 * (1.0 / Math.sin(x_m))) * (0.5 - (Math.cos(x_m) / 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 <= 0.0037:
		tmp = ((math.pow(x_m, 3.0) * 0.020833333333333332) + (x_m * 0.25)) / 0.375
	else:
		tmp = (2.6666666666666665 * (1.0 / math.sin(x_m))) * (0.5 - (math.cos(x_m) / 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 <= 0.0037)
		tmp = Float64(Float64(Float64((x_m ^ 3.0) * 0.020833333333333332) + Float64(x_m * 0.25)) / 0.375);
	else
		tmp = Float64(Float64(2.6666666666666665 * Float64(1.0 / sin(x_m))) * Float64(0.5 - Float64(cos(x_m) / 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 <= 0.0037)
		tmp = (((x_m ^ 3.0) * 0.020833333333333332) + (x_m * 0.25)) / 0.375;
	else
		tmp = (2.6666666666666665 * (1.0 / sin(x_m))) * (0.5 - (cos(x_m) / 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, 0.0037], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 0.020833333333333332), $MachinePrecision] + N[(x$95$m * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(2.6666666666666665 * N[(1.0 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[x$95$m], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0037000000000000002

   1. Initial program 67.0%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r/ 66.8%

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

      2. clear-num 66.8%

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

      3. pow2 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. clear-num 66.8%

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

      2. associate-/l* 66.9%

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

      3. rem-square-sqrt 66.9%

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

      4. pow2 66.9%

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

      5. unpow-prod-down 66.9%

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

      6. clear-num 66.9%

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

      7. *-un-lft-identity 66.9%

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

      8. unpow-prod-down 66.8%

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

      9. pow2 66.8%

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

      10. rem-square-sqrt 66.8%

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

      11. times-frac 66.9%

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

      12. metadata-eval 66.9%

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

      13. unpow2 66.9%

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

      14. associate-/r* 99.3%

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

      15. associate-/l* 99.4%

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

      16. clear-num 99.6%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 66.0%

      \[\leadsto \frac{\color{blue}{x \cdot \left(0.25 + 0.020833333333333332 \cdot {x}^{2}\right)}}{0.375} \]
   10. Step-by-step derivation

       1. +-commutative 66.0%

          \[\leadsto \frac{x \cdot \color{blue}{\left(0.020833333333333332 \cdot {x}^{2} + 0.25\right)}}{0.375} \]

       2. distribute-lft-in 66.0%

          \[\leadsto \frac{\color{blue}{x \cdot \left(0.020833333333333332 \cdot {x}^{2}\right) + x \cdot 0.25}}{0.375} \]

       3. *-commutative 66.0%

          \[\leadsto \frac{x \cdot \color{blue}{\left({x}^{2} \cdot 0.020833333333333332\right)} + x \cdot 0.25}{0.375} \]

       4. associate-*r* 66.0%

          \[\leadsto \frac{\color{blue}{\left(x \cdot {x}^{2}\right) \cdot 0.020833333333333332} + x \cdot 0.25}{0.375} \]

       5. unpow2 66.0%

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot 0.020833333333333332 + x \cdot 0.25}{0.375} \]

       6. associate-*l* 66.0%

          \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot 0.020833333333333332 + x \cdot 0.25}{0.375} \]

       7. pow3 66.0%

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

   11. Applied egg-rr 66.0%

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

   if 0.0037000000000000002 < 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* 98.7%

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

      2. 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. metadata-eval 99.1%

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

   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* 98.7%

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

      2. *-commutative 98.7%

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

      3. div-inv 98.8%

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

      4. associate-*l* 98.9%

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

      5. associate-/r/ 99.2%

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

      6. un-div-inv 99.1%

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

      7. *-un-lft-identity 99.1%

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

      8. times-frac 99.1%

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

      9. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. clear-num 98.9%

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

      2. associate-/l* 98.9%

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

      3. metadata-eval 98.9%

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

      4. associate-/r* 98.9%

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

      5. unpow2 98.9%

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

      6. times-frac 99.1%

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

      7. *-un-lft-identity 99.1%

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

      8. rem-square-sqrt 99.1%

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

      9. pow2 99.1%

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

      10. unpow-prod-down 99.1%

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

      11. clear-num 99.2%

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

      12. unpow-prod-down 99.1%

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

      13. pow2 99.1%

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

      14. rem-square-sqrt 99.1%

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

      15. associate-/l* 99.1%

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

      16. clear-num 99.1%

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

   8. Applied egg-rr 99.3%

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

      1. unpow2 99.3%

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

      2. sin-mult 98.1%

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

   10. Applied egg-rr 98.1%

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

       1. div-sub 98.1%

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

       2. +-inverses 98.1%

          \[\leadsto \left(2.6666666666666665 \cdot \frac{1}{\sin x}\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}\right) \]

       3. cos-0 98.1%

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

       4. metadata-eval 98.1%

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

       5. distribute-lft-out 98.1%

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

       6. metadata-eval 98.1%

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

       7. *-rgt-identity 98.1%

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

   12. Simplified 98.1%

       \[\leadsto \left(2.6666666666666665 \cdot \frac{1}{\sin x}\right) \cdot \color{blue}{\left(0.5 - \frac{\cos x}{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0037:\\ \;\;\;\;\frac{{x\_m}^{3} \cdot 0.020833333333333332 + x\_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2.6666666666666665}{\sin x\_m} \cdot \left(-0.5 + 0.5 \cdot \cos x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0037)
    (/ (+ (* (pow x_m 3.0) 0.020833333333333332) (* x_m 0.25)) 0.375)
    (* (/ -2.6666666666666665 (sin x_m)) (+ -0.5 (* 0.5 (cos 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 <= 0.0037) {
		tmp = ((pow(x_m, 3.0) * 0.020833333333333332) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = (-2.6666666666666665 / sin(x_m)) * (-0.5 + (0.5 * cos(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 <= 0.0037d0) then
        tmp = (((x_m ** 3.0d0) * 0.020833333333333332d0) + (x_m * 0.25d0)) / 0.375d0
    else
        tmp = ((-2.6666666666666665d0) / sin(x_m)) * ((-0.5d0) + (0.5d0 * cos(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 <= 0.0037) {
		tmp = ((Math.pow(x_m, 3.0) * 0.020833333333333332) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = (-2.6666666666666665 / Math.sin(x_m)) * (-0.5 + (0.5 * Math.cos(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 <= 0.0037:
		tmp = ((math.pow(x_m, 3.0) * 0.020833333333333332) + (x_m * 0.25)) / 0.375
	else:
		tmp = (-2.6666666666666665 / math.sin(x_m)) * (-0.5 + (0.5 * math.cos(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 <= 0.0037)
		tmp = Float64(Float64(Float64((x_m ^ 3.0) * 0.020833333333333332) + Float64(x_m * 0.25)) / 0.375);
	else
		tmp = Float64(Float64(-2.6666666666666665 / sin(x_m)) * Float64(-0.5 + Float64(0.5 * cos(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 <= 0.0037)
		tmp = (((x_m ^ 3.0) * 0.020833333333333332) + (x_m * 0.25)) / 0.375;
	else
		tmp = (-2.6666666666666665 / sin(x_m)) * (-0.5 + (0.5 * cos(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, 0.0037], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 0.020833333333333332), $MachinePrecision] + N[(x$95$m * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(-2.6666666666666665 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] * N[(-0.5 + N[(0.5 * N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0037000000000000002

   1. Initial program 67.0%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r/ 66.8%

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

      2. clear-num 66.8%

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

      3. pow2 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. clear-num 66.8%

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

      2. associate-/l* 66.9%

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

      3. rem-square-sqrt 66.9%

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

      4. pow2 66.9%

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

      5. unpow-prod-down 66.9%

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

      6. clear-num 66.9%

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

      7. *-un-lft-identity 66.9%

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

      8. unpow-prod-down 66.8%

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

      9. pow2 66.8%

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

      10. rem-square-sqrt 66.8%

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

      11. times-frac 66.9%

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

      12. metadata-eval 66.9%

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

      13. unpow2 66.9%

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

      14. associate-/r* 99.3%

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

      15. associate-/l* 99.4%

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

      16. clear-num 99.6%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 66.0%

      \[\leadsto \frac{\color{blue}{x \cdot \left(0.25 + 0.020833333333333332 \cdot {x}^{2}\right)}}{0.375} \]
   10. Step-by-step derivation

       1. +-commutative 66.0%

          \[\leadsto \frac{x \cdot \color{blue}{\left(0.020833333333333332 \cdot {x}^{2} + 0.25\right)}}{0.375} \]

       2. distribute-lft-in 66.0%

          \[\leadsto \frac{\color{blue}{x \cdot \left(0.020833333333333332 \cdot {x}^{2}\right) + x \cdot 0.25}}{0.375} \]

       3. *-commutative 66.0%

          \[\leadsto \frac{x \cdot \color{blue}{\left({x}^{2} \cdot 0.020833333333333332\right)} + x \cdot 0.25}{0.375} \]

       4. associate-*r* 66.0%

          \[\leadsto \frac{\color{blue}{\left(x \cdot {x}^{2}\right) \cdot 0.020833333333333332} + x \cdot 0.25}{0.375} \]

       5. unpow2 66.0%

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot 0.020833333333333332 + x \cdot 0.25}{0.375} \]

       6. associate-*l* 66.0%

          \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot 0.020833333333333332 + x \cdot 0.25}{0.375} \]

       7. pow3 66.0%

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

   11. Applied egg-rr 66.0%

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

   if 0.0037000000000000002 < 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* 98.7%

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

      2. 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. metadata-eval 99.1%

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

   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. clear-num 99.1%

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

      3. pow2 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. unpow2 99.3%

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

      2. sin-mult 98.1%

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

   8. Applied egg-rr 97.9%

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

      1. div-sub 98.1%

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

      2. +-inverses 98.1%

         \[\leadsto \left(2.6666666666666665 \cdot \frac{1}{\sin x}\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}\right) \]

      3. cos-0 98.1%

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

      4. metadata-eval 98.1%

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

      5. distribute-lft-out 98.1%

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

      6. metadata-eval 98.1%

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

      7. *-rgt-identity 98.1%

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

   10. Simplified 97.9%

       \[\leadsto 2.6666666666666665 \cdot \frac{1}{\frac{\sin x}{\color{blue}{0.5 - \frac{\cos x}{2}}}} \]
   11. Step-by-step derivation

       1. un-div-inv 98.0%

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

       2. frac-2neg 98.0%

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

       3. metadata-eval 98.0%

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

       4. distribute-neg-frac2 98.0%

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

       5. sub-neg 98.0%

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

       6. distribute-neg-in 98.0%

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

       7. metadata-eval 98.0%

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

       8. distribute-neg-frac2 98.0%

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

       9. distribute-frac-neg 98.0%

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

       10. frac-2neg 98.0%

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

       11. div-inv 98.0%

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

       12. metadata-eval 98.0%

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

   12. Applied egg-rr 98.0%

       \[\leadsto \color{blue}{\frac{-2.6666666666666665}{\frac{\sin x}{-0.5 + \cos x \cdot 0.5}}} \]
   13. Step-by-step derivation

       1. associate-/r/ 98.1%

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

       2. *-commutative 98.1%

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

   14. Simplified 98.1%

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

Alternative 10: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0037:\\ \;\;\;\;\frac{{x\_m}^{3} \cdot 0.020833333333333332 + x\_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x\_m}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0037)
    (/ (+ (* (pow x_m 3.0) 0.020833333333333332) (* x_m 0.25)) 0.375)
    (* 2.6666666666666665 (/ (- 0.5 (* 0.5 (cos x_m))) (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 <= 0.0037) {
		tmp = ((pow(x_m, 3.0) * 0.020833333333333332) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x_m))) / 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 <= 0.0037d0) then
        tmp = (((x_m ** 3.0d0) * 0.020833333333333332d0) + (x_m * 0.25d0)) / 0.375d0
    else
        tmp = 2.6666666666666665d0 * ((0.5d0 - (0.5d0 * cos(x_m))) / 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 <= 0.0037) {
		tmp = ((Math.pow(x_m, 3.0) * 0.020833333333333332) + (x_m * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * Math.cos(x_m))) / 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 <= 0.0037:
		tmp = ((math.pow(x_m, 3.0) * 0.020833333333333332) + (x_m * 0.25)) / 0.375
	else:
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * math.cos(x_m))) / 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 <= 0.0037)
		tmp = Float64(Float64(Float64((x_m ^ 3.0) * 0.020833333333333332) + Float64(x_m * 0.25)) / 0.375);
	else
		tmp = Float64(2.6666666666666665 * Float64(Float64(0.5 - Float64(0.5 * cos(x_m))) / 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 <= 0.0037)
		tmp = (((x_m ^ 3.0) * 0.020833333333333332) + (x_m * 0.25)) / 0.375;
	else
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x_m))) / 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, 0.0037], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 0.020833333333333332), $MachinePrecision] + N[(x$95$m * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(2.6666666666666665 * N[(N[(0.5 - N[(0.5 * N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0037000000000000002

   1. Initial program 67.0%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r/ 66.8%

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

      2. clear-num 66.8%

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

      3. pow2 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. clear-num 66.8%

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

      2. associate-/l* 66.9%

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

      3. rem-square-sqrt 66.9%

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

      4. pow2 66.9%

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

      5. unpow-prod-down 66.9%

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

      6. clear-num 66.9%

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

      7. *-un-lft-identity 66.9%

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

      8. unpow-prod-down 66.8%

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

      9. pow2 66.8%

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

      10. rem-square-sqrt 66.8%

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

      11. times-frac 66.9%

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

      12. metadata-eval 66.9%

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

      13. unpow2 66.9%

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

      14. associate-/r* 99.3%

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

      15. associate-/l* 99.4%

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

      16. clear-num 99.6%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 66.0%

      \[\leadsto \frac{\color{blue}{x \cdot \left(0.25 + 0.020833333333333332 \cdot {x}^{2}\right)}}{0.375} \]
   10. Step-by-step derivation

       1. +-commutative 66.0%

          \[\leadsto \frac{x \cdot \color{blue}{\left(0.020833333333333332 \cdot {x}^{2} + 0.25\right)}}{0.375} \]

       2. distribute-lft-in 66.0%

          \[\leadsto \frac{\color{blue}{x \cdot \left(0.020833333333333332 \cdot {x}^{2}\right) + x \cdot 0.25}}{0.375} \]

       3. *-commutative 66.0%

          \[\leadsto \frac{x \cdot \color{blue}{\left({x}^{2} \cdot 0.020833333333333332\right)} + x \cdot 0.25}{0.375} \]

       4. associate-*r* 66.0%

          \[\leadsto \frac{\color{blue}{\left(x \cdot {x}^{2}\right) \cdot 0.020833333333333332} + x \cdot 0.25}{0.375} \]

       5. unpow2 66.0%

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot 0.020833333333333332 + x \cdot 0.25}{0.375} \]

       6. associate-*l* 66.0%

          \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot 0.020833333333333332 + x \cdot 0.25}{0.375} \]

       7. pow3 66.0%

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

   11. Applied egg-rr 66.0%

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

   if 0.0037000000000000002 < 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* 98.7%

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

      2. 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. metadata-eval 99.1%

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

   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. clear-num 99.1%

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

      3. pow2 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. unpow2 99.3%

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

      2. sin-mult 98.1%

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

   8. Applied egg-rr 97.9%

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

      1. div-sub 98.1%

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

      2. +-inverses 98.1%

         \[\leadsto \left(2.6666666666666665 \cdot \frac{1}{\sin x}\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}\right) \]

      3. cos-0 98.1%

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

      4. metadata-eval 98.1%

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

      5. distribute-lft-out 98.1%

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

      6. metadata-eval 98.1%

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

      7. *-rgt-identity 98.1%

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

   10. Simplified 97.9%

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

   11. Taylor expanded in x around inf 98.0%

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

Alternative 11: 55.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. associate-*r* 99.2%

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

   2. *-commutative 99.2%

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

   3. div-inv 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-/r/ 99.2%

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

   6. un-div-inv 99.2%

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

   7. *-un-lft-identity 99.2%

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

   8. times-frac 99.5%

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

   9. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0 54.6%

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

Alternative 12: 55.3% 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 #s(literal 1 binary64) 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 74.5%

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

   1. *-commutative 74.5%

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

   2. associate-/l* 99.3%

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

   3. remove-double-neg 99.3%

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

   4. sin-neg 99.3%

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

   5. distribute-lft-neg-out 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

   7. distribute-frac-neg 99.3%

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

   8. distribute-frac-neg2 99.3%

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

   9. neg-mul-1 99.3%

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

   10. associate-/r* 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 54.4%

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

Alternative 13: 51.5% 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 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ (* x_m 0.25) 0.375)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. associate-*r/ 74.4%

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

   2. clear-num 74.4%

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

   3. pow2 74.4%

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

6. Applied egg-rr 74.4%

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

   1. clear-num 74.4%

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

   2. associate-/l* 74.4%

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

   3. rem-square-sqrt 74.4%

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

   4. pow2 74.4%

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

   5. unpow-prod-down 74.5%

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

   6. clear-num 74.4%

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

   7. *-un-lft-identity 74.4%

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

   8. unpow-prod-down 74.4%

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

   9. pow2 74.4%

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

   10. rem-square-sqrt 74.4%

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

   11. times-frac 74.4%

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

   12. metadata-eval 74.4%

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

   13. unpow2 74.4%

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

   14. associate-/r* 99.2%

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

   15. associate-/l* 99.3%

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

   16. clear-num 99.5%

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

8. Applied egg-rr 99.5%

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

9. Taylor expanded in x around 0 50.9%

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

10. Final simplification 50.9%

    \[\leadsto \frac{x \cdot 0.25}{0.375} \]
11. Add Preprocessing

Alternative 14: 51.2% 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 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* x_m 0.6666666666666666)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 50.7%

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

6. Final simplification 50.7%

   \[\leadsto x \cdot 0.6666666666666666 \]
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 2024084 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :alt
  (/ (/ (* 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)))