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

Percentage Accurate: 76.9% → 99.5%

Time: 20.4s

Alternatives: 20

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \mathbf{if}\;x \leq -0.001:\\ \;\;\;\;\frac{1}{0.375 \cdot \left(\sin x \cdot {t_0}^{-2}\right)}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \left({t_0}^{2} \cdot \frac{1}{\sin x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))))
   (if (<= x -0.001)
     (/ 1.0 (* 0.375 (* (sin x) (pow t_0 -2.0))))
     (if (<= x 5.2e-7)
       (/ t_0 (+ 0.75 (* -0.09375 (* x x))))
       (* 2.6666666666666665 (* (pow t_0 2.0) (/ 1.0 (sin x))))))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	double tmp;
	if (x <= -0.001) {
		tmp = 1.0 / (0.375 * (sin(x) * pow(t_0, -2.0)));
	} else if (x <= 5.2e-7) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * (pow(t_0, 2.0) * (1.0 / sin(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x * 0.5d0))
    if (x <= (-0.001d0)) then
        tmp = 1.0d0 / (0.375d0 * (sin(x) * (t_0 ** (-2.0d0))))
    else if (x <= 5.2d-7) then
        tmp = t_0 / (0.75d0 + ((-0.09375d0) * (x * x)))
    else
        tmp = 2.6666666666666665d0 * ((t_0 ** 2.0d0) * (1.0d0 / sin(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	double tmp;
	if (x <= -0.001) {
		tmp = 1.0 / (0.375 * (Math.sin(x) * Math.pow(t_0, -2.0)));
	} else if (x <= 5.2e-7) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * (Math.pow(t_0, 2.0) * (1.0 / Math.sin(x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	tmp = 0
	if x <= -0.001:
		tmp = 1.0 / (0.375 * (math.sin(x) * math.pow(t_0, -2.0)))
	elif x <= 5.2e-7:
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)))
	else:
		tmp = 2.6666666666666665 * (math.pow(t_0, 2.0) * (1.0 / math.sin(x)))
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	tmp = 0.0
	if (x <= -0.001)
		tmp = Float64(1.0 / Float64(0.375 * Float64(sin(x) * (t_0 ^ -2.0))));
	elseif (x <= 5.2e-7)
		tmp = Float64(t_0 / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	else
		tmp = Float64(2.6666666666666665 * Float64((t_0 ^ 2.0) * Float64(1.0 / sin(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5));
	tmp = 0.0;
	if (x <= -0.001)
		tmp = 1.0 / (0.375 * (sin(x) * (t_0 ^ -2.0)));
	elseif (x <= 5.2e-7)
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	else
		tmp = 2.6666666666666665 * ((t_0 ^ 2.0) * (1.0 / sin(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -0.001], N[(1.0 / N[(0.375 * N[(N[Sin[x], $MachinePrecision] * N[Power[t$95$0, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e-7], N[(t$95$0 / N[(0.75 + N[(-0.09375 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(1.0 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e-3

   1. Initial program 99.1%

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

      1. associate-*r/ 99.1%

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

      2. *-commutative 99.1%

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

      3. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Applied egg-rr 98.7%

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

      1. add-log-exp 99.1%

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

      2. associate-/r/ 99.0%

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

      3. metadata-eval 99.0%

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

      4. associate-/r* 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-/r/ 99.0%

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

      7. div-inv 99.0%

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

      8. pow-flip 99.1%

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

      9. metadata-eval 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. expm1-log1p-u 82.0%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sin x \cdot 0.375\right) \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)\right)}} \]

      2. expm1-udef 81.3%

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\left(\sin x \cdot 0.375\right) \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)} - 1}} \]

      3. associate-*l* 81.3%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\color{blue}{\sin x \cdot \left(0.375 \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)}\right)} - 1} \]

   9. Applied egg-rr 81.3%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sin x \cdot \left(0.375 \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 81.9%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \left(0.375 \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)\right)\right)}} \]

       2. expm1-log1p 99.1%

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

       3. associate-*r* 99.1%

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

       4. *-commutative 99.1%

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

       5. associate-*l* 99.2%

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

   11. Simplified 99.2%

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

   if -1e-3 < x < 5.19999999999999998e-7

   1. Initial program 55.5%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.6%

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

      3. div-inv 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/l* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   8. Simplified 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}} \]

   if 5.19999999999999998e-7 < x

   1. Initial program 99.2%

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

      1. associate-/l* 99.1%

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

      2. associate-*r/ 99.2%

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

      3. metadata-eval 99.2%

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

      4. remove-double-neg 99.2%

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

      5. sin-neg 99.2%

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

      6. distribute-lft-neg-out 99.2%

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

      7. neg-mul-1 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-/l* 99.2%

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

      10. distribute-lft-neg-out 99.2%

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

      11. distribute-rgt-neg-in 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-/l/ 99.2%

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

      14. neg-mul-1 99.2%

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

      15. sin-neg 99.2%

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

      16. distribute-lft-neg-out 99.2%

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

      17. distribute-lft-neg-out 99.2%

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

      18. distribute-rgt-neg-in 99.2%

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

      19. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. associate-/l* 99.2%

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

      2. div-inv 99.3%

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

      3. sqr-sin-a 98.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 52.3%

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

      6. swap-sqr 52.3%

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

      7. metadata-eval 52.3%

         \[\leadsto 2.6666666666666665 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\left(x \cdot x\right) \cdot \color{blue}{0.25}}\right)\right) \cdot \frac{1}{\sin x}\right) \]

      8. metadata-eval 52.3%

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

      9. swap-sqr 52.3%

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

      10. sqrt-unprod 63.2%

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

      11. add-sqr-sqrt 98.1%

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

      12. sqr-sin-a 99.3%

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

      13. pow2 99.3%

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

   5. Applied egg-rr 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.001:\\ \;\;\;\;\frac{1}{0.375 \cdot \left(\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \left({\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{1}{\sin x}\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t_1 := {t_0}^{2}\\ \mathbf{if}\;x \leq -0.001:\\ \;\;\;\;\frac{\frac{t_1}{0.375}}{\sin x}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \left(t_1 \cdot \frac{1}{\sin x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))) (t_1 (pow t_0 2.0)))
   (if (<= x -0.001)
     (/ (/ t_1 0.375) (sin x))
     (if (<= x 5.2e-7)
       (/ t_0 (+ 0.75 (* -0.09375 (* x x))))
       (* 2.6666666666666665 (* t_1 (/ 1.0 (sin x))))))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	double t_1 = pow(t_0, 2.0);
	double tmp;
	if (x <= -0.001) {
		tmp = (t_1 / 0.375) / sin(x);
	} else if (x <= 5.2e-7) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * (t_1 * (1.0 / sin(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((x * 0.5d0))
    t_1 = t_0 ** 2.0d0
    if (x <= (-0.001d0)) then
        tmp = (t_1 / 0.375d0) / sin(x)
    else if (x <= 5.2d-7) then
        tmp = t_0 / (0.75d0 + ((-0.09375d0) * (x * x)))
    else
        tmp = 2.6666666666666665d0 * (t_1 * (1.0d0 / sin(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	double t_1 = Math.pow(t_0, 2.0);
	double tmp;
	if (x <= -0.001) {
		tmp = (t_1 / 0.375) / Math.sin(x);
	} else if (x <= 5.2e-7) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * (t_1 * (1.0 / Math.sin(x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	t_1 = math.pow(t_0, 2.0)
	tmp = 0
	if x <= -0.001:
		tmp = (t_1 / 0.375) / math.sin(x)
	elif x <= 5.2e-7:
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)))
	else:
		tmp = 2.6666666666666665 * (t_1 * (1.0 / math.sin(x)))
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	t_1 = t_0 ^ 2.0
	tmp = 0.0
	if (x <= -0.001)
		tmp = Float64(Float64(t_1 / 0.375) / sin(x));
	elseif (x <= 5.2e-7)
		tmp = Float64(t_0 / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	else
		tmp = Float64(2.6666666666666665 * Float64(t_1 * Float64(1.0 / sin(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5));
	t_1 = t_0 ^ 2.0;
	tmp = 0.0;
	if (x <= -0.001)
		tmp = (t_1 / 0.375) / sin(x);
	elseif (x <= 5.2e-7)
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	else
		tmp = 2.6666666666666665 * (t_1 * (1.0 / sin(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[LessEqual[x, -0.001], N[(N[(t$95$1 / 0.375), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e-7], N[(t$95$0 / N[(0.75 + N[(-0.09375 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(t$95$1 * N[(1.0 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e-3

   1. Initial program 99.1%

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

      1. associate-*r/ 99.1%

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

      2. *-commutative 99.1%

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

      3. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Applied egg-rr 98.7%

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

      1. add-log-exp 99.1%

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

      2. div-inv 99.0%

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

      3. metadata-eval 99.0%

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

      4. clear-num 99.0%

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

      5. times-frac 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. associate-/r* 99.2%

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

   7. Applied egg-rr 99.2%

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

   if -1e-3 < x < 5.19999999999999998e-7

   1. Initial program 55.5%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.6%

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

      3. div-inv 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/l* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   8. Simplified 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}} \]

   if 5.19999999999999998e-7 < x

   1. Initial program 99.2%

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

      1. associate-/l* 99.1%

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

      2. associate-*r/ 99.2%

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

      3. metadata-eval 99.2%

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

      4. remove-double-neg 99.2%

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

      5. sin-neg 99.2%

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

      6. distribute-lft-neg-out 99.2%

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

      7. neg-mul-1 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-/l* 99.2%

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

      10. distribute-lft-neg-out 99.2%

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

      11. distribute-rgt-neg-in 99.2%

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

      12. metadata-eval 99.2%

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

      13. associate-/l/ 99.2%

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

      14. neg-mul-1 99.2%

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

      15. sin-neg 99.2%

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

      16. distribute-lft-neg-out 99.2%

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

      17. distribute-lft-neg-out 99.2%

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

      18. distribute-rgt-neg-in 99.2%

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

      19. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. associate-/l* 99.2%

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

      2. div-inv 99.3%

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

      3. sqr-sin-a 98.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 52.3%

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

      6. swap-sqr 52.3%

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

      7. metadata-eval 52.3%

         \[\leadsto 2.6666666666666665 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\left(x \cdot x\right) \cdot \color{blue}{0.25}}\right)\right) \cdot \frac{1}{\sin x}\right) \]

      8. metadata-eval 52.3%

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

      9. swap-sqr 52.3%

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

      10. sqrt-unprod 63.2%

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

      11. add-sqr-sqrt 98.1%

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

      12. sqr-sin-a 99.3%

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

      13. pow2 99.3%

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

   5. Applied egg-rr 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.001:\\ \;\;\;\;\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375}}{\sin x}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \left({\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{1}{\sin x}\right)\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \mathbf{if}\;x \leq -0.001 \lor \neg \left(x \leq 5 \cdot 10^{-19}\right):\\ \;\;\;\;2.6666666666666665 \cdot \frac{{t_0}^{2}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))))
   (if (or (<= x -0.001) (not (<= x 5e-19)))
     (* 2.6666666666666665 (/ (pow t_0 2.0) (sin x)))
     (/ t_0 (+ 0.75 (* -0.09375 (* x x)))))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	double tmp;
	if ((x <= -0.001) || !(x <= 5e-19)) {
		tmp = 2.6666666666666665 * (pow(t_0, 2.0) / sin(x));
	} else {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x * 0.5d0))
    if ((x <= (-0.001d0)) .or. (.not. (x <= 5d-19))) then
        tmp = 2.6666666666666665d0 * ((t_0 ** 2.0d0) / sin(x))
    else
        tmp = t_0 / (0.75d0 + ((-0.09375d0) * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	double tmp;
	if ((x <= -0.001) || !(x <= 5e-19)) {
		tmp = 2.6666666666666665 * (Math.pow(t_0, 2.0) / Math.sin(x));
	} else {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	tmp = 0
	if (x <= -0.001) or not (x <= 5e-19):
		tmp = 2.6666666666666665 * (math.pow(t_0, 2.0) / math.sin(x))
	else:
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)))
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	tmp = 0.0
	if ((x <= -0.001) || !(x <= 5e-19))
		tmp = Float64(2.6666666666666665 * Float64((t_0 ^ 2.0) / sin(x)));
	else
		tmp = Float64(t_0 / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5));
	tmp = 0.0;
	if ((x <= -0.001) || ~((x <= 5e-19)))
		tmp = 2.6666666666666665 * ((t_0 ^ 2.0) / sin(x));
	else
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -0.001], N[Not[LessEqual[x, 5e-19]], $MachinePrecision]], N[(2.6666666666666665 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(0.75 + N[(-0.09375 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1e-3 or 5.0000000000000004e-19 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Applied egg-rr 99.2%

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

   if -1e-3 < x < 5.0000000000000004e-19

   1. Initial program 54.7%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.6%

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

      3. div-inv 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/l* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   8. Simplified 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.001 \lor \neg \left(x \leq 5 \cdot 10^{-19}\right):\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t_1 := {t_0}^{2}\\ \mathbf{if}\;x \leq -0.001:\\ \;\;\;\;t_1 \cdot \frac{2.6666666666666665}{\sin x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{t_0}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{t_1}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))) (t_1 (pow t_0 2.0)))
   (if (<= x -0.001)
     (* t_1 (/ 2.6666666666666665 (sin x)))
     (if (<= x 5e-19)
       (/ t_0 (+ 0.75 (* -0.09375 (* x x))))
       (* 2.6666666666666665 (/ t_1 (sin x)))))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	double t_1 = pow(t_0, 2.0);
	double tmp;
	if (x <= -0.001) {
		tmp = t_1 * (2.6666666666666665 / sin(x));
	} else if (x <= 5e-19) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * (t_1 / sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((x * 0.5d0))
    t_1 = t_0 ** 2.0d0
    if (x <= (-0.001d0)) then
        tmp = t_1 * (2.6666666666666665d0 / sin(x))
    else if (x <= 5d-19) then
        tmp = t_0 / (0.75d0 + ((-0.09375d0) * (x * x)))
    else
        tmp = 2.6666666666666665d0 * (t_1 / sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	double t_1 = Math.pow(t_0, 2.0);
	double tmp;
	if (x <= -0.001) {
		tmp = t_1 * (2.6666666666666665 / Math.sin(x));
	} else if (x <= 5e-19) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * (t_1 / Math.sin(x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	t_1 = math.pow(t_0, 2.0)
	tmp = 0
	if x <= -0.001:
		tmp = t_1 * (2.6666666666666665 / math.sin(x))
	elif x <= 5e-19:
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)))
	else:
		tmp = 2.6666666666666665 * (t_1 / math.sin(x))
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	t_1 = t_0 ^ 2.0
	tmp = 0.0
	if (x <= -0.001)
		tmp = Float64(t_1 * Float64(2.6666666666666665 / sin(x)));
	elseif (x <= 5e-19)
		tmp = Float64(t_0 / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	else
		tmp = Float64(2.6666666666666665 * Float64(t_1 / sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5));
	t_1 = t_0 ^ 2.0;
	tmp = 0.0;
	if (x <= -0.001)
		tmp = t_1 * (2.6666666666666665 / sin(x));
	elseif (x <= 5e-19)
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	else
		tmp = 2.6666666666666665 * (t_1 / sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[LessEqual[x, -0.001], N[(t$95$1 * N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-19], N[(t$95$0 / N[(0.75 + N[(-0.09375 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(t$95$1 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e-3

   1. Initial program 99.1%

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

      1. associate-/l* 99.0%

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

      2. metadata-eval 99.0%

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

   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-*r/ 98.9%

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

      4. *-commutative 98.9%

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

      5. associate-*r* 99.0%

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

      6. pow2 99.0%

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

   5. Applied egg-rr 99.0%

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

   if -1e-3 < x < 5.0000000000000004e-19

   1. Initial program 54.7%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.6%

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

      3. div-inv 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/l* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   8. Simplified 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}} \]

   if 5.0000000000000004e-19 < x

   1. Initial program 99.2%

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

      1. associate-/l* 99.1%

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

      2. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Applied egg-rr 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.001:\\ \;\;\;\;{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \mathbf{if}\;x \leq -0.001:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x \cdot {t_0}^{-2}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{t_0}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{t_0}^{2}}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))))
   (if (<= x -0.001)
     (/ 2.6666666666666665 (* (sin x) (pow t_0 -2.0)))
     (if (<= x 5e-19)
       (/ t_0 (+ 0.75 (* -0.09375 (* x x))))
       (* 2.6666666666666665 (/ (pow t_0 2.0) (sin x)))))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	double tmp;
	if (x <= -0.001) {
		tmp = 2.6666666666666665 / (sin(x) * pow(t_0, -2.0));
	} else if (x <= 5e-19) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * (pow(t_0, 2.0) / sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x * 0.5d0))
    if (x <= (-0.001d0)) then
        tmp = 2.6666666666666665d0 / (sin(x) * (t_0 ** (-2.0d0)))
    else if (x <= 5d-19) then
        tmp = t_0 / (0.75d0 + ((-0.09375d0) * (x * x)))
    else
        tmp = 2.6666666666666665d0 * ((t_0 ** 2.0d0) / sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	double tmp;
	if (x <= -0.001) {
		tmp = 2.6666666666666665 / (Math.sin(x) * Math.pow(t_0, -2.0));
	} else if (x <= 5e-19) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * (Math.pow(t_0, 2.0) / Math.sin(x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	tmp = 0
	if x <= -0.001:
		tmp = 2.6666666666666665 / (math.sin(x) * math.pow(t_0, -2.0))
	elif x <= 5e-19:
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)))
	else:
		tmp = 2.6666666666666665 * (math.pow(t_0, 2.0) / math.sin(x))
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	tmp = 0.0
	if (x <= -0.001)
		tmp = Float64(2.6666666666666665 / Float64(sin(x) * (t_0 ^ -2.0)));
	elseif (x <= 5e-19)
		tmp = Float64(t_0 / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	else
		tmp = Float64(2.6666666666666665 * Float64((t_0 ^ 2.0) / sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5));
	tmp = 0.0;
	if (x <= -0.001)
		tmp = 2.6666666666666665 / (sin(x) * (t_0 ^ -2.0));
	elseif (x <= 5e-19)
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	else
		tmp = 2.6666666666666665 * ((t_0 ^ 2.0) / sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -0.001], N[(2.6666666666666665 / N[(N[Sin[x], $MachinePrecision] * N[Power[t$95$0, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-19], N[(t$95$0 / N[(0.75 + N[(-0.09375 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e-3

   1. Initial program 99.1%

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

      1. associate-*r/ 99.1%

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

      2. *-commutative 99.1%

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

      3. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. clear-num 99.1%

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

      3. div-inv 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. *-commutative 99.0%

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

      5. associate-/l* 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in x around inf 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*r/ 99.1%

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

      3. associate-/l* 99.1%

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

      4. unpow2 99.1%

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

      5. associate-/r* 99.1%

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

      6. *-lft-identity 99.1%

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

      7. associate-*l/ 99.1%

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

      8. *-lft-identity 99.1%

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

      9. associate-*l/ 99.0%

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

      10. associate-*l* 98.9%

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

      11. unpow-1 98.9%

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

      12. unpow-1 98.9%

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

      13. pow-sqr 99.2%

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

      14. metadata-eval 99.2%

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

   8. Simplified 99.2%

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

   if -1e-3 < x < 5.0000000000000004e-19

   1. Initial program 54.7%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.6%

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

      3. div-inv 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/l* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   8. Simplified 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}} \]

   if 5.0000000000000004e-19 < x

   1. Initial program 99.2%

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

      1. associate-/l* 99.1%

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

      2. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Applied egg-rr 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.001:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{-2}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]

Alternative 6: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ t_1 := {t_0}^{2}\\ \mathbf{if}\;x \leq -0.001:\\ \;\;\;\;\frac{\frac{t_1}{0.375}}{\sin x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{t_0}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{t_1}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))) (t_1 (pow t_0 2.0)))
   (if (<= x -0.001)
     (/ (/ t_1 0.375) (sin x))
     (if (<= x 5e-19)
       (/ t_0 (+ 0.75 (* -0.09375 (* x x))))
       (* 2.6666666666666665 (/ t_1 (sin x)))))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	double t_1 = pow(t_0, 2.0);
	double tmp;
	if (x <= -0.001) {
		tmp = (t_1 / 0.375) / sin(x);
	} else if (x <= 5e-19) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * (t_1 / sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((x * 0.5d0))
    t_1 = t_0 ** 2.0d0
    if (x <= (-0.001d0)) then
        tmp = (t_1 / 0.375d0) / sin(x)
    else if (x <= 5d-19) then
        tmp = t_0 / (0.75d0 + ((-0.09375d0) * (x * x)))
    else
        tmp = 2.6666666666666665d0 * (t_1 / sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	double t_1 = Math.pow(t_0, 2.0);
	double tmp;
	if (x <= -0.001) {
		tmp = (t_1 / 0.375) / Math.sin(x);
	} else if (x <= 5e-19) {
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	} else {
		tmp = 2.6666666666666665 * (t_1 / Math.sin(x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	t_1 = math.pow(t_0, 2.0)
	tmp = 0
	if x <= -0.001:
		tmp = (t_1 / 0.375) / math.sin(x)
	elif x <= 5e-19:
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)))
	else:
		tmp = 2.6666666666666665 * (t_1 / math.sin(x))
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	t_1 = t_0 ^ 2.0
	tmp = 0.0
	if (x <= -0.001)
		tmp = Float64(Float64(t_1 / 0.375) / sin(x));
	elseif (x <= 5e-19)
		tmp = Float64(t_0 / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	else
		tmp = Float64(2.6666666666666665 * Float64(t_1 / sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5));
	t_1 = t_0 ^ 2.0;
	tmp = 0.0;
	if (x <= -0.001)
		tmp = (t_1 / 0.375) / sin(x);
	elseif (x <= 5e-19)
		tmp = t_0 / (0.75 + (-0.09375 * (x * x)));
	else
		tmp = 2.6666666666666665 * (t_1 / sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[LessEqual[x, -0.001], N[(N[(t$95$1 / 0.375), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-19], N[(t$95$0 / N[(0.75 + N[(-0.09375 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(t$95$1 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e-3

   1. Initial program 99.1%

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

      1. associate-*r/ 99.1%

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

      2. *-commutative 99.1%

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

      3. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Applied egg-rr 98.7%

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

      1. add-log-exp 99.1%

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

      2. div-inv 99.0%

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

      3. metadata-eval 99.0%

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

      4. clear-num 99.0%

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

      5. times-frac 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. associate-/r* 99.2%

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

   7. Applied egg-rr 99.2%

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

   if -1e-3 < x < 5.0000000000000004e-19

   1. Initial program 54.7%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.6%

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

      3. div-inv 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/l* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   8. Simplified 100.0%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}} \]

   if 5.0000000000000004e-19 < x

   1. Initial program 99.2%

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

      1. associate-/l* 99.1%

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

      2. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Applied egg-rr 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.001:\\ \;\;\;\;\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375}}{\sin x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]

Alternative 7: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{t_0}{\frac{\frac{\sin x}{t_0}}{2.6666666666666665}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ t_0 (/ (/ (sin x) t_0) 2.6666666666666665))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return t_0 / ((sin(x) / t_0) / 2.6666666666666665);
}

Fortran

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

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return t_0 / ((Math.sin(x) / t_0) / 2.6666666666666665);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return t_0 / ((math.sin(x) / t_0) / 2.6666666666666665)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(t_0 / Float64(Float64(sin(x) / t_0) / 2.6666666666666665))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = t_0 / ((sin(x) / t_0) / 2.6666666666666665);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 / N[(N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision] / 2.6666666666666665), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.4%

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

   1. associate-*r/ 99.3%

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

   2. *-commutative 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. clear-num 99.3%

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

   3. div-inv 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. *-commutative 99.3%

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

   5. associate-/l* 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 8: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot -0.5\right)\\ 2.6666666666666665 \cdot \left(t_0 \cdot \frac{t_0}{\sin x}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x -0.5))))
   (* 2.6666666666666665 (* t_0 (/ t_0 (sin x))))))

C

double code(double x) {
	double t_0 = sin((x * -0.5));
	return 2.6666666666666665 * (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 = 2.6666666666666665d0 * (t_0 * (t_0 / sin(x)))
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * -0.5));
	return 2.6666666666666665 * (t_0 * (t_0 / Math.sin(x)));
}

Python

def code(x):
	t_0 = math.sin((x * -0.5))
	return 2.6666666666666665 * (t_0 * (t_0 / math.sin(x)))

Julia

function code(x)
	t_0 = sin(Float64(x * -0.5))
	return Float64(2.6666666666666665 * Float64(t_0 * Float64(t_0 / sin(x))))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * -0.5));
	tmp = 2.6666666666666665 * (t_0 * (t_0 / sin(x)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * -0.5), $MachinePrecision]], $MachinePrecision]}, N[(2.6666666666666665 * N[(t$95$0 * N[(t$95$0 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.4%

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

   1. associate-/l* 99.3%

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

   2. associate-*r/ 99.3%

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

   3. metadata-eval 99.3%

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

   4. associate-/l* 80.4%

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

   5. sqr-neg 80.4%

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

   6. sin-neg 80.4%

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

   7. distribute-lft-neg-out 80.4%

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

   8. sin-neg 80.4%

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

   9. distribute-lft-neg-out 80.4%

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

   10. associate-*r/ 99.3%

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

   11. distribute-lft-neg-out 99.3%

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

   12. distribute-rgt-neg-in 99.3%

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

   13. metadata-eval 99.3%

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

   14. distribute-lft-neg-out 99.3%

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

   15. distribute-rgt-neg-in 99.3%

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

   16. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 9: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot -0.5\right)\\ 2.6666666666666665 \cdot \frac{t_0}{\frac{\sin x}{t_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x -0.5))))
   (* 2.6666666666666665 (/ t_0 (/ (sin x) t_0)))))

C

double code(double x) {
	double t_0 = sin((x * -0.5));
	return 2.6666666666666665 * (t_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 = 2.6666666666666665d0 * (t_0 / (sin(x) / t_0))
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * -0.5));
	return 2.6666666666666665 * (t_0 / (Math.sin(x) / t_0));
}

Python

def code(x):
	t_0 = math.sin((x * -0.5))
	return 2.6666666666666665 * (t_0 / (math.sin(x) / t_0))

Julia

function code(x)
	t_0 = sin(Float64(x * -0.5))
	return Float64(2.6666666666666665 * Float64(t_0 / Float64(sin(x) / t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * -0.5));
	tmp = 2.6666666666666665 * (t_0 / (sin(x) / t_0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * -0.5), $MachinePrecision]], $MachinePrecision]}, N[(2.6666666666666665 * N[(t$95$0 / N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.4%

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

   1. associate-/l* 99.3%

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

   2. associate-*r/ 99.3%

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

   3. metadata-eval 99.3%

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

   4. remove-double-neg 99.3%

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

   5. sin-neg 99.3%

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

   6. distribute-lft-neg-out 99.3%

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

   7. neg-mul-1 99.3%

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

   8. *-commutative 99.3%

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

   9. associate-/l* 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. distribute-rgt-neg-in 99.3%

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

   12. metadata-eval 99.3%

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

   13. associate-/l/ 99.3%

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

   14. neg-mul-1 99.3%

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

   15. sin-neg 99.3%

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

   16. distribute-lft-neg-out 99.3%

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

   17. distribute-lft-neg-out 99.3%

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

   18. distribute-rgt-neg-in 99.3%

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

   19. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 10: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{t_0}{\sin x} \cdot \left(t_0 \cdot 2.6666666666666665\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (* (/ t_0 (sin x)) (* t_0 2.6666666666666665))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return (t_0 / sin(x)) * (t_0 * 2.6666666666666665);
}

Fortran

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

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return (t_0 / Math.sin(x)) * (t_0 * 2.6666666666666665);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return (t_0 / math.sin(x)) * (t_0 * 2.6666666666666665)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(t_0 / sin(x)) * Float64(t_0 * 2.6666666666666665))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = (t_0 / sin(x)) * (t_0 * 2.6666666666666665);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 / N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * 2.6666666666666665), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.4%

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

   1. associate-*r/ 99.3%

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

   2. *-commutative 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 11: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0054 \lor \neg \left(x \leq 0.0043\right):\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - \frac{\cos x}{2}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.0054) (not (<= x 0.0043)))
   (* 2.6666666666666665 (/ (- 0.5 (/ (cos x) 2.0)) (sin x)))
   (/ (sin (* x 0.5)) (+ 0.75 (* -0.09375 (* x x))))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.0054) || !(x <= 0.0043)) {
		tmp = 2.6666666666666665 * ((0.5 - (cos(x) / 2.0)) / sin(x));
	} else {
		tmp = sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.0054d0)) .or. (.not. (x <= 0.0043d0))) then
        tmp = 2.6666666666666665d0 * ((0.5d0 - (cos(x) / 2.0d0)) / sin(x))
    else
        tmp = sin((x * 0.5d0)) / (0.75d0 + ((-0.09375d0) * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.0054) || !(x <= 0.0043)) {
		tmp = 2.6666666666666665 * ((0.5 - (Math.cos(x) / 2.0)) / Math.sin(x));
	} else {
		tmp = Math.sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.0054) or not (x <= 0.0043):
		tmp = 2.6666666666666665 * ((0.5 - (math.cos(x) / 2.0)) / math.sin(x))
	else:
		tmp = math.sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.0054) || !(x <= 0.0043))
		tmp = Float64(2.6666666666666665 * Float64(Float64(0.5 - Float64(cos(x) / 2.0)) / sin(x)));
	else
		tmp = Float64(sin(Float64(x * 0.5)) / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.0054) || ~((x <= 0.0043)))
		tmp = 2.6666666666666665 * ((0.5 - (cos(x) / 2.0)) / sin(x));
	else
		tmp = sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.0054], N[Not[LessEqual[x, 0.0043]], $MachinePrecision]], N[(2.6666666666666665 * N[(N[(0.5 - N[(N[Cos[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[(0.75 + N[(-0.09375 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0054000000000000003 or 0.0043 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Applied egg-rr 99.2%

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

      1. unpow2 99.2%

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

      2. sin-mult 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. div-sub 98.6%

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

      2. +-inverses 98.6%

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

      3. cos-0 98.6%

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

      4. metadata-eval 98.6%

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

      5. distribute-lft-out 98.6%

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

      6. metadata-eval 98.6%

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

      7. *-rgt-identity 98.6%

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

   9. Simplified 98.6%

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

   if -0.0054000000000000003 < x < 0.0043

   1. Initial program 56.3%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.5%

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

      3. div-inv 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/l* 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 99.9%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 99.9%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   8. Simplified 99.9%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0054 \lor \neg \left(x \leq 0.0043\right):\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - \frac{\cos x}{2}}{\sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 12: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0054 \lor \neg \left(x \leq 0.0043\right):\\ \;\;\;\;\frac{0.5 - \frac{\cos x}{2}}{\sin x \cdot 0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.0054) (not (<= x 0.0043)))
   (/ (- 0.5 (/ (cos x) 2.0)) (* (sin x) 0.375))
   (/ (sin (* x 0.5)) (+ 0.75 (* -0.09375 (* x x))))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.0054) || !(x <= 0.0043)) {
		tmp = (0.5 - (cos(x) / 2.0)) / (sin(x) * 0.375);
	} else {
		tmp = sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.0054d0)) .or. (.not. (x <= 0.0043d0))) then
        tmp = (0.5d0 - (cos(x) / 2.0d0)) / (sin(x) * 0.375d0)
    else
        tmp = sin((x * 0.5d0)) / (0.75d0 + ((-0.09375d0) * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.0054) || !(x <= 0.0043)) {
		tmp = (0.5 - (Math.cos(x) / 2.0)) / (Math.sin(x) * 0.375);
	} else {
		tmp = Math.sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.0054) or not (x <= 0.0043):
		tmp = (0.5 - (math.cos(x) / 2.0)) / (math.sin(x) * 0.375)
	else:
		tmp = math.sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.0054) || !(x <= 0.0043))
		tmp = Float64(Float64(0.5 - Float64(cos(x) / 2.0)) / Float64(sin(x) * 0.375));
	else
		tmp = Float64(sin(Float64(x * 0.5)) / Float64(0.75 + Float64(-0.09375 * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.0054) || ~((x <= 0.0043)))
		tmp = (0.5 - (cos(x) / 2.0)) / (sin(x) * 0.375);
	else
		tmp = sin((x * 0.5)) / (0.75 + (-0.09375 * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.0054], N[Not[LessEqual[x, 0.0043]], $MachinePrecision]], N[(N[(0.5 - N[(N[Cos[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[(0.75 + N[(-0.09375 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0054000000000000003 or 0.0043 < x

   1. Initial program 99.1%

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

      1. associate-*r/ 99.1%

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

      2. *-commutative 99.1%

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

      3. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*l/ 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*r* 99.1%

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

      4. associate-*r/ 99.1%

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

      5. clear-num 99.1%

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

      6. un-div-inv 99.1%

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

      7. pow2 99.1%

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

      8. div-inv 99.2%

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

      9. 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%

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

      1. unpow2 99.2%

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

      2. sin-mult 98.6%

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

   7. Applied egg-rr 98.7%

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

      1. div-sub 98.6%

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

      2. +-inverses 98.6%

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

      3. cos-0 98.6%

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

      4. metadata-eval 98.6%

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

      5. distribute-lft-out 98.6%

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

      6. metadata-eval 98.6%

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

      7. *-rgt-identity 98.6%

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

   9. Simplified 98.7%

      \[\leadsto \frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{\sin x \cdot 0.375} \]

   if -0.0054000000000000003 < x < 0.0043

   1. Initial program 56.3%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.5%

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

      3. div-inv 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/l* 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 99.9%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 99.9%

         \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   8. Simplified 99.9%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0054 \lor \neg \left(x \leq 0.0043\right):\\ \;\;\;\;\frac{0.5 - \frac{\cos x}{2}}{\sin x \cdot 0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 13: 56.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-18}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.375 \cdot \left(\sin x \cdot \left(0.3333333333333333 + \frac{\frac{4}{x}}{x}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e-18)
   (/ (sin (* x 0.5)) 0.75)
   (/ 1.0 (* 0.375 (* (sin x) (+ 0.3333333333333333 (/ (/ 4.0 x) x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1e-18) {
		tmp = sin((x * 0.5)) / 0.75;
	} else {
		tmp = 1.0 / (0.375 * (sin(x) * (0.3333333333333333 + ((4.0 / x) / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d-18) then
        tmp = sin((x * 0.5d0)) / 0.75d0
    else
        tmp = 1.0d0 / (0.375d0 * (sin(x) * (0.3333333333333333d0 + ((4.0d0 / x) / x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e-18) {
		tmp = Math.sin((x * 0.5)) / 0.75;
	} else {
		tmp = 1.0 / (0.375 * (Math.sin(x) * (0.3333333333333333 + ((4.0 / x) / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e-18:
		tmp = math.sin((x * 0.5)) / 0.75
	else:
		tmp = 1.0 / (0.375 * (math.sin(x) * (0.3333333333333333 + ((4.0 / x) / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e-18)
		tmp = Float64(sin(Float64(x * 0.5)) / 0.75);
	else
		tmp = Float64(1.0 / Float64(0.375 * Float64(sin(x) * Float64(0.3333333333333333 + Float64(Float64(4.0 / x) / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e-18)
		tmp = sin((x * 0.5)) / 0.75;
	else
		tmp = 1.0 / (0.375 * (sin(x) * (0.3333333333333333 + ((4.0 / x) / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e-18], N[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision], N[(1.0 / N[(0.375 * N[(N[Sin[x], $MachinePrecision] * N[(0.3333333333333333 + N[(N[(4.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0000000000000001e-18

   1. Initial program 71.4%

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

      1. associate-*r/ 99.4%

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

      2. *-commutative 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. clear-num 99.4%

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

      3. div-inv 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-/l* 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around 0 68.1%

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

   if 1.0000000000000001e-18 < x

   1. Initial program 99.2%

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

      1. associate-*r/ 99.2%

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

      2. *-commutative 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Applied egg-rr 98.1%

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

      1. add-log-exp 99.1%

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

      2. associate-/r/ 99.2%

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

      3. metadata-eval 99.2%

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

      4. associate-/r* 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-/r/ 98.8%

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

      7. div-inv 99.1%

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

      8. pow-flip 99.0%

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

      9. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. expm1-log1p-u 82.3%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sin x \cdot 0.375\right) \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)\right)}} \]

      2. expm1-udef 81.9%

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\left(\sin x \cdot 0.375\right) \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)} - 1}} \]

      3. associate-*l* 81.9%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\color{blue}{\sin x \cdot \left(0.375 \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)}\right)} - 1} \]

   9. Applied egg-rr 81.9%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sin x \cdot \left(0.375 \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 82.2%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \left(0.375 \cdot {\sin \left(x \cdot 0.5\right)}^{-2}\right)\right)\right)}} \]

       2. expm1-log1p 99.1%

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

       3. associate-*r* 99.0%

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

       4. *-commutative 99.0%

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

       5. associate-*l* 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in x around 0 18.4%

       \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot \color{blue}{\left(0.3333333333333333 + 4 \cdot \frac{1}{{x}^{2}}\right)}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 18.4%

          \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot \left(0.3333333333333333 + \color{blue}{\frac{4 \cdot 1}{{x}^{2}}}\right)\right)} \]

       2. metadata-eval 18.4%

          \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot \left(0.3333333333333333 + \frac{\color{blue}{4}}{{x}^{2}}\right)\right)} \]

       3. unpow2 18.4%

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

       4. associate-/r* 18.4%

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

   14. Simplified 18.4%

       \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot \color{blue}{\left(0.3333333333333333 + \frac{\frac{4}{x}}{x}\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-18}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.375 \cdot \left(\sin x \cdot \left(0.3333333333333333 + \frac{\frac{4}{x}}{x}\right)\right)}\\ \end{array} \]

Alternative 14: 56.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-18}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\sin x \cdot 0.375\right) \cdot \left(0.3333333333333333 + \frac{\frac{4}{x}}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e-18)
   (/ (sin (* x 0.5)) 0.75)
   (/ 1.0 (* (* (sin x) 0.375) (+ 0.3333333333333333 (/ (/ 4.0 x) x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1e-18) {
		tmp = sin((x * 0.5)) / 0.75;
	} else {
		tmp = 1.0 / ((sin(x) * 0.375) * (0.3333333333333333 + ((4.0 / x) / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d-18) then
        tmp = sin((x * 0.5d0)) / 0.75d0
    else
        tmp = 1.0d0 / ((sin(x) * 0.375d0) * (0.3333333333333333d0 + ((4.0d0 / x) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e-18) {
		tmp = Math.sin((x * 0.5)) / 0.75;
	} else {
		tmp = 1.0 / ((Math.sin(x) * 0.375) * (0.3333333333333333 + ((4.0 / x) / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e-18:
		tmp = math.sin((x * 0.5)) / 0.75
	else:
		tmp = 1.0 / ((math.sin(x) * 0.375) * (0.3333333333333333 + ((4.0 / x) / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e-18)
		tmp = Float64(sin(Float64(x * 0.5)) / 0.75);
	else
		tmp = Float64(1.0 / Float64(Float64(sin(x) * 0.375) * Float64(0.3333333333333333 + Float64(Float64(4.0 / x) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e-18)
		tmp = sin((x * 0.5)) / 0.75;
	else
		tmp = 1.0 / ((sin(x) * 0.375) * (0.3333333333333333 + ((4.0 / x) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e-18], N[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision], N[(1.0 / N[(N[(N[Sin[x], $MachinePrecision] * 0.375), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(4.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0000000000000001e-18

   1. Initial program 71.4%

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

      1. associate-*r/ 99.4%

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

      2. *-commutative 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. clear-num 99.4%

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

      3. div-inv 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-/l* 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around 0 68.1%

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

   if 1.0000000000000001e-18 < x

   1. Initial program 99.2%

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

      1. associate-*r/ 99.2%

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

      2. *-commutative 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Applied egg-rr 98.1%

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

      1. add-log-exp 99.1%

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

      2. associate-/r/ 99.2%

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

      3. metadata-eval 99.2%

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

      4. associate-/r* 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-/r/ 98.8%

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

      7. div-inv 99.1%

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

      8. pow-flip 99.0%

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

      9. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in x around 0 18.4%

      \[\leadsto \frac{1}{\left(\sin x \cdot 0.375\right) \cdot \color{blue}{\left(0.3333333333333333 + 4 \cdot \frac{1}{{x}^{2}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 18.4%

         \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot \left(0.3333333333333333 + \color{blue}{\frac{4 \cdot 1}{{x}^{2}}}\right)\right)} \]

      2. metadata-eval 18.4%

         \[\leadsto \frac{1}{0.375 \cdot \left(\sin x \cdot \left(0.3333333333333333 + \frac{\color{blue}{4}}{{x}^{2}}\right)\right)} \]

      3. unpow2 18.4%

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

      4. associate-/r* 18.4%

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

   10. Simplified 18.5%

       \[\leadsto \frac{1}{\left(\sin x \cdot 0.375\right) \cdot \color{blue}{\left(0.3333333333333333 + \frac{\frac{4}{x}}{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-18}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\sin x \cdot 0.375\right) \cdot \left(0.3333333333333333 + \frac{\frac{4}{x}}{x}\right)}\\ \end{array} \]

Alternative 15: 52.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\frac{1}{x \cdot -0.125 + 1.5 \cdot \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x \cdot 0.6666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.1)
   (/ 1.0 (+ (* x -0.125) (* 1.5 (/ 1.0 x))))
   (log (+ 1.0 (* x 0.6666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
	} else {
		tmp = log((1.0 + (x * 0.6666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.1d0) then
        tmp = 1.0d0 / ((x * (-0.125d0)) + (1.5d0 * (1.0d0 / x)))
    else
        tmp = log((1.0d0 + (x * 0.6666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
	} else {
		tmp = Math.log((1.0 + (x * 0.6666666666666666)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.1:
		tmp = 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)))
	else:
		tmp = math.log((1.0 + (x * 0.6666666666666666)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.1)
		tmp = Float64(1.0 / Float64(Float64(x * -0.125) + Float64(1.5 * Float64(1.0 / x))));
	else
		tmp = log(Float64(1.0 + Float64(x * 0.6666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.1)
		tmp = 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
	else
		tmp = log((1.0 + (x * 0.6666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.1], N[(1.0 / N[(N[(x * -0.125), $MachinePrecision] + N[(1.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(1.0 + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000009

   1. Initial program 71.7%

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

      1. associate-*r/ 99.4%

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

      2. *-commutative 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Applied egg-rr 41.1%

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

      1. add-log-exp 71.6%

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

      2. associate-/r/ 71.7%

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

      3. metadata-eval 71.7%

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

      4. associate-/r* 71.7%

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

      5. *-commutative 71.7%

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

      6. associate-/r/ 71.7%

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

      7. div-inv 71.2%

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

      8. pow-flip 71.2%

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

      9. metadata-eval 71.2%

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

   7. Applied egg-rr 71.2%

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

   8. Taylor expanded in x around 0 65.2%

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

   if 3.10000000000000009 < x

   1. Initial program 99.2%

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

      1. associate-*r/ 99.2%

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

      2. *-commutative 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in x around 0 8.7%

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

      1. +-commutative 8.7%

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

      2. *-commutative 8.7%

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

   8. Simplified 8.7%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\frac{1}{x \cdot -0.125 + 1.5 \cdot \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x \cdot 0.6666666666666666\right)\\ \end{array} \]

Alternative 16: 55.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(x \cdot 0.5\right) \cdot 1.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sin (* x 0.5)) 1.3333333333333333))

C

double code(double x) {
	return sin((x * 0.5)) * 1.3333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sin((x * 0.5d0)) * 1.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.sin((x * 0.5)) * 1.3333333333333333;
}

Python

def code(x):
	return math.sin((x * 0.5)) * 1.3333333333333333

Julia

function code(x)
	return Float64(sin(Float64(x * 0.5)) * 1.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = sin((x * 0.5)) * 1.3333333333333333;
end

Wolfram

code[x_] := N[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * 1.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 80.4%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 50.0%

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

5. Final simplification 50.0%

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

Alternative 17: 55.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(x \cdot 0.5\right)}{0.75} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (sin (* x 0.5)) 0.75))

C

double code(double x) {
	return sin((x * 0.5)) / 0.75;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sin((x * 0.5d0)) / 0.75d0
end function

Java

public static double code(double x) {
	return Math.sin((x * 0.5)) / 0.75;
}

Python

def code(x):
	return math.sin((x * 0.5)) / 0.75

Julia

function code(x)
	return Float64(sin(Float64(x * 0.5)) / 0.75)
end

MATLAB

function tmp = code(x)
	tmp = sin((x * 0.5)) / 0.75;
end

Wolfram

code[x_] := N[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision]

Derivation

1. Initial program 80.4%

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

   1. associate-*r/ 99.3%

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

   2. *-commutative 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. clear-num 99.3%

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

   3. div-inv 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. *-commutative 99.3%

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

   5. associate-/l* 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in x around 0 50.2%

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

7. Final simplification 50.2%

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

Alternative 18: 51.5% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot -0.125 + 1.5 \cdot \frac{1}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ (* x -0.125) (* 1.5 (/ 1.0 x)))))

C

double code(double x) {
	return 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((x * (-0.125d0)) + (1.5d0 * (1.0d0 / x)))
end function

Java

public static double code(double x) {
	return 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
}

Python

def code(x):
	return 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(x * -0.125) + Float64(1.5 * Float64(1.0 / x))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((x * -0.125) + (1.5 * (1.0 / x)));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(x * -0.125), $MachinePrecision] + N[(1.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.4%

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

   1. associate-*r/ 99.3%

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

   2. *-commutative 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

5. Applied egg-rr 59.4%

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

   1. add-log-exp 80.3%

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

   2. associate-/r/ 80.4%

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

   3. metadata-eval 80.4%

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

   4. associate-/r* 80.4%

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

   5. *-commutative 80.4%

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

   6. associate-/r/ 80.3%

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

   7. div-inv 80.0%

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

   8. pow-flip 80.0%

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

   9. metadata-eval 80.0%

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

7. Applied egg-rr 80.0%

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

8. Taylor expanded in x around 0 45.7%

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

9. Final simplification 45.7%

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

Alternative 19: 51.1% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1.5}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ 1.5 x)))

C

double code(double x) {
	return 1.0 / (1.5 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.5d0 / x)
end function

Java

public static double code(double x) {
	return 1.0 / (1.5 / x);
}

Python

def code(x):
	return 1.0 / (1.5 / x)

Julia

function code(x)
	return Float64(1.0 / Float64(1.5 / x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.5 / x);
end

Wolfram

code[x_] := N[(1.0 / N[(1.5 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.4%

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

   1. associate-*r/ 99.3%

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

   2. *-commutative 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

5. Applied egg-rr 59.4%

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

   1. add-log-exp 80.3%

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

   2. associate-/r/ 80.4%

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

   3. metadata-eval 80.4%

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

   4. associate-/r* 80.4%

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

   5. *-commutative 80.4%

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

   6. associate-/r/ 80.3%

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

   7. div-inv 80.0%

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

   8. pow-flip 80.0%

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

   9. metadata-eval 80.0%

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

7. Applied egg-rr 80.0%

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

8. Taylor expanded in x around 0 45.3%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]

9. Final simplification 45.3%

   \[\leadsto \frac{1}{\frac{1.5}{x}} \]

Alternative 20: 51.0% accurate, 104.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.6666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.6666666666666666))

C

double code(double x) {
	return x * 0.6666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.6666666666666666d0
end function

Java

public static double code(double x) {
	return x * 0.6666666666666666;
}

Python

def code(x):
	return x * 0.6666666666666666

Julia

function code(x)
	return Float64(x * 0.6666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.6666666666666666;
end

Wolfram

code[x_] := N[(x * 0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 80.4%

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

   1. associate-*r/ 99.3%

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

   2. *-commutative 99.3%

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

   3. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in x around 0 45.2%

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

   1. *-commutative 45.2%

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

6. Simplified 45.2%

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

7. Final simplification 45.2%

   \[\leadsto x \cdot 0.6666666666666666 \]

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 2023297 
(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)))