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

Percentage Accurate: 76.6% → 99.5%

Time: 18.5s

Alternatives: 15

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x_m \cdot 0.5\right)\\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.0002:\\ \;\;\;\;\frac{t_0}{0.375} \cdot \left(0.5 + {x_m}^{2} \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{t_0}^{2}}{\sin x_m \cdot 0.375}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5))))
   (*
    x_s
    (if (<= x_m 0.0002)
      (* (/ t_0 0.375) (+ 0.5 (* (pow x_m 2.0) 0.0625)))
      (/ (pow t_0 2.0) (* (sin x_m) 0.375))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((x_m * 0.5));
	double tmp;
	if (x_m <= 0.0002) {
		tmp = (t_0 / 0.375) * (0.5 + (pow(x_m, 2.0) * 0.0625));
	} else {
		tmp = pow(t_0, 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) :: t_0
    real(8) :: tmp
    t_0 = sin((x_m * 0.5d0))
    if (x_m <= 0.0002d0) then
        tmp = (t_0 / 0.375d0) * (0.5d0 + ((x_m ** 2.0d0) * 0.0625d0))
    else
        tmp = (t_0 ** 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 t_0 = Math.sin((x_m * 0.5));
	double tmp;
	if (x_m <= 0.0002) {
		tmp = (t_0 / 0.375) * (0.5 + (Math.pow(x_m, 2.0) * 0.0625));
	} else {
		tmp = Math.pow(t_0, 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):
	t_0 = math.sin((x_m * 0.5))
	tmp = 0
	if x_m <= 0.0002:
		tmp = (t_0 / 0.375) * (0.5 + (math.pow(x_m, 2.0) * 0.0625))
	else:
		tmp = math.pow(t_0, 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)
	t_0 = sin(Float64(x_m * 0.5))
	tmp = 0.0
	if (x_m <= 0.0002)
		tmp = Float64(Float64(t_0 / 0.375) * Float64(0.5 + Float64((x_m ^ 2.0) * 0.0625)));
	else
		tmp = Float64((t_0 ^ 2.0) / Float64(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)
	t_0 = sin((x_m * 0.5));
	tmp = 0.0;
	if (x_m <= 0.0002)
		tmp = (t_0 / 0.375) * (0.5 + ((x_m ^ 2.0) * 0.0625));
	else
		tmp = (t_0 ^ 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_] := Block[{t$95$0 = N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 0.0002], N[(N[(t$95$0 / 0.375), $MachinePrecision] * N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[(N[Sin[x$95$m], $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-4

   1. Initial program 68.3%

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

      1. associate-/l* 99.3%

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

      2. *-commutative 99.3%

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

      3. *-lft-identity 99.3%

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

      4. metadata-eval 99.3%

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

      5. times-frac 99.3%

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

      6. associate-/l* 99.3%

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

      7. *-commutative 99.3%

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

      8. neg-mul-1 99.3%

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

      9. sin-neg 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. times-frac 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/l/ 99.3%

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

      14. associate-/r* 99.3%

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

   3. Simplified 99.3%

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

      1. associate-/r/ 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l/ 99.2%

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

      4. associate-/r/ 99.2%

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

      5. associate-*l/ 68.3%

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

      6. div-inv 68.5%

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

      7. times-frac 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around 0 62.5%

      \[\leadsto \color{blue}{\left(0.5 + 0.0625 \cdot {x}^{2}\right)} \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375} \]
   7. Step-by-step derivation

      1. *-commutative 62.5%

         \[\leadsto \left(0.5 + \color{blue}{{x}^{2} \cdot 0.0625}\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375} \]

   8. Simplified 62.5%

      \[\leadsto \color{blue}{\left(0.5 + {x}^{2} \cdot 0.0625\right)} \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375} \]

   if 2.0000000000000001e-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. *-commutative 98.9%

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

      2. remove-double-neg 98.9%

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

      3. sin-neg 98.9%

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

      4. distribute-lft-neg-out 98.9%

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

      5. distribute-rgt-neg-in 98.9%

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

      6. associate-*l/ 98.9%

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

      7. *-commutative 98.9%

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

      8. distribute-rgt-neg-in 98.9%

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

      9. distribute-lft-neg-out 98.9%

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

      10. sin-neg 98.9%

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

      11. remove-double-neg 98.9%

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

      12. associate-*l* 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. 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. Step-by-step derivation

      1. *-commutative 98.9%

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

      2. associate-*r/ 99.1%

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

      3. associate-/r/ 99.1%

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

      4. pow2 99.1%

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

      5. div-inv 99.2%

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

      6. metadata-eval 99.2%

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

   5. Applied egg-rr 99.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0002:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.375} \cdot \left(0.5 + {x}^{2} \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x \cdot 0.375}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5))))
   (* x_s (* (/ 1.0 (/ (sin x_m) t_0)) (/ t_0 0.375)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.2%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. *-lft-identity 99.2%

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

   4. metadata-eval 99.2%

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

   5. times-frac 99.2%

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

   6. associate-/l* 99.2%

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

   7. *-commutative 99.2%

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

   8. neg-mul-1 99.2%

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

   9. sin-neg 99.2%

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

   10. distribute-lft-neg-out 99.2%

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

   11. times-frac 99.2%

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

   12. *-commutative 99.2%

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

   13. associate-/l/ 99.2%

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

   14. associate-/r* 99.2%

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

3. Simplified 99.2%

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

   1. associate-/r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.1%

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

   4. associate-/r/ 99.2%

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

   5. associate-*l/ 75.3%

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

   6. div-inv 75.4%

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

   7. times-frac 99.4%

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

   8. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

   1. clear-num 99.4%

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

   2. inv-pow 99.4%

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

7. Applied egg-rr 99.4%

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

   1. unpow-1 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5))))
   (* x_s (* 2.6666666666666665 (* t_0 (/ t_0 (sin x_m)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.2%

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

   1. *-commutative 75.2%

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

   2. remove-double-neg 75.2%

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

   3. sin-neg 75.2%

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

   4. distribute-lft-neg-out 75.2%

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

   5. distribute-rgt-neg-in 75.2%

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

   6. associate-*l/ 99.1%

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

   7. *-commutative 99.1%

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

   8. distribute-rgt-neg-in 99.1%

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

   9. distribute-lft-neg-out 99.1%

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

   10. sin-neg 99.1%

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

   11. remove-double-neg 99.1%

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

   12. associate-*l* 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 75.2%

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

   1. *-commutative 75.2%

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

   2. remove-double-neg 75.2%

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

   3. sin-neg 75.2%

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

   4. distribute-lft-neg-out 75.2%

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

   5. distribute-rgt-neg-in 75.2%

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

   6. associate-*r/ 99.2%

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

   7. *-commutative 99.2%

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

   8. distribute-lft-neg-in 99.2%

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

   9. distribute-lft-neg-out 99.2%

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

   10. sin-neg 99.2%

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

   11. remove-double-neg 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.2%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. *-lft-identity 99.2%

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

   4. metadata-eval 99.2%

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

   5. times-frac 99.2%

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

   6. associate-/l* 99.2%

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

   7. *-commutative 99.2%

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

   8. neg-mul-1 99.2%

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

   9. sin-neg 99.2%

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

   10. distribute-lft-neg-out 99.2%

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

   11. times-frac 99.2%

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

   12. *-commutative 99.2%

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

   13. associate-/l/ 99.2%

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

   14. associate-/r* 99.2%

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

3. Simplified 99.2%

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

   1. associate-/r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.1%

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

   4. associate-/r/ 99.2%

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

   5. associate-*l/ 75.3%

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

   6. div-inv 75.4%

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

   7. times-frac 99.4%

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

   8. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x_m \cdot 0.5\right)\\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{0.375} \cdot \left(0.5 + {x_m}^{2} \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{t_0}^{2}}{\sin x_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5))))
   (*
    x_s
    (if (<= x_m 5e-7)
      (* (/ t_0 0.375) (+ 0.5 (* (pow x_m 2.0) 0.0625)))
      (* 2.6666666666666665 (/ (pow t_0 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 t_0 = sin((x_m * 0.5));
	double tmp;
	if (x_m <= 5e-7) {
		tmp = (t_0 / 0.375) * (0.5 + (pow(x_m, 2.0) * 0.0625));
	} else {
		tmp = 2.6666666666666665 * (pow(t_0, 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) :: t_0
    real(8) :: tmp
    t_0 = sin((x_m * 0.5d0))
    if (x_m <= 5d-7) then
        tmp = (t_0 / 0.375d0) * (0.5d0 + ((x_m ** 2.0d0) * 0.0625d0))
    else
        tmp = 2.6666666666666665d0 * ((t_0 ** 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 t_0 = Math.sin((x_m * 0.5));
	double tmp;
	if (x_m <= 5e-7) {
		tmp = (t_0 / 0.375) * (0.5 + (Math.pow(x_m, 2.0) * 0.0625));
	} else {
		tmp = 2.6666666666666665 * (Math.pow(t_0, 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):
	t_0 = math.sin((x_m * 0.5))
	tmp = 0
	if x_m <= 5e-7:
		tmp = (t_0 / 0.375) * (0.5 + (math.pow(x_m, 2.0) * 0.0625))
	else:
		tmp = 2.6666666666666665 * (math.pow(t_0, 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)
	t_0 = sin(Float64(x_m * 0.5))
	tmp = 0.0
	if (x_m <= 5e-7)
		tmp = Float64(Float64(t_0 / 0.375) * Float64(0.5 + Float64((x_m ^ 2.0) * 0.0625)));
	else
		tmp = Float64(2.6666666666666665 * Float64((t_0 ^ 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)
	t_0 = sin((x_m * 0.5));
	tmp = 0.0;
	if (x_m <= 5e-7)
		tmp = (t_0 / 0.375) * (0.5 + ((x_m ^ 2.0) * 0.0625));
	else
		tmp = 2.6666666666666665 * ((t_0 ^ 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_] := Block[{t$95$0 = N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 5e-7], N[(N[(t$95$0 / 0.375), $MachinePrecision] * N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999977e-7

   1. Initial program 68.3%

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

      1. associate-/l* 99.3%

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

      2. *-commutative 99.3%

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

      3. *-lft-identity 99.3%

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

      4. metadata-eval 99.3%

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

      5. times-frac 99.3%

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

      6. associate-/l* 99.3%

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

      7. *-commutative 99.3%

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

      8. neg-mul-1 99.3%

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

      9. sin-neg 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. times-frac 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/l/ 99.3%

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

      14. associate-/r* 99.3%

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

   3. Simplified 99.3%

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

      1. associate-/r/ 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l/ 99.2%

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

      4. associate-/r/ 99.2%

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

      5. associate-*l/ 68.3%

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

      6. div-inv 68.5%

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

      7. times-frac 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around 0 62.5%

      \[\leadsto \color{blue}{\left(0.5 + 0.0625 \cdot {x}^{2}\right)} \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375} \]
   7. Step-by-step derivation

      1. *-commutative 62.5%

         \[\leadsto \left(0.5 + \color{blue}{{x}^{2} \cdot 0.0625}\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375} \]

   8. Simplified 62.5%

      \[\leadsto \color{blue}{\left(0.5 + {x}^{2} \cdot 0.0625\right)} \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375} \]

   if 4.99999999999999977e-7 < 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* 99.0%

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

      2. *-commutative 99.0%

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

      3. *-lft-identity 99.0%

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

      4. metadata-eval 99.0%

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

      5. times-frac 98.9%

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

      6. associate-/l* 98.9%

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

      7. *-commutative 98.9%

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

      8. neg-mul-1 98.9%

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

      9. sin-neg 98.9%

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

      10. distribute-lft-neg-out 98.9%

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

      11. times-frac 99.0%

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

      12. *-commutative 99.0%

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

      13. associate-/l/ 99.0%

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

      14. associate-/r* 99.0%

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

   3. Simplified 99.0%

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

      1. div-inv 98.9%

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

      2. clear-num 98.9%

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

      3. associate-*r* 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%

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

   5. 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 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.375} \cdot \left(0.5 + {x}^{2} \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]

Alternative 7: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x_m \cdot 0.5\right)\\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{0.375}\\ \mathbf{else}:\\ \;\;\;\;{t_0}^{2} \cdot \frac{2.6666666666666665}{\sin x_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5))))
   (*
    x_s
    (if (<= x_m 5e-9)
      (* 0.5 (/ t_0 0.375))
      (* (pow t_0 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 t_0 = sin((x_m * 0.5));
	double tmp;
	if (x_m <= 5e-9) {
		tmp = 0.5 * (t_0 / 0.375);
	} else {
		tmp = pow(t_0, 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) :: t_0
    real(8) :: tmp
    t_0 = sin((x_m * 0.5d0))
    if (x_m <= 5d-9) then
        tmp = 0.5d0 * (t_0 / 0.375d0)
    else
        tmp = (t_0 ** 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 t_0 = Math.sin((x_m * 0.5));
	double tmp;
	if (x_m <= 5e-9) {
		tmp = 0.5 * (t_0 / 0.375);
	} else {
		tmp = Math.pow(t_0, 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):
	t_0 = math.sin((x_m * 0.5))
	tmp = 0
	if x_m <= 5e-9:
		tmp = 0.5 * (t_0 / 0.375)
	else:
		tmp = math.pow(t_0, 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)
	t_0 = sin(Float64(x_m * 0.5))
	tmp = 0.0
	if (x_m <= 5e-9)
		tmp = Float64(0.5 * Float64(t_0 / 0.375));
	else
		tmp = Float64((t_0 ^ 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)
	t_0 = sin((x_m * 0.5));
	tmp = 0.0;
	if (x_m <= 5e-9)
		tmp = 0.5 * (t_0 / 0.375);
	else
		tmp = (t_0 ^ 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_] := Block[{t$95$0 = N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 5e-9], N[(0.5 * N[(t$95$0 / 0.375), $MachinePrecision]), $MachinePrecision], N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(2.6666666666666665 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-9

   1. Initial program 68.2%

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

      1. associate-/l* 99.3%

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

      2. *-commutative 99.3%

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

      3. *-lft-identity 99.3%

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

      4. metadata-eval 99.3%

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

      5. times-frac 99.3%

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

      6. associate-/l* 99.3%

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

      7. *-commutative 99.3%

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

      8. neg-mul-1 99.3%

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

      9. sin-neg 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. times-frac 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/l/ 99.3%

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

      14. associate-/r* 99.3%

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

   3. Simplified 99.3%

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

      1. associate-/r/ 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l/ 99.2%

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

      4. associate-/r/ 99.2%

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

      5. associate-*l/ 68.2%

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

      6. div-inv 68.3%

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

      7. times-frac 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around 0 65.3%

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

   if 5.0000000000000001e-9 < 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* 99.0%

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

      2. *-commutative 99.0%

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

      3. *-lft-identity 99.0%

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

      4. metadata-eval 99.0%

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

      5. times-frac 98.9%

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

      6. associate-/l* 98.9%

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

      7. *-commutative 98.9%

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

      8. neg-mul-1 98.9%

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

      9. sin-neg 98.9%

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

      10. distribute-lft-neg-out 98.9%

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

      11. times-frac 99.0%

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

      12. *-commutative 99.0%

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

      13. associate-/l/ 99.0%

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

      14. associate-/r* 99.0%

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

   3. Simplified 99.0%

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

      1. associate-/r/ 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l/ 98.8%

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

      4. associate-/r/ 99.0%

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

      5. associate-*l/ 99.1%

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

      6. div-inv 99.0%

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

      7. pow2 99.0%

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

      8. clear-num 99.1%

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

   5. Applied egg-rr 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. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}\\ \end{array} \]

Alternative 8: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x_m \cdot 0.5\right)\\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{t_0}^{2}}{\sin x_m}}{0.375}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5))))
   (*
    x_s
    (if (<= x_m 5e-9)
      (* 0.5 (/ t_0 0.375))
      (/ (/ (pow t_0 2.0) (sin x_m)) 0.375)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((x_m * 0.5));
	double tmp;
	if (x_m <= 5e-9) {
		tmp = 0.5 * (t_0 / 0.375);
	} else {
		tmp = (pow(t_0, 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) :: t_0
    real(8) :: tmp
    t_0 = sin((x_m * 0.5d0))
    if (x_m <= 5d-9) then
        tmp = 0.5d0 * (t_0 / 0.375d0)
    else
        tmp = ((t_0 ** 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 t_0 = Math.sin((x_m * 0.5));
	double tmp;
	if (x_m <= 5e-9) {
		tmp = 0.5 * (t_0 / 0.375);
	} else {
		tmp = (Math.pow(t_0, 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):
	t_0 = math.sin((x_m * 0.5))
	tmp = 0
	if x_m <= 5e-9:
		tmp = 0.5 * (t_0 / 0.375)
	else:
		tmp = (math.pow(t_0, 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)
	t_0 = sin(Float64(x_m * 0.5))
	tmp = 0.0
	if (x_m <= 5e-9)
		tmp = Float64(0.5 * Float64(t_0 / 0.375));
	else
		tmp = Float64(Float64((t_0 ^ 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)
	t_0 = sin((x_m * 0.5));
	tmp = 0.0;
	if (x_m <= 5e-9)
		tmp = 0.5 * (t_0 / 0.375);
	else
		tmp = ((t_0 ^ 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_] := Block[{t$95$0 = N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 5e-9], N[(0.5 * N[(t$95$0 / 0.375), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-9

   1. Initial program 68.2%

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

      1. associate-/l* 99.3%

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

      2. *-commutative 99.3%

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

      3. *-lft-identity 99.3%

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

      4. metadata-eval 99.3%

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

      5. times-frac 99.3%

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

      6. associate-/l* 99.3%

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

      7. *-commutative 99.3%

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

      8. neg-mul-1 99.3%

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

      9. sin-neg 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. times-frac 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/l/ 99.3%

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

      14. associate-/r* 99.3%

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

   3. Simplified 99.3%

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

      1. associate-/r/ 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l/ 99.2%

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

      4. associate-/r/ 99.2%

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

      5. associate-*l/ 68.2%

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

      6. div-inv 68.3%

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

      7. times-frac 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around 0 65.3%

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

   if 5.0000000000000001e-9 < 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. *-commutative 98.9%

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

      2. remove-double-neg 98.9%

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

      3. sin-neg 98.9%

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

      4. distribute-lft-neg-out 98.9%

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

      5. distribute-rgt-neg-in 98.9%

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

      6. associate-*l/ 98.9%

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

      7. *-commutative 98.9%

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

      8. distribute-rgt-neg-in 98.9%

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

      9. distribute-lft-neg-out 98.9%

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

      10. sin-neg 98.9%

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

      11. remove-double-neg 98.9%

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

      12. associate-*l* 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. 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. Step-by-step derivation

      1. *-commutative 98.9%

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

      2. associate-*r/ 99.1%

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

      3. associate-/r/ 99.1%

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

      4. div-inv 99.2%

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

      5. associate-/r* 99.2%

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

      6. pow2 99.2%

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

      7. metadata-eval 99.2%

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

   5. Applied egg-rr 99.2%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}\\ \end{array} \]

Alternative 9: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.0036:\\ \;\;\;\;\frac{\sin \left(x_m \cdot 0.5\right)}{0.375} \cdot \left(0.5 + {x_m}^{2} \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x_m, -1.3333333333333333, 1.3333333333333333\right)}{\sin x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0036)
    (* (/ (sin (* x_m 0.5)) 0.375) (+ 0.5 (* (pow x_m 2.0) 0.0625)))
    (/ (fma (cos x_m) -1.3333333333333333 1.3333333333333333) (sin x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0036) {
		tmp = (sin((x_m * 0.5)) / 0.375) * (0.5 + (pow(x_m, 2.0) * 0.0625));
	} else {
		tmp = fma(cos(x_m), -1.3333333333333333, 1.3333333333333333) / sin(x_m);
	}
	return x_s * tmp;
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0036)
		tmp = Float64(Float64(sin(Float64(x_m * 0.5)) / 0.375) * Float64(0.5 + Float64((x_m ^ 2.0) * 0.0625)));
	else
		tmp = Float64(fma(cos(x_m), -1.3333333333333333, 1.3333333333333333) / sin(x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0036], N[(N[(N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] / 0.375), $MachinePrecision] * N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x$95$m], $MachinePrecision] * -1.3333333333333333 + 1.3333333333333333), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0035999999999999999

   1. Initial program 68.3%

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

      1. associate-/l* 99.3%

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

      2. *-commutative 99.3%

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

      3. *-lft-identity 99.3%

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

      4. metadata-eval 99.3%

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

      5. times-frac 99.3%

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

      6. associate-/l* 99.3%

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

      7. *-commutative 99.3%

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

      8. neg-mul-1 99.3%

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

      9. sin-neg 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. times-frac 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/l/ 99.3%

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

      14. associate-/r* 99.3%

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

   3. Simplified 99.3%

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

      1. associate-/r/ 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l/ 99.2%

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

      4. associate-/r/ 99.2%

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

      5. associate-*l/ 68.3%

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

      6. div-inv 68.5%

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

      7. times-frac 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around 0 62.5%

      \[\leadsto \color{blue}{\left(0.5 + 0.0625 \cdot {x}^{2}\right)} \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375} \]
   7. Step-by-step derivation

      1. *-commutative 62.5%

         \[\leadsto \left(0.5 + \color{blue}{{x}^{2} \cdot 0.0625}\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375} \]

   8. Simplified 62.5%

      \[\leadsto \color{blue}{\left(0.5 + {x}^{2} \cdot 0.0625\right)} \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375} \]

   if 0.0035999999999999999 < 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* 99.0%

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

      2. *-commutative 99.0%

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

      3. *-lft-identity 99.0%

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

      4. metadata-eval 99.0%

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

      5. times-frac 98.9%

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

      6. associate-/l* 98.9%

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

      7. *-commutative 98.9%

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

      8. neg-mul-1 98.9%

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

      9. sin-neg 98.9%

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

      10. distribute-lft-neg-out 98.9%

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

      11. times-frac 99.0%

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

      12. *-commutative 99.0%

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

      13. associate-/l/ 99.0%

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

      14. associate-/r* 99.0%

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

   3. Simplified 99.0%

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

      1. associate-/l* 98.9%

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

      2. metadata-eval 98.9%

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

      3. clear-num 98.9%

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

      4. associate-/r/ 98.9%

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

      5. metadata-eval 98.9%

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

      6. associate-*l* 98.9%

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

      7. pow2 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. unpow2 98.9%

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

      2. sin-mult 98.4%

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

   7. Applied egg-rr 98.4%

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

      1. div-sub 98.4%

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

      2. +-inverses 98.4%

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

      3. cos-0 98.4%

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

      4. metadata-eval 98.4%

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

      5. distribute-lft-out 98.4%

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

      6. metadata-eval 98.4%

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

      7. *-rgt-identity 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in x around inf 98.6%

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

       1. associate-*r/ 98.6%

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

       2. cancel-sign-sub-inv 98.6%

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

       3. metadata-eval 98.6%

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

       4. *-commutative 98.6%

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

       5. +-commutative 98.6%

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

       6. distribute-rgt-in 98.3%

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

       7. metadata-eval 98.3%

          \[\leadsto \frac{\left(\cos x \cdot -0.5\right) \cdot 2.6666666666666665 + \color{blue}{1.3333333333333333}}{\sin x} \]

       8. associate-*l* 98.3%

          \[\leadsto \frac{\color{blue}{\cos x \cdot \left(-0.5 \cdot 2.6666666666666665\right)} + 1.3333333333333333}{\sin x} \]

       9. metadata-eval 98.3%

          \[\leadsto \frac{\cos x \cdot \color{blue}{-1.3333333333333333} + 1.3333333333333333}{\sin x} \]

       10. metadata-eval 98.3%

           \[\leadsto \frac{\cos x \cdot \color{blue}{\left(-1.3333333333333333\right)} + 1.3333333333333333}{\sin x} \]

       11. fma-def 98.7%

           \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x, -1.3333333333333333, 1.3333333333333333\right)}}{\sin x} \]

       12. metadata-eval 98.7%

           \[\leadsto \frac{\mathsf{fma}\left(\cos x, \color{blue}{-1.3333333333333333}, 1.3333333333333333\right)}{\sin x} \]

   12. Simplified 98.7%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0036:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.375} \cdot \left(0.5 + {x}^{2} \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, -1.3333333333333333, 1.3333333333333333\right)}{\sin x}\\ \end{array} \]

Alternative 10: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.0036:\\ \;\;\;\;\frac{\sin \left(x_m \cdot 0.5\right)}{0.375} \cdot \left(0.5 + {x_m}^{2} \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x_m}{0.5 + \cos x_m \cdot -0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0036)
    (* (/ (sin (* x_m 0.5)) 0.375) (+ 0.5 (* (pow x_m 2.0) 0.0625)))
    (/ 2.6666666666666665 (/ (sin x_m) (+ 0.5 (* (cos x_m) -0.5)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0036) {
		tmp = (sin((x_m * 0.5)) / 0.375) * (0.5 + (pow(x_m, 2.0) * 0.0625));
	} else {
		tmp = 2.6666666666666665 / (sin(x_m) / (0.5 + (cos(x_m) * -0.5)));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0036d0) then
        tmp = (sin((x_m * 0.5d0)) / 0.375d0) * (0.5d0 + ((x_m ** 2.0d0) * 0.0625d0))
    else
        tmp = 2.6666666666666665d0 / (sin(x_m) / (0.5d0 + (cos(x_m) * (-0.5d0))))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0036) {
		tmp = (Math.sin((x_m * 0.5)) / 0.375) * (0.5 + (Math.pow(x_m, 2.0) * 0.0625));
	} else {
		tmp = 2.6666666666666665 / (Math.sin(x_m) / (0.5 + (Math.cos(x_m) * -0.5)));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.0036:
		tmp = (math.sin((x_m * 0.5)) / 0.375) * (0.5 + (math.pow(x_m, 2.0) * 0.0625))
	else:
		tmp = 2.6666666666666665 / (math.sin(x_m) / (0.5 + (math.cos(x_m) * -0.5)))
	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 <= 0.0036)
		tmp = Float64(Float64(sin(Float64(x_m * 0.5)) / 0.375) * Float64(0.5 + Float64((x_m ^ 2.0) * 0.0625)));
	else
		tmp = Float64(2.6666666666666665 / Float64(sin(x_m) / Float64(0.5 + Float64(cos(x_m) * -0.5))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.0036)
		tmp = (sin((x_m * 0.5)) / 0.375) * (0.5 + ((x_m ^ 2.0) * 0.0625));
	else
		tmp = 2.6666666666666665 / (sin(x_m) / (0.5 + (cos(x_m) * -0.5)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0036], N[(N[(N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] / 0.375), $MachinePrecision] * N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 / N[(N[Sin[x$95$m], $MachinePrecision] / N[(0.5 + N[(N[Cos[x$95$m], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0035999999999999999

   1. Initial program 68.3%

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

      1. associate-/l* 99.3%

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

      2. *-commutative 99.3%

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

      3. *-lft-identity 99.3%

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

      4. metadata-eval 99.3%

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

      5. times-frac 99.3%

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

      6. associate-/l* 99.3%

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

      7. *-commutative 99.3%

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

      8. neg-mul-1 99.3%

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

      9. sin-neg 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. times-frac 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/l/ 99.3%

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

      14. associate-/r* 99.3%

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

   3. Simplified 99.3%

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

      1. associate-/r/ 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l/ 99.2%

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

      4. associate-/r/ 99.2%

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

      5. associate-*l/ 68.3%

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

      6. div-inv 68.5%

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

      7. times-frac 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around 0 62.5%

      \[\leadsto \color{blue}{\left(0.5 + 0.0625 \cdot {x}^{2}\right)} \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375} \]
   7. Step-by-step derivation

      1. *-commutative 62.5%

         \[\leadsto \left(0.5 + \color{blue}{{x}^{2} \cdot 0.0625}\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375} \]

   8. Simplified 62.5%

      \[\leadsto \color{blue}{\left(0.5 + {x}^{2} \cdot 0.0625\right)} \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375} \]

   if 0.0035999999999999999 < 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* 99.0%

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

      2. *-commutative 99.0%

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

      3. *-lft-identity 99.0%

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

      4. metadata-eval 99.0%

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

      5. times-frac 98.9%

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

      6. associate-/l* 98.9%

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

      7. *-commutative 98.9%

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

      8. neg-mul-1 98.9%

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

      9. sin-neg 98.9%

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

      10. distribute-lft-neg-out 98.9%

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

      11. times-frac 99.0%

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

      12. *-commutative 99.0%

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

      13. associate-/l/ 99.0%

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

      14. associate-/r* 99.0%

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

   3. Simplified 99.0%

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

      1. associate-/l* 98.9%

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

      2. metadata-eval 98.9%

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

      3. clear-num 98.9%

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

      4. associate-/r/ 98.9%

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

      5. metadata-eval 98.9%

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

      6. associate-*l* 98.9%

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

      7. pow2 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. unpow2 98.9%

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

      2. sin-mult 98.4%

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

   7. Applied egg-rr 98.4%

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

      1. div-sub 98.4%

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

      2. +-inverses 98.4%

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

      3. cos-0 98.4%

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

      4. metadata-eval 98.4%

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

      5. distribute-lft-out 98.4%

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

      6. metadata-eval 98.4%

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

      7. *-rgt-identity 98.4%

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

   9. Simplified 98.4%

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

       1. associate-*l/ 98.6%

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

       2. *-un-lft-identity 98.6%

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

       3. associate-/l* 98.8%

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

       4. sub-neg 98.8%

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

       5. div-inv 98.8%

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

       6. metadata-eval 98.8%

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

       7. distribute-rgt-neg-in 98.8%

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

       8. metadata-eval 98.8%

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

   11. Applied egg-rr 98.8%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0036:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.375} \cdot \left(0.5 + {x}^{2} \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{0.5 + \cos x \cdot -0.5}}\\ \end{array} \]

Alternative 11: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.000115:\\ \;\;\;\;0.5 \cdot \frac{\sin \left(x_m \cdot 0.5\right)}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x_m}{\sin x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.000115)
    (* 0.5 (/ (sin (* x_m 0.5)) 0.375))
    (* 2.6666666666666665 (/ (- 0.5 (* 0.5 (cos x_m))) (sin x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.000115) {
		tmp = 0.5 * (sin((x_m * 0.5)) / 0.375);
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x_m))) / sin(x_m));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.000115d0) then
        tmp = 0.5d0 * (sin((x_m * 0.5d0)) / 0.375d0)
    else
        tmp = 2.6666666666666665d0 * ((0.5d0 - (0.5d0 * cos(x_m))) / sin(x_m))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.000115) {
		tmp = 0.5 * (Math.sin((x_m * 0.5)) / 0.375);
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * Math.cos(x_m))) / Math.sin(x_m));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.000115:
		tmp = 0.5 * (math.sin((x_m * 0.5)) / 0.375)
	else:
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * math.cos(x_m))) / math.sin(x_m))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.000115)
		tmp = Float64(0.5 * Float64(sin(Float64(x_m * 0.5)) / 0.375));
	else
		tmp = Float64(2.6666666666666665 * Float64(Float64(0.5 - Float64(0.5 * cos(x_m))) / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.000115)
		tmp = 0.5 * (sin((x_m * 0.5)) / 0.375);
	else
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x_m))) / sin(x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.000115], N[(0.5 * N[(N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] / 0.375), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[(0.5 - N[(0.5 * N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e-4

   1. Initial program 68.3%

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

      1. associate-/l* 99.3%

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

      2. *-commutative 99.3%

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

      3. *-lft-identity 99.3%

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

      4. metadata-eval 99.3%

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

      5. times-frac 99.3%

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

      6. associate-/l* 99.3%

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

      7. *-commutative 99.3%

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

      8. neg-mul-1 99.3%

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

      9. sin-neg 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. times-frac 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/l/ 99.3%

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

      14. associate-/r* 99.3%

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

   3. Simplified 99.3%

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

      1. associate-/r/ 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l/ 99.2%

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

      4. associate-/r/ 99.2%

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

      5. associate-*l/ 68.3%

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

      6. div-inv 68.5%

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

      7. times-frac 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around 0 65.4%

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

   if 1.15e-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* 99.0%

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

      2. *-commutative 99.0%

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

      3. *-lft-identity 99.0%

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

      4. metadata-eval 99.0%

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

      5. times-frac 98.9%

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

      6. associate-/l* 98.9%

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

      7. *-commutative 98.9%

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

      8. neg-mul-1 98.9%

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

      9. sin-neg 98.9%

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

      10. distribute-lft-neg-out 98.9%

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

      11. times-frac 99.0%

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

      12. *-commutative 99.0%

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

      13. associate-/l/ 99.0%

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

      14. associate-/r* 99.0%

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

   3. Simplified 99.0%

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

      1. associate-/l* 98.9%

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

      2. metadata-eval 98.9%

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

      3. clear-num 98.9%

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

      4. associate-/r/ 98.9%

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

      5. metadata-eval 98.9%

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

      6. associate-*l* 98.9%

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

      7. pow2 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. unpow2 98.9%

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

      2. sin-mult 98.4%

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

   7. Applied egg-rr 98.4%

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

      1. div-sub 98.4%

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

      2. +-inverses 98.4%

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

      3. cos-0 98.4%

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

      4. metadata-eval 98.4%

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

      5. distribute-lft-out 98.4%

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

      6. metadata-eval 98.4%

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

      7. *-rgt-identity 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in x around inf 98.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000115:\\ \;\;\;\;0.5 \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}\\ \end{array} \]

Alternative 12: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.000115:\\ \;\;\;\;0.5 \cdot \frac{\sin \left(x_m \cdot 0.5\right)}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x_m}{0.5 + \cos x_m \cdot -0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.000115)
    (* 0.5 (/ (sin (* x_m 0.5)) 0.375))
    (/ 2.6666666666666665 (/ (sin x_m) (+ 0.5 (* (cos x_m) -0.5)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.000115) {
		tmp = 0.5 * (sin((x_m * 0.5)) / 0.375);
	} else {
		tmp = 2.6666666666666665 / (sin(x_m) / (0.5 + (cos(x_m) * -0.5)));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.000115d0) then
        tmp = 0.5d0 * (sin((x_m * 0.5d0)) / 0.375d0)
    else
        tmp = 2.6666666666666665d0 / (sin(x_m) / (0.5d0 + (cos(x_m) * (-0.5d0))))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.000115) {
		tmp = 0.5 * (Math.sin((x_m * 0.5)) / 0.375);
	} else {
		tmp = 2.6666666666666665 / (Math.sin(x_m) / (0.5 + (Math.cos(x_m) * -0.5)));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.000115:
		tmp = 0.5 * (math.sin((x_m * 0.5)) / 0.375)
	else:
		tmp = 2.6666666666666665 / (math.sin(x_m) / (0.5 + (math.cos(x_m) * -0.5)))
	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 <= 0.000115)
		tmp = Float64(0.5 * Float64(sin(Float64(x_m * 0.5)) / 0.375));
	else
		tmp = Float64(2.6666666666666665 / Float64(sin(x_m) / Float64(0.5 + Float64(cos(x_m) * -0.5))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.000115)
		tmp = 0.5 * (sin((x_m * 0.5)) / 0.375);
	else
		tmp = 2.6666666666666665 / (sin(x_m) / (0.5 + (cos(x_m) * -0.5)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.000115], N[(0.5 * N[(N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] / 0.375), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 / N[(N[Sin[x$95$m], $MachinePrecision] / N[(0.5 + N[(N[Cos[x$95$m], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e-4

   1. Initial program 68.3%

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

      1. associate-/l* 99.3%

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

      2. *-commutative 99.3%

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

      3. *-lft-identity 99.3%

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

      4. metadata-eval 99.3%

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

      5. times-frac 99.3%

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

      6. associate-/l* 99.3%

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

      7. *-commutative 99.3%

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

      8. neg-mul-1 99.3%

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

      9. sin-neg 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. times-frac 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/l/ 99.3%

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

      14. associate-/r* 99.3%

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

   3. Simplified 99.3%

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

      1. associate-/r/ 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l/ 99.2%

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

      4. associate-/r/ 99.2%

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

      5. associate-*l/ 68.3%

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

      6. div-inv 68.5%

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

      7. times-frac 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around 0 65.4%

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

   if 1.15e-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* 99.0%

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

      2. *-commutative 99.0%

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

      3. *-lft-identity 99.0%

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

      4. metadata-eval 99.0%

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

      5. times-frac 98.9%

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

      6. associate-/l* 98.9%

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

      7. *-commutative 98.9%

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

      8. neg-mul-1 98.9%

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

      9. sin-neg 98.9%

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

      10. distribute-lft-neg-out 98.9%

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

      11. times-frac 99.0%

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

      12. *-commutative 99.0%

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

      13. associate-/l/ 99.0%

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

      14. associate-/r* 99.0%

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

   3. Simplified 99.0%

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

      1. associate-/l* 98.9%

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

      2. metadata-eval 98.9%

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

      3. clear-num 98.9%

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

      4. associate-/r/ 98.9%

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

      5. metadata-eval 98.9%

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

      6. associate-*l* 98.9%

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

      7. pow2 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. unpow2 98.9%

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

      2. sin-mult 98.4%

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

   7. Applied egg-rr 98.4%

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

      1. div-sub 98.4%

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

      2. +-inverses 98.4%

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

      3. cos-0 98.4%

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

      4. metadata-eval 98.4%

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

      5. distribute-lft-out 98.4%

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

      6. metadata-eval 98.4%

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

      7. *-rgt-identity 98.4%

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

   9. Simplified 98.4%

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

       1. associate-*l/ 98.6%

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

       2. *-un-lft-identity 98.6%

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

       3. associate-/l* 98.8%

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

       4. sub-neg 98.8%

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

       5. div-inv 98.8%

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

       6. metadata-eval 98.8%

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

       7. distribute-rgt-neg-in 98.8%

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

       8. metadata-eval 98.8%

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

   11. Applied egg-rr 98.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000115:\\ \;\;\;\;0.5 \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{0.5 + \cos x \cdot -0.5}}\\ \end{array} \]

Alternative 13: 55.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.2%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. *-lft-identity 99.2%

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

   4. metadata-eval 99.2%

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

   5. times-frac 99.2%

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

   6. associate-/l* 99.2%

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

   7. *-commutative 99.2%

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

   8. neg-mul-1 99.2%

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

   9. sin-neg 99.2%

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

   10. distribute-lft-neg-out 99.2%

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

   11. times-frac 99.2%

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

   12. *-commutative 99.2%

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

   13. associate-/l/ 99.2%

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

   14. associate-/r* 99.2%

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

3. Simplified 99.2%

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

   1. associate-/r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.1%

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

   4. associate-/r/ 99.2%

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

   5. associate-*l/ 75.3%

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

   6. div-inv 75.4%

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

   7. times-frac 99.4%

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

   8. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in x around 0 53.7%

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

7. Final simplification 53.7%

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

Alternative 14: 55.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.2%

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

   1. *-commutative 75.2%

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

   2. remove-double-neg 75.2%

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

   3. sin-neg 75.2%

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

   4. distribute-lft-neg-out 75.2%

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

   5. distribute-rgt-neg-in 75.2%

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

   6. associate-*r/ 99.2%

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

   7. *-commutative 99.2%

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

   8. distribute-lft-neg-in 99.2%

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

   9. distribute-lft-neg-out 99.2%

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

   10. sin-neg 99.2%

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

   11. remove-double-neg 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 53.4%

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

5. Final simplification 53.4%

   \[\leadsto \sin \left(x \cdot 0.5\right) \cdot 1.3333333333333333 \]

Alternative 15: 51.1% accurate, 104.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \left(x_m \cdot 0.6666666666666666\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* x_m 0.6666666666666666)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.2%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. *-lft-identity 99.2%

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

   4. metadata-eval 99.2%

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

   5. times-frac 99.2%

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

   6. associate-/l* 99.2%

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

   7. *-commutative 99.2%

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

   8. neg-mul-1 99.2%

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

   9. sin-neg 99.2%

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

   10. distribute-lft-neg-out 99.2%

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

   11. times-frac 99.2%

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

   12. *-commutative 99.2%

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

   13. associate-/l/ 99.2%

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

   14. associate-/r* 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 49.4%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
5. Step-by-step derivation

   1. *-commutative 49.4%

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

6. Simplified 49.4%

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

7. Final simplification 49.4%

   \[\leadsto x \cdot 0.6666666666666666 \]

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 2023322 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :herbie-target
  (/ (/ (* 8.0 (sin (* x 0.5))) 3.0) (/ (sin x) (sin (* x 0.5))))

  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))