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

Percentage Accurate: 77.4% → 99.6%

Time: 15.8s

Alternatives: 11

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x\_m \cdot -0.5\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0001:\\ \;\;\;\;\frac{t\_0}{0.09375 \cdot {x\_m}^{2} - 0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{t\_0}^{2}}{0.375}}{\sin x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m -0.5))))
   (*
    x_s
    (if (<= x_m 0.0001)
      (/ t_0 (- (* 0.09375 (pow x_m 2.0)) 0.75))
      (/ (/ (pow t_0 2.0) 0.375) (sin x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((x_m * -0.5));
	double tmp;
	if (x_m <= 0.0001) {
		tmp = t_0 / ((0.09375 * pow(x_m, 2.0)) - 0.75);
	} else {
		tmp = (pow(t_0, 2.0) / 0.375) / sin(x_m);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x_m * (-0.5d0)))
    if (x_m <= 0.0001d0) then
        tmp = t_0 / ((0.09375d0 * (x_m ** 2.0d0)) - 0.75d0)
    else
        tmp = ((t_0 ** 2.0d0) / 0.375d0) / sin(x_m)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin((x_m * -0.5));
	double tmp;
	if (x_m <= 0.0001) {
		tmp = t_0 / ((0.09375 * Math.pow(x_m, 2.0)) - 0.75);
	} else {
		tmp = (Math.pow(t_0, 2.0) / 0.375) / Math.sin(x_m);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin((x_m * -0.5))
	tmp = 0
	if x_m <= 0.0001:
		tmp = t_0 / ((0.09375 * math.pow(x_m, 2.0)) - 0.75)
	else:
		tmp = (math.pow(t_0, 2.0) / 0.375) / math.sin(x_m)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = sin(Float64(x_m * -0.5))
	tmp = 0.0
	if (x_m <= 0.0001)
		tmp = Float64(t_0 / Float64(Float64(0.09375 * (x_m ^ 2.0)) - 0.75));
	else
		tmp = Float64(Float64((t_0 ^ 2.0) / 0.375) / sin(x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = sin((x_m * -0.5));
	tmp = 0.0;
	if (x_m <= 0.0001)
		tmp = t_0 / ((0.09375 * (x_m ^ 2.0)) - 0.75);
	else
		tmp = ((t_0 ^ 2.0) / 0.375) / sin(x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Sin[N[(x$95$m * -0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 0.0001], N[(t$95$0 / N[(N[(0.09375 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.75), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] / 0.375), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000005e-4

   1. Initial program 66.9%

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

      1. associate-*l* 66.8%

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

      2. associate-/l* 66.9%

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

      3. sqr-neg 66.9%

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

      4. sin-neg 66.9%

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

      5. distribute-lft-neg-out 66.9%

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

      6. sin-neg 66.9%

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

      7. distribute-lft-neg-out 66.9%

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

      8. metadata-eval 66.9%

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

      9. associate-/l* 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. distribute-rgt-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*r* 99.3%

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

      3. associate-/l* 99.3%

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

      4. *-commutative 99.3%

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

      5. clear-num 99.1%

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

      6. un-div-inv 99.4%

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

      7. associate-/r* 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 69.8%

      \[\leadsto \frac{\sin \left(x \cdot -0.5\right)}{\color{blue}{0.09375 \cdot {x}^{2} - 0.75}} \]

   if 1.00000000000000005e-4 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. associate-/l* 98.9%

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

      3. sqr-neg 98.9%

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

      4. sin-neg 98.9%

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

      5. distribute-lft-neg-out 98.9%

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

      6. sin-neg 98.9%

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

      7. distribute-lft-neg-out 98.9%

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

      8. metadata-eval 98.9%

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

      9. associate-/l* 99.0%

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

      10. distribute-lft-neg-out 99.0%

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

      11. distribute-rgt-neg-in 99.0%

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

      12. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*r* 99.0%

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

      3. associate-/l* 98.9%

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

      4. *-commutative 98.9%

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

      5. clear-num 98.9%

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

      6. un-div-inv 99.1%

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

      7. associate-/r* 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in x around inf 99.0%

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

      1. *-commutative 99.0%

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

   9. Simplified 99.0%

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

       1. clear-num 98.9%

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

       2. associate-/r/ 99.0%

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

   11. Applied egg-rr 99.0%

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

       1. *-un-lft-identity 99.0%

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

       2. associate-*r* 99.0%

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

       3. times-frac 98.9%

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

       4. un-div-inv 98.9%

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

       5. clear-num 99.0%

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

       6. times-frac 99.0%

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

       7. unpow2 99.0%

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

       8. *-un-lft-identity 99.0%

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

       9. *-commutative 99.0%

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

       10. times-frac 99.1%

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

   13. Applied egg-rr 99.1%

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

       1. associate-*l/ 99.1%

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

       2. *-lft-identity 99.1%

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

   15. Simplified 99.1%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0001:\\ \;\;\;\;\frac{\sin \left(x \cdot -0.5\right)}{0.09375 \cdot {x}^{2} - 0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\sin \left(x \cdot -0.5\right)}^{2}}{0.375}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x\_m \cdot -0.5\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0005:\\ \;\;\;\;\frac{t\_0}{0.09375 \cdot {x\_m}^{2} - 0.75}\\ \mathbf{else}:\\ \;\;\;\;{t\_0}^{2} \cdot \frac{2.6666666666666665}{\sin x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m -0.5))))
   (*
    x_s
    (if (<= x_m 0.0005)
      (/ t_0 (- (* 0.09375 (pow x_m 2.0)) 0.75))
      (* (pow t_0 2.0) (/ 2.6666666666666665 (sin x_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((x_m * -0.5));
	double tmp;
	if (x_m <= 0.0005) {
		tmp = t_0 / ((0.09375 * pow(x_m, 2.0)) - 0.75);
	} else {
		tmp = pow(t_0, 2.0) * (2.6666666666666665 / sin(x_m));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x_m * (-0.5d0)))
    if (x_m <= 0.0005d0) then
        tmp = t_0 / ((0.09375d0 * (x_m ** 2.0d0)) - 0.75d0)
    else
        tmp = (t_0 ** 2.0d0) * (2.6666666666666665d0 / sin(x_m))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin((x_m * -0.5));
	double tmp;
	if (x_m <= 0.0005) {
		tmp = t_0 / ((0.09375 * Math.pow(x_m, 2.0)) - 0.75);
	} else {
		tmp = Math.pow(t_0, 2.0) * (2.6666666666666665 / Math.sin(x_m));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin((x_m * -0.5))
	tmp = 0
	if x_m <= 0.0005:
		tmp = t_0 / ((0.09375 * math.pow(x_m, 2.0)) - 0.75)
	else:
		tmp = math.pow(t_0, 2.0) * (2.6666666666666665 / math.sin(x_m))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = sin(Float64(x_m * -0.5))
	tmp = 0.0
	if (x_m <= 0.0005)
		tmp = Float64(t_0 / Float64(Float64(0.09375 * (x_m ^ 2.0)) - 0.75));
	else
		tmp = Float64((t_0 ^ 2.0) * Float64(2.6666666666666665 / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = sin((x_m * -0.5));
	tmp = 0.0;
	if (x_m <= 0.0005)
		tmp = t_0 / ((0.09375 * (x_m ^ 2.0)) - 0.75);
	else
		tmp = (t_0 ^ 2.0) * (2.6666666666666665 / sin(x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Sin[N[(x$95$m * -0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 0.0005], N[(t$95$0 / N[(N[(0.09375 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.75), $MachinePrecision]), $MachinePrecision], N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(2.6666666666666665 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-4

   1. Initial program 67.0%

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

      1. associate-*l* 67.0%

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

      2. associate-/l* 67.1%

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

      3. sqr-neg 67.1%

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

      4. sin-neg 67.1%

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

      5. distribute-lft-neg-out 67.1%

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

      6. sin-neg 67.1%

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

      7. distribute-lft-neg-out 67.1%

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

      8. metadata-eval 67.1%

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

      9. associate-/l* 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. distribute-rgt-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*r* 99.3%

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

      3. associate-/l* 99.3%

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

      4. *-commutative 99.3%

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

      5. clear-num 99.1%

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

      6. un-div-inv 99.4%

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

      7. associate-/r* 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 70.0%

      \[\leadsto \frac{\sin \left(x \cdot -0.5\right)}{\color{blue}{0.09375 \cdot {x}^{2} - 0.75}} \]

   if 5.0000000000000001e-4 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. associate-/l* 98.9%

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

      3. sqr-neg 98.9%

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

      4. sin-neg 98.9%

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

      5. distribute-lft-neg-out 98.9%

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

      6. sin-neg 98.9%

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

      7. distribute-lft-neg-out 98.9%

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

      8. metadata-eval 98.9%

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

      9. associate-/l* 99.0%

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

      10. distribute-lft-neg-out 99.0%

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

      11. distribute-rgt-neg-in 99.0%

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

      12. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*r/ 98.9%

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

      3. associate-*l/ 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;\frac{\sin \left(x \cdot -0.5\right)}{0.09375 \cdot {x}^{2} - 0.75}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x \cdot -0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x\_m \cdot -0.5\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0005:\\ \;\;\;\;\frac{t\_0}{0.09375 \cdot {x\_m}^{2} - 0.75}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{t\_0}^{2}}{\sin x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m -0.5))))
   (*
    x_s
    (if (<= x_m 0.0005)
      (/ t_0 (- (* 0.09375 (pow x_m 2.0)) 0.75))
      (* 2.6666666666666665 (/ (pow t_0 2.0) (sin x_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((x_m * -0.5));
	double tmp;
	if (x_m <= 0.0005) {
		tmp = t_0 / ((0.09375 * pow(x_m, 2.0)) - 0.75);
	} else {
		tmp = 2.6666666666666665 * (pow(t_0, 2.0) / sin(x_m));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x_m * (-0.5d0)))
    if (x_m <= 0.0005d0) then
        tmp = t_0 / ((0.09375d0 * (x_m ** 2.0d0)) - 0.75d0)
    else
        tmp = 2.6666666666666665d0 * ((t_0 ** 2.0d0) / sin(x_m))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin((x_m * -0.5));
	double tmp;
	if (x_m <= 0.0005) {
		tmp = t_0 / ((0.09375 * Math.pow(x_m, 2.0)) - 0.75);
	} else {
		tmp = 2.6666666666666665 * (Math.pow(t_0, 2.0) / Math.sin(x_m));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin((x_m * -0.5))
	tmp = 0
	if x_m <= 0.0005:
		tmp = t_0 / ((0.09375 * math.pow(x_m, 2.0)) - 0.75)
	else:
		tmp = 2.6666666666666665 * (math.pow(t_0, 2.0) / math.sin(x_m))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = sin(Float64(x_m * -0.5))
	tmp = 0.0
	if (x_m <= 0.0005)
		tmp = Float64(t_0 / Float64(Float64(0.09375 * (x_m ^ 2.0)) - 0.75));
	else
		tmp = Float64(2.6666666666666665 * Float64((t_0 ^ 2.0) / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = sin((x_m * -0.5));
	tmp = 0.0;
	if (x_m <= 0.0005)
		tmp = t_0 / ((0.09375 * (x_m ^ 2.0)) - 0.75);
	else
		tmp = 2.6666666666666665 * ((t_0 ^ 2.0) / sin(x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Sin[N[(x$95$m * -0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 0.0005], N[(t$95$0 / N[(N[(0.09375 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.75), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-4

   1. Initial program 67.0%

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

      1. associate-*l* 67.0%

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

      2. associate-/l* 67.1%

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

      3. sqr-neg 67.1%

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

      4. sin-neg 67.1%

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

      5. distribute-lft-neg-out 67.1%

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

      6. sin-neg 67.1%

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

      7. distribute-lft-neg-out 67.1%

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

      8. metadata-eval 67.1%

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

      9. associate-/l* 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. distribute-rgt-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*r* 99.3%

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

      3. associate-/l* 99.3%

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

      4. *-commutative 99.3%

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

      5. clear-num 99.1%

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

      6. un-div-inv 99.4%

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

      7. associate-/r* 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 70.0%

      \[\leadsto \frac{\sin \left(x \cdot -0.5\right)}{\color{blue}{0.09375 \cdot {x}^{2} - 0.75}} \]

   if 5.0000000000000001e-4 < x

   1. Initial program 98.9%

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

      1. metadata-eval 98.9%

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

      2. associate-*r/ 98.9%

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

      3. associate-*l* 99.0%

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

      4. *-commutative 99.0%

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

   4. Applied egg-rr 98.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;\frac{\sin \left(x \cdot -0.5\right)}{0.09375 \cdot {x}^{2} - 0.75}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot -0.5\right)}^{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x\_m \cdot -0.5\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0001:\\ \;\;\;\;\frac{t\_0}{0.09375 \cdot {x\_m}^{2} - 0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x\_m}{{t\_0}^{2}}}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m -0.5))))
   (*
    x_s
    (if (<= x_m 0.0001)
      (/ t_0 (- (* 0.09375 (pow x_m 2.0)) 0.75))
      (/ 2.6666666666666665 (/ (sin x_m) (pow t_0 2.0)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((x_m * -0.5));
	double tmp;
	if (x_m <= 0.0001) {
		tmp = t_0 / ((0.09375 * pow(x_m, 2.0)) - 0.75);
	} else {
		tmp = 2.6666666666666665 / (sin(x_m) / pow(t_0, 2.0));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x_m * (-0.5d0)))
    if (x_m <= 0.0001d0) then
        tmp = t_0 / ((0.09375d0 * (x_m ** 2.0d0)) - 0.75d0)
    else
        tmp = 2.6666666666666665d0 / (sin(x_m) / (t_0 ** 2.0d0))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin((x_m * -0.5));
	double tmp;
	if (x_m <= 0.0001) {
		tmp = t_0 / ((0.09375 * Math.pow(x_m, 2.0)) - 0.75);
	} else {
		tmp = 2.6666666666666665 / (Math.sin(x_m) / Math.pow(t_0, 2.0));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin((x_m * -0.5))
	tmp = 0
	if x_m <= 0.0001:
		tmp = t_0 / ((0.09375 * math.pow(x_m, 2.0)) - 0.75)
	else:
		tmp = 2.6666666666666665 / (math.sin(x_m) / math.pow(t_0, 2.0))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = sin(Float64(x_m * -0.5))
	tmp = 0.0
	if (x_m <= 0.0001)
		tmp = Float64(t_0 / Float64(Float64(0.09375 * (x_m ^ 2.0)) - 0.75));
	else
		tmp = Float64(2.6666666666666665 / Float64(sin(x_m) / (t_0 ^ 2.0)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = sin((x_m * -0.5));
	tmp = 0.0;
	if (x_m <= 0.0001)
		tmp = t_0 / ((0.09375 * (x_m ^ 2.0)) - 0.75);
	else
		tmp = 2.6666666666666665 / (sin(x_m) / (t_0 ^ 2.0));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Sin[N[(x$95$m * -0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 0.0001], N[(t$95$0 / N[(N[(0.09375 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.75), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 / N[(N[Sin[x$95$m], $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000005e-4

   1. Initial program 66.9%

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

      1. associate-*l* 66.8%

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

      2. associate-/l* 66.9%

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

      3. sqr-neg 66.9%

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

      4. sin-neg 66.9%

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

      5. distribute-lft-neg-out 66.9%

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

      6. sin-neg 66.9%

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

      7. distribute-lft-neg-out 66.9%

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

      8. metadata-eval 66.9%

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

      9. associate-/l* 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. distribute-rgt-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*r* 99.3%

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

      3. associate-/l* 99.3%

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

      4. *-commutative 99.3%

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

      5. clear-num 99.1%

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

      6. un-div-inv 99.4%

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

      7. associate-/r* 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 69.8%

      \[\leadsto \frac{\sin \left(x \cdot -0.5\right)}{\color{blue}{0.09375 \cdot {x}^{2} - 0.75}} \]

   if 1.00000000000000005e-4 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. associate-/l* 98.9%

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

      3. sqr-neg 98.9%

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

      4. sin-neg 98.9%

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

      5. distribute-lft-neg-out 98.9%

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

      6. sin-neg 98.9%

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

      7. distribute-lft-neg-out 98.9%

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

      8. metadata-eval 98.9%

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

      9. associate-/l* 99.0%

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

      10. distribute-lft-neg-out 99.0%

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

      11. distribute-rgt-neg-in 99.0%

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

      12. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   6. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(2.6666666666666665 \cdot \frac{{\sin \left(x \cdot -0.5\right)}^{2}}{\sin x}\right)\right)} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 98.9%

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

      2. clear-num 98.9%

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

      3. un-div-inv 99.0%

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

   8. Applied egg-rr 99.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0001:\\ \;\;\;\;\frac{\sin \left(x \cdot -0.5\right)}{0.09375 \cdot {x}^{2} - 0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{{\sin \left(x \cdot -0.5\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. associate-*l* 75.6%

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

   2. associate-/l* 75.7%

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

   3. sqr-neg 75.7%

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

   4. sin-neg 75.7%

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

   5. distribute-lft-neg-out 75.7%

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

   6. sin-neg 75.7%

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

   7. distribute-lft-neg-out 75.7%

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

   8. metadata-eval 75.7%

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

   9. associate-/l* 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. distribute-rgt-neg-in 99.3%

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

   12. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.2%

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

   3. associate-/l* 99.2%

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

   4. *-commutative 99.2%

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

   5. clear-num 99.1%

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

   6. un-div-inv 99.3%

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

   7. associate-/r* 99.3%

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in x around inf 99.5%

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

   1. *-commutative 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. associate-*l* 75.6%

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

   2. associate-/l* 75.7%

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

   3. sqr-neg 75.7%

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

   4. sin-neg 75.7%

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

   5. distribute-lft-neg-out 75.7%

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

   6. sin-neg 75.7%

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

   7. distribute-lft-neg-out 75.7%

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

   8. metadata-eval 75.7%

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

   9. associate-/l* 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. distribute-rgt-neg-in 99.3%

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

   12. metadata-eval 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 7: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0048:\\ \;\;\;\;\frac{\sin \left(x\_m \cdot -0.5\right)}{0.09375 \cdot {x\_m}^{2} - 0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665 \cdot \frac{1 - \cos \left(-x\_m\right)}{2}}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0048)
    (/ (sin (* x_m -0.5)) (- (* 0.09375 (pow x_m 2.0)) 0.75))
    (/ (* 2.6666666666666665 (/ (- 1.0 (cos (- x_m))) 2.0)) (sin x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0048) {
		tmp = sin((x_m * -0.5)) / ((0.09375 * pow(x_m, 2.0)) - 0.75);
	} else {
		tmp = (2.6666666666666665 * ((1.0 - cos(-x_m)) / 2.0)) / sin(x_m);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0048d0) then
        tmp = sin((x_m * (-0.5d0))) / ((0.09375d0 * (x_m ** 2.0d0)) - 0.75d0)
    else
        tmp = (2.6666666666666665d0 * ((1.0d0 - cos(-x_m)) / 2.0d0)) / sin(x_m)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0048) {
		tmp = Math.sin((x_m * -0.5)) / ((0.09375 * Math.pow(x_m, 2.0)) - 0.75);
	} else {
		tmp = (2.6666666666666665 * ((1.0 - Math.cos(-x_m)) / 2.0)) / Math.sin(x_m);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.0048:
		tmp = math.sin((x_m * -0.5)) / ((0.09375 * math.pow(x_m, 2.0)) - 0.75)
	else:
		tmp = (2.6666666666666665 * ((1.0 - math.cos(-x_m)) / 2.0)) / math.sin(x_m)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0048)
		tmp = Float64(sin(Float64(x_m * -0.5)) / Float64(Float64(0.09375 * (x_m ^ 2.0)) - 0.75));
	else
		tmp = Float64(Float64(2.6666666666666665 * Float64(Float64(1.0 - cos(Float64(-x_m))) / 2.0)) / sin(x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.0048)
		tmp = sin((x_m * -0.5)) / ((0.09375 * (x_m ^ 2.0)) - 0.75);
	else
		tmp = (2.6666666666666665 * ((1.0 - cos(-x_m)) / 2.0)) / sin(x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0048], N[(N[Sin[N[(x$95$m * -0.5), $MachinePrecision]], $MachinePrecision] / N[(N[(0.09375 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.75), $MachinePrecision]), $MachinePrecision], N[(N[(2.6666666666666665 * N[(N[(1.0 - N[Cos[(-x$95$m)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.00479999999999999958

   1. Initial program 67.0%

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

      1. associate-*l* 67.0%

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

      2. associate-/l* 67.1%

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

      3. sqr-neg 67.1%

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

      4. sin-neg 67.1%

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

      5. distribute-lft-neg-out 67.1%

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

      6. sin-neg 67.1%

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

      7. distribute-lft-neg-out 67.1%

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

      8. metadata-eval 67.1%

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

      9. associate-/l* 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. distribute-rgt-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*r* 99.3%

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

      3. associate-/l* 99.3%

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

      4. *-commutative 99.3%

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

      5. clear-num 99.1%

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

      6. un-div-inv 99.4%

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

      7. associate-/r* 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 70.0%

      \[\leadsto \frac{\sin \left(x \cdot -0.5\right)}{\color{blue}{0.09375 \cdot {x}^{2} - 0.75}} \]

   if 0.00479999999999999958 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. *-commutative 98.9%

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

   5. Simplified 98.9%

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

      1. unpow2 98.9%

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

      2. sqr-sin-a 98.1%

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

      3. add-sqr-sqrt 69.1%

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

      4. sqrt-unprod 48.2%

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

      5. swap-sqr 46.9%

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

      6. metadata-eval 46.9%

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

      7. metadata-eval 46.9%

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

      8. swap-sqr 48.2%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 98.1%

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

      11. sqr-sin-a 98.9%

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

      12. sin-mult 98.1%

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

   7. Applied egg-rr 98.1%

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

      1. +-inverses 98.1%

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

      2. cos-0 98.1%

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

      3. distribute-lft-out 98.1%

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

      4. metadata-eval 98.1%

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

   9. Simplified 98.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0048:\\ \;\;\;\;\frac{\sin \left(x \cdot -0.5\right)}{0.09375 \cdot {x}^{2} - 0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665 \cdot \frac{1 - \cos \left(-x\right)}{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 44000000:\\ \;\;\;\;x\_m \cdot 0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin \left(x\_m \cdot -0.5\right) \cdot 1.3333333333333333\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 44000000.0)
    (* x_m 0.6666666666666666)
    (* (sin (* x_m -0.5)) 1.3333333333333333))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 44000000.0) {
		tmp = x_m * 0.6666666666666666;
	} else {
		tmp = sin((x_m * -0.5)) * 1.3333333333333333;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 44000000.0d0) then
        tmp = x_m * 0.6666666666666666d0
    else
        tmp = sin((x_m * (-0.5d0))) * 1.3333333333333333d0
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 44000000.0) {
		tmp = x_m * 0.6666666666666666;
	} else {
		tmp = Math.sin((x_m * -0.5)) * 1.3333333333333333;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 44000000.0:
		tmp = x_m * 0.6666666666666666
	else:
		tmp = math.sin((x_m * -0.5)) * 1.3333333333333333
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 44000000.0)
		tmp = Float64(x_m * 0.6666666666666666);
	else
		tmp = Float64(sin(Float64(x_m * -0.5)) * 1.3333333333333333);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 44000000.0)
		tmp = x_m * 0.6666666666666666;
	else
		tmp = sin((x_m * -0.5)) * 1.3333333333333333;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 44000000.0], N[(x$95$m * 0.6666666666666666), $MachinePrecision], N[(N[Sin[N[(x$95$m * -0.5), $MachinePrecision]], $MachinePrecision] * 1.3333333333333333), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.4e7

   1. Initial program 67.5%

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

      1. associate-*l* 67.5%

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

      2. associate-/l* 67.6%

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

      3. sqr-neg 67.6%

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

      4. sin-neg 67.6%

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

      5. distribute-lft-neg-out 67.6%

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

      6. sin-neg 67.6%

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

      7. distribute-lft-neg-out 67.6%

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

      8. metadata-eval 67.6%

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

      9. associate-/l* 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. distribute-rgt-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 68.7%

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

   if 4.4e7 < x

   1. Initial program 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-/l* 98.9%

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

      3. associate-*r/ 99.0%

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

      4. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around 0 12.2%

      \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{1.3333333333333333} \]
   6. Step-by-step derivation

      1. add0 12.2%

         \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right) \cdot 1.3333333333333333 + 0} \]

      2. add-sqr-sqrt 11.5%

         \[\leadsto \sin \color{blue}{\left(\sqrt{x \cdot 0.5} \cdot \sqrt{x \cdot 0.5}\right)} \cdot 1.3333333333333333 + 0 \]

      3. sqrt-unprod 5.8%

         \[\leadsto \sin \color{blue}{\left(\sqrt{\left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)}\right)} \cdot 1.3333333333333333 + 0 \]

      4. swap-sqr 5.3%

         \[\leadsto \sin \left(\sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(0.5 \cdot 0.5\right)}}\right) \cdot 1.3333333333333333 + 0 \]

      5. metadata-eval 5.3%

         \[\leadsto \sin \left(\sqrt{\left(x \cdot x\right) \cdot \color{blue}{0.25}}\right) \cdot 1.3333333333333333 + 0 \]

      6. metadata-eval 5.3%

         \[\leadsto \sin \left(\sqrt{\left(x \cdot x\right) \cdot \color{blue}{\left(-0.5 \cdot -0.5\right)}}\right) \cdot 1.3333333333333333 + 0 \]

      7. swap-sqr 5.8%

         \[\leadsto \sin \left(\sqrt{\color{blue}{\left(x \cdot -0.5\right) \cdot \left(x \cdot -0.5\right)}}\right) \cdot 1.3333333333333333 + 0 \]

      8. sqrt-unprod 0.0%

         \[\leadsto \sin \color{blue}{\left(\sqrt{x \cdot -0.5} \cdot \sqrt{x \cdot -0.5}\right)} \cdot 1.3333333333333333 + 0 \]

      9. add-sqr-sqrt 10.8%

         \[\leadsto \sin \color{blue}{\left(x \cdot -0.5\right)} \cdot 1.3333333333333333 + 0 \]

   7. Applied egg-rr 10.8%

      \[\leadsto \color{blue}{\sin \left(x \cdot -0.5\right) \cdot 1.3333333333333333 + 0} \]
   8. Step-by-step derivation

      1. *-commutative 10.8%

         \[\leadsto \color{blue}{1.3333333333333333 \cdot \sin \left(x \cdot -0.5\right)} + 0 \]

      2. add0 10.8%

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

   9. Simplified 10.8%

      \[\leadsto \color{blue}{1.3333333333333333 \cdot \sin \left(x \cdot -0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 44000000:\\ \;\;\;\;x \cdot 0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin \left(x \cdot -0.5\right) \cdot 1.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. *-commutative 75.6%

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

   2. associate-/l* 99.2%

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

   3. associate-*r/ 99.2%

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

   4. metadata-eval 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 56.2%

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

6. Final simplification 56.2%

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

Alternative 10: 55.7% accurate, 3.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ (sin (* x_m -0.5)) -0.75)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. associate-*l* 75.6%

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

   2. associate-/l* 75.7%

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

   3. sqr-neg 75.7%

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

   4. sin-neg 75.7%

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

   5. distribute-lft-neg-out 75.7%

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

   6. sin-neg 75.7%

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

   7. distribute-lft-neg-out 75.7%

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

   8. metadata-eval 75.7%

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

   9. associate-/l* 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. distribute-rgt-neg-in 99.3%

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

   12. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.2%

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

   3. associate-/l* 99.2%

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

   4. *-commutative 99.2%

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

   5. clear-num 99.1%

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

   6. un-div-inv 99.3%

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

   7. associate-/r* 99.3%

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in x around 0 56.4%

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

8. Final simplification 56.4%

   \[\leadsto \frac{\sin \left(x \cdot -0.5\right)}{-0.75} \]
9. Add Preprocessing

Alternative 11: 51.2% accurate, 104.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. associate-*l* 75.6%

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

   2. associate-/l* 75.7%

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

   3. sqr-neg 75.7%

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

   4. sin-neg 75.7%

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

   5. distribute-lft-neg-out 75.7%

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

   6. sin-neg 75.7%

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

   7. distribute-lft-neg-out 75.7%

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

   8. metadata-eval 75.7%

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

   9. associate-/l* 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. distribute-rgt-neg-in 99.3%

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

   12. metadata-eval 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 51.9%

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

6. Final simplification 51.9%

   \[\leadsto x \cdot 0.6666666666666666 \]
7. Add Preprocessing

Developer target: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024046 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

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

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