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

Percentage Accurate: 77.4% → 99.5%

Time: 12.9s

Alternatives: 20

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\frac{x\_m}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\sin \left(x\_m \cdot 0.5\right)}^{2}}{--0.375}}{\sin x\_m}\\ \end{array} \end{array} \]

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-18)
    (/ x_m 1.5)
    (/ (/ (pow (sin (* x_m 0.5)) 2.0) (- -0.375)) (sin x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5e-18) {
		tmp = x_m / 1.5;
	} else {
		tmp = (pow(sin((x_m * 0.5)), 2.0) / -(-0.375)) / sin(x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 5d-18) then
        tmp = x_m / 1.5d0
    else
        tmp = ((sin((x_m * 0.5d0)) ** 2.0d0) / -(-0.375d0)) / sin(x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5e-18) {
		tmp = x_m / 1.5;
	} else {
		tmp = (Math.pow(Math.sin((x_m * 0.5)), 2.0) / -(-0.375)) / Math.sin(x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 5e-18:
		tmp = x_m / 1.5
	else:
		tmp = (math.pow(math.sin((x_m * 0.5)), 2.0) / -(-0.375)) / math.sin(x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 5e-18)
		tmp = Float64(x_m / 1.5);
	else
		tmp = Float64(Float64((sin(Float64(x_m * 0.5)) ^ 2.0) / Float64(-(-0.375))) / sin(x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 5e-18)
		tmp = x_m / 1.5;
	else
		tmp = ((sin((x_m * 0.5)) ^ 2.0) / -(-0.375)) / sin(x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 5e-18], N[(x$95$m / 1.5), $MachinePrecision], N[(N[(N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / (--0.375)), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000036e-18

   1. Initial program 70.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. 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* 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. associate-*r* 99.3%

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

      2. associate-*r/ 70.3%

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

      3. metadata-eval 70.3%

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

      4. clear-num 70.2%

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

      5. *-un-lft-identity 70.2%

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

      6. metadata-eval 70.2%

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

      7. associate-*l* 70.1%

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

      8. times-frac 70.2%

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

      9. metadata-eval 70.2%

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

      10. pow2 70.2%

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

   6. Applied egg-rr 70.2%

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

   7. Taylor expanded in x around 0 63.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
   8. Step-by-step derivation

      1. clear-num 64.2%

         \[\leadsto \color{blue}{\frac{x}{1.5}} \]

      2. add-cbrt-cube 24.4%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}}}{1.5} \]

      3. unpow2 24.4%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{x}^{2}} \cdot x}}{1.5} \]

      4. cbrt-prod 34.6%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{x}}}{1.5} \]

      5. associate-/l* 34.5%

         \[\leadsto \color{blue}{\sqrt[3]{{x}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5}} \]

      6. unpow2 34.5%

         \[\leadsto \sqrt[3]{\color{blue}{x \cdot x}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      7. cbrt-prod 62.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      8. pow2 62.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

   9. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{1.5}} \]
   10. Step-by-step derivation

       1. associate-*r/ 62.8%

          \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{1.5}} \]

       2. unpow2 62.8%

          \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{1.5} \]

       3. rem-3cbrt-lft 64.2%

          \[\leadsto \frac{\color{blue}{x}}{1.5} \]

   11. Simplified 64.2%

       \[\leadsto \color{blue}{\frac{x}{1.5}} \]

   if 5.00000000000000036e-18 < x

   1. Initial program 98.9%

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

      1. associate-/l* 98.8%

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

      2. associate-*l* 98.9%

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

      3. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

      1. associate-*r* 98.8%

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

      2. *-commutative 98.8%

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

      3. div-inv 98.7%

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

      4. associate-*l* 98.8%

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

      5. associate-/r/ 98.8%

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

      6. un-div-inv 98.9%

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

      7. *-un-lft-identity 98.9%

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

      8. times-frac 99.0%

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

      9. metadata-eval 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. clear-num 98.9%

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

      2. un-div-inv 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. frac-2neg 98.9%

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

      2. associate-/r/ 99.1%

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

      3. metadata-eval 99.1%

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

      4. distribute-neg-frac2 99.1%

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

   10. Applied egg-rr 99.1%

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

       1. associate-*r/ 99.1%

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

       2. associate-*l/ 99.0%

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

       3. unpow2 99.0%

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

   12. Simplified 99.0%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{--0.375}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{-27}:\\ \;\;\;\;\frac{x\_m}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\sin \left(x\_m \cdot 0.5\right)}^{2}}{\sin x\_m}}{0.375}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 4e-27)
    (/ x_m 1.5)
    (/ (/ (pow (sin (* x_m 0.5)) 2.0) (sin x_m)) 0.375))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 4e-27) {
		tmp = x_m / 1.5;
	} else {
		tmp = (pow(sin((x_m * 0.5)), 2.0) / sin(x_m)) / 0.375;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 4d-27) then
        tmp = x_m / 1.5d0
    else
        tmp = ((sin((x_m * 0.5d0)) ** 2.0d0) / sin(x_m)) / 0.375d0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 4e-27) {
		tmp = x_m / 1.5;
	} else {
		tmp = (Math.pow(Math.sin((x_m * 0.5)), 2.0) / Math.sin(x_m)) / 0.375;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 4e-27:
		tmp = x_m / 1.5
	else:
		tmp = (math.pow(math.sin((x_m * 0.5)), 2.0) / math.sin(x_m)) / 0.375
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 4e-27)
		tmp = Float64(x_m / 1.5);
	else
		tmp = Float64(Float64((sin(Float64(x_m * 0.5)) ^ 2.0) / sin(x_m)) / 0.375);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 4e-27)
		tmp = x_m / 1.5;
	else
		tmp = ((sin((x_m * 0.5)) ^ 2.0) / sin(x_m)) / 0.375;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 4e-27], N[(x$95$m / 1.5), $MachinePrecision], N[(N[(N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.0000000000000002e-27

   1. Initial program 69.4%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. associate-*r* 99.3%

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

      2. associate-*r/ 69.4%

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

      3. metadata-eval 69.4%

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

      4. clear-num 69.3%

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

      5. *-un-lft-identity 69.3%

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

      6. metadata-eval 69.3%

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

      7. associate-*l* 69.2%

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

      8. times-frac 69.3%

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

      9. metadata-eval 69.3%

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

      10. pow2 69.3%

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

   6. Applied egg-rr 69.3%

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

   7. Taylor expanded in x around 0 62.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
   8. Step-by-step derivation

      1. clear-num 63.0%

         \[\leadsto \color{blue}{\frac{x}{1.5}} \]

      2. add-cbrt-cube 21.9%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}}}{1.5} \]

      3. unpow2 21.9%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{x}^{2}} \cdot x}}{1.5} \]

      4. cbrt-prod 32.5%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{x}}}{1.5} \]

      5. associate-/l* 32.4%

         \[\leadsto \color{blue}{\sqrt[3]{{x}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5}} \]

      6. unpow2 32.4%

         \[\leadsto \sqrt[3]{\color{blue}{x \cdot x}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      7. cbrt-prod 61.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      8. pow2 61.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

   9. Applied egg-rr 61.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{1.5}} \]
   10. Step-by-step derivation

       1. associate-*r/ 61.7%

          \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{1.5}} \]

       2. unpow2 61.7%

          \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{1.5} \]

       3. rem-3cbrt-lft 63.0%

          \[\leadsto \frac{\color{blue}{x}}{1.5} \]

   11. Simplified 63.0%

       \[\leadsto \color{blue}{\frac{x}{1.5}} \]

   if 4.0000000000000002e-27 < x

   1. Initial program 98.9%

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

      1. associate-/l* 98.9%

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

      2. associate-*l* 98.9%

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

      3. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

      1. associate-*r* 98.9%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. add-log-exp 89.8%

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

      5. metadata-eval 89.8%

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

      6. associate-*l* 89.9%

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

      7. pow2 89.9%

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

   6. Applied egg-rr 89.9%

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

      1. clear-num 89.7%

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

      2. add-log-exp 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. times-frac 99.0%

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

      5. metadata-eval 99.0%

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

      6. *-commutative 99.0%

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

      7. associate-/r* 99.0%

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

      8. clear-num 99.1%

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

   8. Applied egg-rr 99.1%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\frac{x\_m}{1.5}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x\_m \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 4e-17)
    (/ x_m 1.5)
    (* (pow (sin (* x_m 0.5)) 2.0) (/ 2.6666666666666665 (sin x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 4e-17) {
		tmp = x_m / 1.5;
	} else {
		tmp = pow(sin((x_m * 0.5)), 2.0) * (2.6666666666666665 / sin(x_m));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 4d-17) then
        tmp = x_m / 1.5d0
    else
        tmp = (sin((x_m * 0.5d0)) ** 2.0d0) * (2.6666666666666665d0 / sin(x_m))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 4e-17) {
		tmp = x_m / 1.5;
	} else {
		tmp = Math.pow(Math.sin((x_m * 0.5)), 2.0) * (2.6666666666666665 / Math.sin(x_m));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 4e-17:
		tmp = x_m / 1.5
	else:
		tmp = math.pow(math.sin((x_m * 0.5)), 2.0) * (2.6666666666666665 / math.sin(x_m))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 4e-17)
		tmp = Float64(x_m / 1.5);
	else
		tmp = Float64((sin(Float64(x_m * 0.5)) ^ 2.0) * Float64(2.6666666666666665 / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 4e-17)
		tmp = x_m / 1.5;
	else
		tmp = (sin((x_m * 0.5)) ^ 2.0) * (2.6666666666666665 / sin(x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 4e-17], N[(x$95$m / 1.5), $MachinePrecision], N[(N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(2.6666666666666665 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000029e-17

   1. Initial program 70.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. 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* 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. associate-*r* 99.3%

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

      2. associate-*r/ 70.3%

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

      3. metadata-eval 70.3%

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

      4. clear-num 70.2%

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

      5. *-un-lft-identity 70.2%

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

      6. metadata-eval 70.2%

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

      7. associate-*l* 70.1%

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

      8. times-frac 70.2%

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

      9. metadata-eval 70.2%

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

      10. pow2 70.2%

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

   6. Applied egg-rr 70.2%

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

   7. Taylor expanded in x around 0 63.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
   8. Step-by-step derivation

      1. clear-num 64.2%

         \[\leadsto \color{blue}{\frac{x}{1.5}} \]

      2. add-cbrt-cube 24.4%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}}}{1.5} \]

      3. unpow2 24.4%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{x}^{2}} \cdot x}}{1.5} \]

      4. cbrt-prod 34.6%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{x}}}{1.5} \]

      5. associate-/l* 34.5%

         \[\leadsto \color{blue}{\sqrt[3]{{x}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5}} \]

      6. unpow2 34.5%

         \[\leadsto \sqrt[3]{\color{blue}{x \cdot x}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      7. cbrt-prod 62.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      8. pow2 62.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

   9. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{1.5}} \]
   10. Step-by-step derivation

       1. associate-*r/ 62.8%

          \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{1.5}} \]

       2. unpow2 62.8%

          \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{1.5} \]

       3. rem-3cbrt-lft 64.2%

          \[\leadsto \frac{\color{blue}{x}}{1.5} \]

   11. Simplified 64.2%

       \[\leadsto \color{blue}{\frac{x}{1.5}} \]

   if 4.00000000000000029e-17 < x

   1. Initial program 98.9%

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

      1. associate-/l* 98.8%

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

      2. associate-*l* 98.9%

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

      3. metadata-eval 98.9%

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

   3. Simplified 98.9%

      \[\leadsto \color{blue}{2.6666666666666665 \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}\right)} \]
   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. *-commutative 99.1%

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

      2. associate-*r/ 99.1%

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

      3. associate-*l/ 99.1%

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

      4. *-commutative 99.1%

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

   7. Simplified 99.1%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\frac{x\_m}{1.5}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x\_m \cdot 0.5\right)}^{2}}{\sin x\_m}\\ \end{array} \end{array} \]

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 4e-17)
    (/ x_m 1.5)
    (* 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 <= 4e-17) {
		tmp = x_m / 1.5;
	} 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 <= 4d-17) then
        tmp = x_m / 1.5d0
    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 <= 4e-17) {
		tmp = x_m / 1.5;
	} 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 <= 4e-17:
		tmp = x_m / 1.5
	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 <= 4e-17)
		tmp = Float64(x_m / 1.5);
	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 <= 4e-17)
		tmp = x_m / 1.5;
	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, 4e-17], N[(x$95$m / 1.5), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000029e-17

   1. Initial program 70.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. 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* 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. associate-*r* 99.3%

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

      2. associate-*r/ 70.3%

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

      3. metadata-eval 70.3%

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

      4. clear-num 70.2%

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

      5. *-un-lft-identity 70.2%

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

      6. metadata-eval 70.2%

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

      7. associate-*l* 70.1%

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

      8. times-frac 70.2%

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

      9. metadata-eval 70.2%

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

      10. pow2 70.2%

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

   6. Applied egg-rr 70.2%

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

   7. Taylor expanded in x around 0 63.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
   8. Step-by-step derivation

      1. clear-num 64.2%

         \[\leadsto \color{blue}{\frac{x}{1.5}} \]

      2. add-cbrt-cube 24.4%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}}}{1.5} \]

      3. unpow2 24.4%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{x}^{2}} \cdot x}}{1.5} \]

      4. cbrt-prod 34.6%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{x}}}{1.5} \]

      5. associate-/l* 34.5%

         \[\leadsto \color{blue}{\sqrt[3]{{x}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5}} \]

      6. unpow2 34.5%

         \[\leadsto \sqrt[3]{\color{blue}{x \cdot x}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      7. cbrt-prod 62.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      8. pow2 62.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

   9. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{1.5}} \]
   10. Step-by-step derivation

       1. associate-*r/ 62.8%

          \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{1.5}} \]

       2. unpow2 62.8%

          \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{1.5} \]

       3. rem-3cbrt-lft 64.2%

          \[\leadsto \frac{\color{blue}{x}}{1.5} \]

   11. Simplified 64.2%

       \[\leadsto \color{blue}{\frac{x}{1.5}} \]

   if 4.00000000000000029e-17 < x

   1. Initial program 98.9%

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

      1. metadata-eval 98.9%

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

      2. associate-*r/ 98.8%

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

      3. associate-*r* 98.9%

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

      4. *-commutative 98.9%

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

      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 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. 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* 99.1%

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

   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* 99.2%

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

   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 99.0%

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

   4. associate-*l* 99.0%

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

   5. associate-/r/ 99.0%

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

   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 99.5%

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

   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. Step-by-step derivation

   1. clear-num 99.4%

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

   2. un-div-inv 99.5%

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

8. Applied egg-rr 99.5%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. 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* 99.1%

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

   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* 99.2%

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

   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 99.0%

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

   4. associate-*l* 99.0%

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

   5. associate-/r/ 99.0%

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

   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 99.5%

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

   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. Add Preprocessing

Alternative 7: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((x_m * 0.5));
	return x_s * (t_0 * ((t_0 / sin(x_m)) * 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 * ((t_0 / sin(x_m)) * 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 * ((t_0 / Math.sin(x_m)) * 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 * ((t_0 / math.sin(x_m)) * 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(t_0 * Float64(Float64(t_0 / sin(x_m)) * 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 * ((t_0 / sin(x_m)) * 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[(t$95$0 * N[(N[(t$95$0 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] * 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 77.9%

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

   1. metadata-eval 77.9%

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

   2. associate-*l/ 99.2%

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

   3. associate-/l* 99.2%

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

4. Applied egg-rr 99.2%

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

5. Final simplification 99.2%

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

Alternative 8: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. *-commutative 77.9%

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

   2. associate-/l* 99.2%

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

   3. remove-double-neg 99.2%

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

   4. sin-neg 99.2%

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

   5. distribute-lft-neg-out 99.2%

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

   6. distribute-rgt-neg-in 99.2%

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

   7. distribute-frac-neg 99.2%

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

   8. distribute-frac-neg2 99.2%

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

   9. neg-mul-1 99.2%

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

   10. associate-/r* 99.2%

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

3. Simplified 99.2%

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

   1. clear-num 99.0%

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

   2. inv-pow 99.0%

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

   3. *-un-lft-identity 99.0%

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

   4. times-frac 99.1%

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

   5. metadata-eval 99.1%

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

6. Applied egg-rr 99.1%

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

   1. unpow-1 99.1%

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

   2. associate-/r* 99.2%

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

   3. metadata-eval 99.2%

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

8. Simplified 99.2%

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

Alternative 9: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. 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* 99.1%

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

   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. clear-num 99.1%

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

   2. un-div-inv 99.2%

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

6. Applied egg-rr 99.2%

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

Alternative 10: 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 77.9%

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

   1. 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* 99.1%

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

   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. Add Preprocessing

Alternative 11: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0305:\\ \;\;\;\;\frac{1}{\frac{1.5 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot -0.0020833333333333333 - 0.125\right)}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.375 \cdot \sin x\_m}{\frac{1 - \cos x\_m}{2}}}\\ \end{array} \end{array} \]

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 0.0305)
    (/
     1.0
     (/
      (+
       1.5
       (* (pow x_m 2.0) (- (* (pow x_m 2.0) -0.0020833333333333333) 0.125)))
      x_m))
    (/ 1.0 (/ (* 0.375 (sin x_m)) (/ (- 1.0 (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 <= 0.0305) {
		tmp = 1.0 / ((1.5 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * -0.0020833333333333333) - 0.125))) / x_m);
	} else {
		tmp = 1.0 / ((0.375 * sin(x_m)) / ((1.0 - 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 <= 0.0305d0) then
        tmp = 1.0d0 / ((1.5d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * (-0.0020833333333333333d0)) - 0.125d0))) / x_m)
    else
        tmp = 1.0d0 / ((0.375d0 * sin(x_m)) / ((1.0d0 - 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 <= 0.0305) {
		tmp = 1.0 / ((1.5 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * -0.0020833333333333333) - 0.125))) / x_m);
	} else {
		tmp = 1.0 / ((0.375 * Math.sin(x_m)) / ((1.0 - 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 <= 0.0305:
		tmp = 1.0 / ((1.5 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * -0.0020833333333333333) - 0.125))) / x_m)
	else:
		tmp = 1.0 / ((0.375 * math.sin(x_m)) / ((1.0 - 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 <= 0.0305)
		tmp = Float64(1.0 / Float64(Float64(1.5 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * -0.0020833333333333333) - 0.125))) / x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(0.375 * sin(x_m)) / Float64(Float64(1.0 - 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 <= 0.0305)
		tmp = 1.0 / ((1.5 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * -0.0020833333333333333) - 0.125))) / x_m);
	else
		tmp = 1.0 / ((0.375 * sin(x_m)) / ((1.0 - 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, 0.0305], N[(1.0 / N[(N[(1.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.0020833333333333333), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.375 * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.030499999999999999

   1. Initial program 70.8%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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* 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. associate-*r* 99.3%

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

      2. associate-*r/ 70.8%

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

      3. metadata-eval 70.8%

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

      4. clear-num 70.7%

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

      5. *-un-lft-identity 70.7%

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

      6. metadata-eval 70.7%

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

      7. associate-*l* 70.6%

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

      8. times-frac 70.7%

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

      9. metadata-eval 70.7%

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

      10. pow2 70.7%

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

   6. Applied egg-rr 70.7%

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

   7. Taylor expanded in x around 0 64.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5 + {x}^{2} \cdot \left(-0.0020833333333333333 \cdot {x}^{2} - 0.125\right)}{x}}} \]

   if 0.030499999999999999 < x

   1. Initial program 98.9%

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

      1. associate-/l* 98.8%

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

      2. associate-*l* 98.8%

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

      3. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

      1. associate-*r* 98.8%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. clear-num 98.8%

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

      5. *-un-lft-identity 98.8%

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

      6. metadata-eval 98.8%

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

      7. associate-*l* 98.9%

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

      8. times-frac 98.9%

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

      9. metadata-eval 98.9%

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

      10. pow2 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*l/ 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. unpow2 98.9%

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

      2. sin-mult 98.0%

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

   10. Applied egg-rr 98.0%

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

       1. +-inverses 98.0%

          \[\leadsto \frac{1}{\frac{\sin x \cdot 0.375}{\frac{\cos \color{blue}{0} - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}} \]

       2. cos-0 98.0%

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

       3. distribute-lft-out 98.0%

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

       4. metadata-eval 98.0%

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

       5. *-rgt-identity 98.0%

          \[\leadsto \frac{1}{\frac{\sin x \cdot 0.375}{\frac{1 - \cos \color{blue}{x}}{2}}} \]

   12. Simplified 98.0%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0305:\\ \;\;\;\;\frac{1}{\frac{1.5 + {x}^{2} \cdot \left({x}^{2} \cdot -0.0020833333333333333 - 0.125\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.375 \cdot \sin x}{\frac{1 - \cos x}{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0225:\\ \;\;\;\;x\_m \cdot \left(0.6666666666666666 + {x\_m}^{2} \cdot \left(0.05555555555555555 + {x\_m}^{2} \cdot 0.005555555555555556\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.375 \cdot \sin x\_m}{\frac{1 - \cos x\_m}{2}}}\\ \end{array} \end{array} \]

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 0.0225)
    (*
     x_m
     (+
      0.6666666666666666
      (*
       (pow x_m 2.0)
       (+ 0.05555555555555555 (* (pow x_m 2.0) 0.005555555555555556)))))
    (/ 1.0 (/ (* 0.375 (sin x_m)) (/ (- 1.0 (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 <= 0.0225) {
		tmp = x_m * (0.6666666666666666 + (pow(x_m, 2.0) * (0.05555555555555555 + (pow(x_m, 2.0) * 0.005555555555555556))));
	} else {
		tmp = 1.0 / ((0.375 * sin(x_m)) / ((1.0 - 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 <= 0.0225d0) then
        tmp = x_m * (0.6666666666666666d0 + ((x_m ** 2.0d0) * (0.05555555555555555d0 + ((x_m ** 2.0d0) * 0.005555555555555556d0))))
    else
        tmp = 1.0d0 / ((0.375d0 * sin(x_m)) / ((1.0d0 - 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 <= 0.0225) {
		tmp = x_m * (0.6666666666666666 + (Math.pow(x_m, 2.0) * (0.05555555555555555 + (Math.pow(x_m, 2.0) * 0.005555555555555556))));
	} else {
		tmp = 1.0 / ((0.375 * Math.sin(x_m)) / ((1.0 - 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 <= 0.0225:
		tmp = x_m * (0.6666666666666666 + (math.pow(x_m, 2.0) * (0.05555555555555555 + (math.pow(x_m, 2.0) * 0.005555555555555556))))
	else:
		tmp = 1.0 / ((0.375 * math.sin(x_m)) / ((1.0 - 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 <= 0.0225)
		tmp = Float64(x_m * Float64(0.6666666666666666 + Float64((x_m ^ 2.0) * Float64(0.05555555555555555 + Float64((x_m ^ 2.0) * 0.005555555555555556)))));
	else
		tmp = Float64(1.0 / Float64(Float64(0.375 * sin(x_m)) / Float64(Float64(1.0 - 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 <= 0.0225)
		tmp = x_m * (0.6666666666666666 + ((x_m ^ 2.0) * (0.05555555555555555 + ((x_m ^ 2.0) * 0.005555555555555556))));
	else
		tmp = 1.0 / ((0.375 * sin(x_m)) / ((1.0 - 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, 0.0225], N[(x$95$m * N[(0.6666666666666666 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.05555555555555555 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.375 * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.022499999999999999

   1. Initial program 70.8%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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* 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 64.3%

      \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + 0.005555555555555556 \cdot {x}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 64.3%

         \[\leadsto x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + \color{blue}{{x}^{2} \cdot 0.005555555555555556}\right)\right) \]

   7. Simplified 64.3%

      \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot 0.005555555555555556\right)\right)} \]

   if 0.022499999999999999 < x

   1. Initial program 98.9%

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

      1. associate-/l* 98.8%

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

      2. associate-*l* 98.8%

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

      3. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

      1. associate-*r* 98.8%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. clear-num 98.8%

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

      5. *-un-lft-identity 98.8%

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

      6. metadata-eval 98.8%

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

      7. associate-*l* 98.9%

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

      8. times-frac 98.9%

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

      9. metadata-eval 98.9%

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

      10. pow2 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*l/ 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. unpow2 98.9%

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

      2. sin-mult 98.0%

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

   10. Applied egg-rr 98.0%

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

       1. +-inverses 98.0%

          \[\leadsto \frac{1}{\frac{\sin x \cdot 0.375}{\frac{\cos \color{blue}{0} - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}} \]

       2. cos-0 98.0%

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

       3. distribute-lft-out 98.0%

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

       4. metadata-eval 98.0%

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

       5. *-rgt-identity 98.0%

          \[\leadsto \frac{1}{\frac{\sin x \cdot 0.375}{\frac{1 - \cos \color{blue}{x}}{2}}} \]

   12. Simplified 98.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0225:\\ \;\;\;\;x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot 0.005555555555555556\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.375 \cdot \sin x}{\frac{1 - \cos x}{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000135:\\ \;\;\;\;\frac{x\_m}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.375 \cdot \sin x\_m}{\frac{1 - \cos x\_m}{2}}}\\ \end{array} \end{array} \]

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 0.000135)
    (/ x_m 1.5)
    (/ 1.0 (/ (* 0.375 (sin x_m)) (/ (- 1.0 (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 <= 0.000135) {
		tmp = x_m / 1.5;
	} else {
		tmp = 1.0 / ((0.375 * sin(x_m)) / ((1.0 - 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 <= 0.000135d0) then
        tmp = x_m / 1.5d0
    else
        tmp = 1.0d0 / ((0.375d0 * sin(x_m)) / ((1.0d0 - 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 <= 0.000135) {
		tmp = x_m / 1.5;
	} else {
		tmp = 1.0 / ((0.375 * Math.sin(x_m)) / ((1.0 - 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 <= 0.000135:
		tmp = x_m / 1.5
	else:
		tmp = 1.0 / ((0.375 * math.sin(x_m)) / ((1.0 - 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 <= 0.000135)
		tmp = Float64(x_m / 1.5);
	else
		tmp = Float64(1.0 / Float64(Float64(0.375 * sin(x_m)) / Float64(Float64(1.0 - 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 <= 0.000135)
		tmp = x_m / 1.5;
	else
		tmp = 1.0 / ((0.375 * sin(x_m)) / ((1.0 - 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, 0.000135], N[(x$95$m / 1.5), $MachinePrecision], N[(1.0 / N[(N[(0.375 * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000002e-4

   1. Initial program 70.6%

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

      1. 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* 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. associate-*r* 99.3%

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

      2. associate-*r/ 70.6%

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

      3. metadata-eval 70.6%

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

      4. clear-num 70.5%

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

      5. *-un-lft-identity 70.5%

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

      6. metadata-eval 70.5%

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

      7. associate-*l* 70.5%

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

      8. times-frac 70.5%

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

      9. metadata-eval 70.5%

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

      10. pow2 70.5%

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

   6. Applied egg-rr 70.5%

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

   7. Taylor expanded in x around 0 64.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
   8. Step-by-step derivation

      1. clear-num 64.5%

         \[\leadsto \color{blue}{\frac{x}{1.5}} \]

      2. add-cbrt-cube 25.1%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}}}{1.5} \]

      3. unpow2 25.1%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{x}^{2}} \cdot x}}{1.5} \]

      4. cbrt-prod 35.2%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{x}}}{1.5} \]

      5. associate-/l* 35.1%

         \[\leadsto \color{blue}{\sqrt[3]{{x}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5}} \]

      6. unpow2 35.1%

         \[\leadsto \sqrt[3]{\color{blue}{x \cdot x}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      7. cbrt-prod 63.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      8. pow2 63.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

   9. Applied egg-rr 63.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{1.5}} \]
   10. Step-by-step derivation

       1. associate-*r/ 63.1%

          \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{1.5}} \]

       2. unpow2 63.1%

          \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{1.5} \]

       3. rem-3cbrt-lft 64.5%

          \[\leadsto \frac{\color{blue}{x}}{1.5} \]

   11. Simplified 64.5%

       \[\leadsto \color{blue}{\frac{x}{1.5}} \]

   if 1.35000000000000002e-4 < x

   1. Initial program 98.9%

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

      1. associate-/l* 98.8%

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

      2. associate-*l* 98.8%

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

      3. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

      1. associate-*r* 98.8%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. clear-num 98.8%

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

      5. *-un-lft-identity 98.8%

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

      6. metadata-eval 98.8%

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

      7. associate-*l* 98.9%

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

      8. times-frac 98.9%

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

      9. metadata-eval 98.9%

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

      10. pow2 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*l/ 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. unpow2 98.9%

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

      2. sin-mult 97.5%

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

   10. Applied egg-rr 97.5%

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

       1. +-inverses 97.5%

          \[\leadsto \frac{1}{\frac{\sin x \cdot 0.375}{\frac{\cos \color{blue}{0} - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}} \]

       2. cos-0 97.5%

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

       3. distribute-lft-out 97.5%

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

       4. metadata-eval 97.5%

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

       5. *-rgt-identity 97.5%

          \[\leadsto \frac{1}{\frac{\sin x \cdot 0.375}{\frac{1 - \cos \color{blue}{x}}{2}}} \]

   12. Simplified 97.5%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000135:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.375 \cdot \sin x}{\frac{1 - \cos x}{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000135:\\ \;\;\;\;\frac{x\_m}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665 \cdot \frac{1 - \cos x\_m}{2}}{\sin x\_m}\\ \end{array} \end{array} \]

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 0.000135)
    (/ x_m 1.5)
    (/ (* 2.6666666666666665 (/ (- 1.0 (cos x_m)) 2.0)) (sin x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.000135) {
		tmp = x_m / 1.5;
	} else {
		tmp = (2.6666666666666665 * ((1.0 - cos(x_m)) / 2.0)) / sin(x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.000135d0) then
        tmp = x_m / 1.5d0
    else
        tmp = (2.6666666666666665d0 * ((1.0d0 - cos(x_m)) / 2.0d0)) / sin(x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.000135) {
		tmp = x_m / 1.5;
	} else {
		tmp = (2.6666666666666665 * ((1.0 - Math.cos(x_m)) / 2.0)) / Math.sin(x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.000135:
		tmp = x_m / 1.5
	else:
		tmp = (2.6666666666666665 * ((1.0 - math.cos(x_m)) / 2.0)) / math.sin(x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.000135)
		tmp = Float64(x_m / 1.5);
	else
		tmp = Float64(Float64(2.6666666666666665 * Float64(Float64(1.0 - cos(x_m)) / 2.0)) / sin(x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.000135)
		tmp = x_m / 1.5;
	else
		tmp = (2.6666666666666665 * ((1.0 - cos(x_m)) / 2.0)) / sin(x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.000135], N[(x$95$m / 1.5), $MachinePrecision], N[(N[(2.6666666666666665 * N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000002e-4

   1. Initial program 70.6%

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

      1. 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* 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. associate-*r* 99.3%

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

      2. associate-*r/ 70.6%

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

      3. metadata-eval 70.6%

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

      4. clear-num 70.5%

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

      5. *-un-lft-identity 70.5%

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

      6. metadata-eval 70.5%

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

      7. associate-*l* 70.5%

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

      8. times-frac 70.5%

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

      9. metadata-eval 70.5%

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

      10. pow2 70.5%

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

   6. Applied egg-rr 70.5%

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

   7. Taylor expanded in x around 0 64.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
   8. Step-by-step derivation

      1. clear-num 64.5%

         \[\leadsto \color{blue}{\frac{x}{1.5}} \]

      2. add-cbrt-cube 25.1%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}}}{1.5} \]

      3. unpow2 25.1%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{x}^{2}} \cdot x}}{1.5} \]

      4. cbrt-prod 35.2%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{x}}}{1.5} \]

      5. associate-/l* 35.1%

         \[\leadsto \color{blue}{\sqrt[3]{{x}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5}} \]

      6. unpow2 35.1%

         \[\leadsto \sqrt[3]{\color{blue}{x \cdot x}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      7. cbrt-prod 63.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      8. pow2 63.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

   9. Applied egg-rr 63.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{1.5}} \]
   10. Step-by-step derivation

       1. associate-*r/ 63.1%

          \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{1.5}} \]

       2. unpow2 63.1%

          \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{1.5} \]

       3. rem-3cbrt-lft 64.5%

          \[\leadsto \frac{\color{blue}{x}}{1.5} \]

   11. Simplified 64.5%

       \[\leadsto \color{blue}{\frac{x}{1.5}} \]

   if 1.35000000000000002e-4 < x

   1. Initial program 98.9%

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

      1. associate-/l* 98.8%

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

      2. associate-*l* 98.8%

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

      3. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

      1. associate-*r* 98.8%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. add-log-exp 98.4%

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

      5. metadata-eval 98.4%

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

      6. associate-*l* 98.4%

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

      7. pow2 98.4%

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

   6. Applied egg-rr 98.4%

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

      1. rem-log-exp 99.0%

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

      2. *-commutative 99.0%

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

   8. Applied egg-rr 99.0%

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

      1. unpow2 98.9%

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

      2. sin-mult 97.5%

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

   10. Applied egg-rr 97.6%

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

       1. +-inverses 97.5%

          \[\leadsto \frac{1}{\frac{\sin x \cdot 0.375}{\frac{\cos \color{blue}{0} - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}} \]

       2. cos-0 97.5%

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

       3. distribute-lft-out 97.5%

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

       4. metadata-eval 97.5%

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

       5. *-rgt-identity 97.5%

          \[\leadsto \frac{1}{\frac{\sin x \cdot 0.375}{\frac{1 - \cos \color{blue}{x}}{2}}} \]

   12. Simplified 97.6%

       \[\leadsto \frac{\color{blue}{\frac{1 - \cos x}{2}} \cdot 2.6666666666666665}{\sin x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000135:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665 \cdot \frac{1 - \cos x}{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000135:\\ \;\;\;\;\frac{x\_m}{1.5}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{\frac{1 - \cos x\_m}{2}}{\sin x\_m}\\ \end{array} \end{array} \]

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 0.000135)
    (/ x_m 1.5)
    (* 2.6666666666666665 (/ (/ (- 1.0 (cos x_m)) 2.0) (sin x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.000135) {
		tmp = x_m / 1.5;
	} else {
		tmp = 2.6666666666666665 * (((1.0 - cos(x_m)) / 2.0) / sin(x_m));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.000135d0) then
        tmp = x_m / 1.5d0
    else
        tmp = 2.6666666666666665d0 * (((1.0d0 - cos(x_m)) / 2.0d0) / sin(x_m))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.000135) {
		tmp = x_m / 1.5;
	} else {
		tmp = 2.6666666666666665 * (((1.0 - Math.cos(x_m)) / 2.0) / Math.sin(x_m));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.000135:
		tmp = x_m / 1.5
	else:
		tmp = 2.6666666666666665 * (((1.0 - math.cos(x_m)) / 2.0) / math.sin(x_m))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.000135)
		tmp = Float64(x_m / 1.5);
	else
		tmp = Float64(2.6666666666666665 * Float64(Float64(Float64(1.0 - cos(x_m)) / 2.0) / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.000135)
		tmp = x_m / 1.5;
	else
		tmp = 2.6666666666666665 * (((1.0 - cos(x_m)) / 2.0) / sin(x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.000135], N[(x$95$m / 1.5), $MachinePrecision], N[(2.6666666666666665 * N[(N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000002e-4

   1. Initial program 70.6%

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

      1. 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* 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. associate-*r* 99.3%

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

      2. associate-*r/ 70.6%

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

      3. metadata-eval 70.6%

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

      4. clear-num 70.5%

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

      5. *-un-lft-identity 70.5%

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

      6. metadata-eval 70.5%

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

      7. associate-*l* 70.5%

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

      8. times-frac 70.5%

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

      9. metadata-eval 70.5%

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

      10. pow2 70.5%

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

   6. Applied egg-rr 70.5%

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

   7. Taylor expanded in x around 0 64.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
   8. Step-by-step derivation

      1. clear-num 64.5%

         \[\leadsto \color{blue}{\frac{x}{1.5}} \]

      2. add-cbrt-cube 25.1%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}}}{1.5} \]

      3. unpow2 25.1%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{x}^{2}} \cdot x}}{1.5} \]

      4. cbrt-prod 35.2%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{x}}}{1.5} \]

      5. associate-/l* 35.1%

         \[\leadsto \color{blue}{\sqrt[3]{{x}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5}} \]

      6. unpow2 35.1%

         \[\leadsto \sqrt[3]{\color{blue}{x \cdot x}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      7. cbrt-prod 63.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      8. pow2 63.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

   9. Applied egg-rr 63.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{1.5}} \]
   10. Step-by-step derivation

       1. associate-*r/ 63.1%

          \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{1.5}} \]

       2. unpow2 63.1%

          \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{1.5} \]

       3. rem-3cbrt-lft 64.5%

          \[\leadsto \frac{\color{blue}{x}}{1.5} \]

   11. Simplified 64.5%

       \[\leadsto \color{blue}{\frac{x}{1.5}} \]

   if 1.35000000000000002e-4 < x

   1. Initial program 98.9%

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

      1. metadata-eval 98.9%

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

      2. associate-*r/ 98.8%

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

      3. associate-*r* 98.8%

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

      4. *-commutative 98.8%

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

      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%

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

      1. unpow2 98.9%

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

      2. sin-mult 97.5%

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

   6. Applied egg-rr 97.4%

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

      1. +-inverses 97.5%

         \[\leadsto \frac{1}{\frac{\sin x \cdot 0.375}{\frac{\cos \color{blue}{0} - \cos \left(x \cdot 0.5 + x \cdot 0.5\right)}{2}}} \]

      2. cos-0 97.5%

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

      3. distribute-lft-out 97.5%

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

      4. metadata-eval 97.5%

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

      5. *-rgt-identity 97.5%

         \[\leadsto \frac{1}{\frac{\sin x \cdot 0.375}{\frac{1 - \cos \color{blue}{x}}{2}}} \]

   8. Simplified 97.4%

      \[\leadsto \frac{\color{blue}{\frac{1 - \cos x}{2}}}{\sin x} \cdot 2.6666666666666665 \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000135:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{\frac{1 - \cos x}{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 53.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 9.5:\\ \;\;\;\;\frac{x\_m}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\_m \cdot 0.6666666666666666\right)\\ \end{array} \end{array} \]

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 9.5) (/ x_m 1.5) (log (+ 1.0 (* x_m 0.6666666666666666))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 9.5) {
		tmp = x_m / 1.5;
	} else {
		tmp = log((1.0 + (x_m * 0.6666666666666666)));
	}
	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 <= 9.5d0) then
        tmp = x_m / 1.5d0
    else
        tmp = log((1.0d0 + (x_m * 0.6666666666666666d0)))
    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 <= 9.5) {
		tmp = x_m / 1.5;
	} else {
		tmp = Math.log((1.0 + (x_m * 0.6666666666666666)));
	}
	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 <= 9.5:
		tmp = x_m / 1.5
	else:
		tmp = math.log((1.0 + (x_m * 0.6666666666666666)))
	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 <= 9.5)
		tmp = Float64(x_m / 1.5);
	else
		tmp = log(Float64(1.0 + Float64(x_m * 0.6666666666666666)));
	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 <= 9.5)
		tmp = x_m / 1.5;
	else
		tmp = log((1.0 + (x_m * 0.6666666666666666)));
	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, 9.5], N[(x$95$m / 1.5), $MachinePrecision], N[Log[N[(1.0 + N[(x$95$m * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.5

   1. Initial program 70.9%

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

      1. 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* 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. associate-*r* 99.3%

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

      2. associate-*r/ 70.9%

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

      3. metadata-eval 70.9%

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

      4. clear-num 70.8%

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

      5. *-un-lft-identity 70.8%

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

      6. metadata-eval 70.8%

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

      7. associate-*l* 70.8%

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

      8. times-frac 70.8%

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

      9. metadata-eval 70.8%

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

      10. pow2 70.8%

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

   6. Applied egg-rr 70.8%

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

   7. Taylor expanded in x around 0 63.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
   8. Step-by-step derivation

      1. clear-num 64.2%

         \[\leadsto \color{blue}{\frac{x}{1.5}} \]

      2. add-cbrt-cube 25.2%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}}}{1.5} \]

      3. unpow2 25.2%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{x}^{2}} \cdot x}}{1.5} \]

      4. cbrt-prod 35.3%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{x}}}{1.5} \]

      5. associate-/l* 35.1%

         \[\leadsto \color{blue}{\sqrt[3]{{x}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5}} \]

      6. unpow2 35.1%

         \[\leadsto \sqrt[3]{\color{blue}{x \cdot x}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      7. cbrt-prod 62.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\sqrt[3]{x}}{1.5} \]

      8. pow2 62.8%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

   9. Applied egg-rr 62.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{1.5}} \]
   10. Step-by-step derivation

       1. associate-*r/ 62.8%

          \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{1.5}} \]

       2. unpow2 62.8%

          \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{1.5} \]

       3. rem-3cbrt-lft 64.2%

          \[\leadsto \frac{\color{blue}{x}}{1.5} \]

   11. Simplified 64.2%

       \[\leadsto \color{blue}{\frac{x}{1.5}} \]

   if 9.5 < x

   1. Initial program 98.9%

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

      1. associate-/l* 98.8%

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

      2. associate-*l* 98.8%

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

      3. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

      1. associate-*r* 98.8%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. add-log-exp 98.6%

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

      5. metadata-eval 98.6%

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

      6. associate-*l* 98.6%

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

      7. pow2 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in x around 0 7.6%

      \[\leadsto \log \color{blue}{\left(1 + 0.6666666666666666 \cdot x\right)} \]
   8. Step-by-step derivation

      1. *-commutative 7.6%

         \[\leadsto \log \left(1 + \color{blue}{x \cdot 0.6666666666666666}\right) \]

   9. Simplified 7.6%

      \[\leadsto \log \color{blue}{\left(1 + x \cdot 0.6666666666666666\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 54.9% 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 77.9%

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

   1. 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* 99.1%

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

   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* 99.2%

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

   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 99.0%

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

   4. associate-*l* 99.0%

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

   5. associate-/r/ 99.0%

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

   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 99.5%

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

   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 52.5%

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

Alternative 18: 54.7% 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 77.9%

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

   1. *-commutative 77.9%

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

   2. associate-/l* 99.2%

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

   3. remove-double-neg 99.2%

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

   4. sin-neg 99.2%

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

   5. distribute-lft-neg-out 99.2%

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

   6. distribute-rgt-neg-in 99.2%

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

   7. distribute-frac-neg 99.2%

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

   8. distribute-frac-neg2 99.2%

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

   9. neg-mul-1 99.2%

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

   10. associate-/r* 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 52.3%

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

Alternative 19: 50.7% accurate, 104.3× speedup

Math

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

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 1.5)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. 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* 99.1%

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

   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* 99.2%

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

   2. associate-*r/ 77.9%

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

   3. metadata-eval 77.9%

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

   4. clear-num 77.8%

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

   5. *-un-lft-identity 77.8%

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

   6. metadata-eval 77.8%

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

   7. associate-*l* 77.8%

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

   8. times-frac 77.9%

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

   9. metadata-eval 77.9%

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

   10. pow2 77.9%

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

6. Applied egg-rr 77.9%

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

7. Taylor expanded in x around 0 48.8%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
8. Step-by-step derivation

   1. clear-num 49.0%

      \[\leadsto \color{blue}{\frac{x}{1.5}} \]

   2. add-cbrt-cube 19.7%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}}}{1.5} \]

   3. unpow2 19.7%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{x}^{2}} \cdot x}}{1.5} \]

   4. cbrt-prod 27.2%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{x}}}{1.5} \]

   5. associate-/l* 27.1%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5}} \]

   6. unpow2 27.1%

      \[\leadsto \sqrt[3]{\color{blue}{x \cdot x}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

   7. cbrt-prod 47.9%

      \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\sqrt[3]{x}}{1.5} \]

   8. pow2 47.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{1.5} \]

9. Applied egg-rr 47.9%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{1.5}} \]
10. Step-by-step derivation

    1. associate-*r/ 48.0%

       \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{1.5}} \]

    2. unpow2 48.0%

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{1.5} \]

    3. rem-3cbrt-lft 49.0%

       \[\leadsto \frac{\color{blue}{x}}{1.5} \]

11. Simplified 49.0%

    \[\leadsto \color{blue}{\frac{x}{1.5}} \]
12. Add Preprocessing

Alternative 20: 50.5% 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 77.9%

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

   1. 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* 99.1%

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

   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 0 48.7%

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

6. Final simplification 48.7%

   \[\leadsto x \cdot 0.6666666666666666 \]
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 2024085 
(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)))