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

Percentage Accurate: 76.7% → 99.5%

Time: 10.0s

Alternatives: 16

Speedup: 1.5×

• Could not converge on a ground truth

  1. x = 1.1087530428591662e-45

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0002:\\ \;\;\;\;\frac{x\_m}{0.375 \cdot \mathsf{fma}\left({x\_m}^{2}, -0.3333333333333333, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.375 \cdot \sin x\_m}{{\sin \left(x\_m \cdot 0.5\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0002)
    (/ x_m (* 0.375 (fma (pow x_m 2.0) -0.3333333333333333 4.0)))
    (/ 1.0 (/ (* 0.375 (sin x_m)) (pow (sin (* x_m 0.5)) 2.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0002) {
		tmp = x_m / (0.375 * fma(pow(x_m, 2.0), -0.3333333333333333, 4.0));
	} else {
		tmp = 1.0 / ((0.375 * sin(x_m)) / pow(sin((x_m * 0.5)), 2.0));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0002)
		tmp = Float64(x_m / Float64(0.375 * fma((x_m ^ 2.0), -0.3333333333333333, 4.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(0.375 * sin(x_m)) / (sin(Float64(x_m * 0.5)) ^ 2.0)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0002], N[(x$95$m / N[(0.375 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.3333333333333333 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.375 * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-4

   1. Initial program 68.0%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.3%

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

      2. associate-*r/ 68.0%

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

      3. metadata-eval 68.0%

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

      4. clear-num 68.0%

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

      5. *-un-lft-identity 68.0%

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

      6. metadata-eval 68.0%

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

      7. associate-*l* 68.0%

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

      8. times-frac 67.9%

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

      9. metadata-eval 67.9%

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

      10. pow2 67.9%

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

   6. Applied egg-rr 67.9%

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

   7. Taylor expanded in x around 0 63.9%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + -0.3333333333333333 \cdot {x}^{2}}{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 63.9%

         \[\leadsto \frac{1}{0.375 \cdot \frac{4 + \color{blue}{{x}^{2} \cdot -0.3333333333333333}}{x}} \]

   9. Simplified 63.9%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]
   10. Step-by-step derivation

       1. associate-/r* 64.0%

          \[\leadsto \color{blue}{\frac{\frac{1}{0.375}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       2. metadata-eval 64.0%

          \[\leadsto \frac{\color{blue}{2.6666666666666665}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}} \]

       3. div-inv 64.0%

          \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{1}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       4. clear-num 64.0%

          \[\leadsto 2.6666666666666665 \cdot \color{blue}{\frac{x}{4 + {x}^{2} \cdot -0.3333333333333333}} \]

       5. +-commutative 64.0%

          \[\leadsto 2.6666666666666665 \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.3333333333333333 + 4}} \]

       6. fma-define 64.0%

          \[\leadsto 2.6666666666666665 \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

   11. Applied egg-rr 64.0%

       \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]
   12. Step-by-step derivation

       1. metadata-eval 64.0%

          \[\leadsto \color{blue}{\frac{1}{0.375}} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

       2. times-frac 64.4%

          \[\leadsto \color{blue}{\frac{1 \cdot x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

       3. *-lft-identity 64.4%

          \[\leadsto \frac{\color{blue}{x}}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

   13. Simplified 64.4%

       \[\leadsto \color{blue}{\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

   if 2.0000000000000001e-4 < x

   1. Initial program 98.9%

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

      1. associate-/l* 99.0%

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

      2. associate-*l* 99.0%

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

      3. metadata-eval 99.0%

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

   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. associate-*r* 99.0%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. clear-num 98.9%

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

      5. *-un-lft-identity 98.9%

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

      6. metadata-eval 98.9%

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

      7. associate-*l* 98.9%

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

      8. times-frac 99.2%

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

      9. metadata-eval 99.2%

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

      10. pow2 99.2%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in x around inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. *-commutative 99.2%

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

      3. unpow2 99.2%

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

      4. *-commutative 99.2%

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

      5. unpow2 99.2%

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

   9. Simplified 99.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin x \cdot 0.375}{{\sin \left(x \cdot 0.5\right)}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0002:\\ \;\;\;\;\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.375 \cdot \sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1e-10)
    (/ (* x_m 0.25) 0.375)
    (/ (/ (pow (sin (* x_m 0.5)) 2.0) (sin x_m)) 0.375))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1e-10) {
		tmp = (x_m * 0.25) / 0.375;
	} else {
		tmp = (pow(sin((x_m * 0.5)), 2.0) / sin(x_m)) / 0.375;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1d-10) then
        tmp = (x_m * 0.25d0) / 0.375d0
    else
        tmp = ((sin((x_m * 0.5d0)) ** 2.0d0) / sin(x_m)) / 0.375d0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1e-10) {
		tmp = (x_m * 0.25) / 0.375;
	} else {
		tmp = (Math.pow(Math.sin((x_m * 0.5)), 2.0) / Math.sin(x_m)) / 0.375;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1e-10:
		tmp = (x_m * 0.25) / 0.375
	else:
		tmp = (math.pow(math.sin((x_m * 0.5)), 2.0) / math.sin(x_m)) / 0.375
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1e-10)
		tmp = Float64(Float64(x_m * 0.25) / 0.375);
	else
		tmp = Float64(Float64((sin(Float64(x_m * 0.5)) ^ 2.0) / sin(x_m)) / 0.375);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1e-10)
		tmp = (x_m * 0.25) / 0.375;
	else
		tmp = ((sin((x_m * 0.5)) ^ 2.0) / sin(x_m)) / 0.375;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1e-10], N[(N[(x$95$m * 0.25), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000004e-10

   1. Initial program 67.7%

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

      1. associate-/l* 99.2%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt 46.5%

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

      2. pow2 46.5%

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

      3. *-commutative 46.5%

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

      4. sqrt-prod 46.5%

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

      5. associate-*r/ 32.2%

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

      6. sqrt-div 30.9%

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

      7. sqrt-unprod 37.3%

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

      8. add-sqr-sqrt 46.4%

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

   6. Applied egg-rr 46.4%

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

      1. *-commutative 46.4%

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

      2. unpow-prod-down 46.3%

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

      3. pow2 46.3%

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

      4. rem-square-sqrt 46.3%

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

      5. pow2 46.3%

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

      6. frac-times 30.9%

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

      7. unpow2 30.9%

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

      8. add-sqr-sqrt 67.6%

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

      9. clear-num 67.6%

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

      10. div-inv 67.7%

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

      11. metadata-eval 67.7%

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

      12. associate-/r* 67.6%

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

      13. *-commutative 67.6%

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

      14. associate-/r* 67.8%

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

      15. clear-num 67.8%

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

   8. Applied egg-rr 67.8%

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

   9. Taylor expanded in x around 0 63.6%

      \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]
   10. Step-by-step derivation

       1. *-commutative 63.6%

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

   11. Simplified 63.6%

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

   if 1.00000000000000004e-10 < x

   1. Initial program 98.9%

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

      1. associate-/l* 99.1%

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

      2. associate-*l* 99.0%

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

      3. metadata-eval 99.0%

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

   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. add-sqr-sqrt 52.9%

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

      2. pow2 52.9%

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

      3. *-commutative 52.9%

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

      4. sqrt-prod 52.8%

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

      5. associate-*r/ 52.8%

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

      6. sqrt-div 52.7%

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

      7. sqrt-unprod 28.7%

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

      8. add-sqr-sqrt 52.7%

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

   6. Applied egg-rr 52.7%

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

      1. *-commutative 52.7%

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

      2. unpow-prod-down 52.6%

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

      3. pow2 52.6%

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

      4. rem-square-sqrt 52.6%

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

      5. pow2 52.6%

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

      6. frac-times 52.7%

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

      7. unpow2 52.7%

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

      8. add-sqr-sqrt 99.0%

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

      9. clear-num 98.9%

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

      10. div-inv 99.0%

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

      11. metadata-eval 99.0%

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

      12. associate-/r* 99.2%

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

      13. *-commutative 99.2%

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

      14. associate-/r* 99.1%

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

      15. clear-num 99.2%

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

   8. Applied egg-rr 99.2%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\frac{x\_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x\_m \cdot {\sin \left(x\_m \cdot 0.5\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 4e-9)
    (/ (* x_m 0.25) 0.375)
    (/ 2.6666666666666665 (* (sin x_m) (pow (sin (* x_m 0.5)) -2.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 4e-9) {
		tmp = (x_m * 0.25) / 0.375;
	} else {
		tmp = 2.6666666666666665 / (sin(x_m) * pow(sin((x_m * 0.5)), -2.0));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 4d-9) then
        tmp = (x_m * 0.25d0) / 0.375d0
    else
        tmp = 2.6666666666666665d0 / (sin(x_m) * (sin((x_m * 0.5d0)) ** (-2.0d0)))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 4e-9) {
		tmp = (x_m * 0.25) / 0.375;
	} else {
		tmp = 2.6666666666666665 / (Math.sin(x_m) * Math.pow(Math.sin((x_m * 0.5)), -2.0));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 4e-9:
		tmp = (x_m * 0.25) / 0.375
	else:
		tmp = 2.6666666666666665 / (math.sin(x_m) * math.pow(math.sin((x_m * 0.5)), -2.0))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 4e-9)
		tmp = Float64(Float64(x_m * 0.25) / 0.375);
	else
		tmp = Float64(2.6666666666666665 / Float64(sin(x_m) * (sin(Float64(x_m * 0.5)) ^ -2.0)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 4e-9)
		tmp = (x_m * 0.25) / 0.375;
	else
		tmp = 2.6666666666666665 / (sin(x_m) * (sin((x_m * 0.5)) ^ -2.0));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 4e-9], N[(N[(x$95$m * 0.25), $MachinePrecision] / 0.375), $MachinePrecision], N[(2.6666666666666665 / N[(N[Sin[x$95$m], $MachinePrecision] * N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000025e-9

   1. Initial program 67.8%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt 46.7%

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

      2. pow2 46.7%

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

      3. *-commutative 46.7%

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

      4. sqrt-prod 46.7%

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

      5. associate-*r/ 32.5%

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

      6. sqrt-div 31.2%

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

      7. sqrt-unprod 37.6%

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

      8. add-sqr-sqrt 46.7%

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

   6. Applied egg-rr 46.7%

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

      1. *-commutative 46.7%

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

      2. unpow-prod-down 46.6%

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

      3. pow2 46.6%

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

      4. rem-square-sqrt 46.6%

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

      5. pow2 46.6%

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

      6. frac-times 31.3%

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

      7. unpow2 31.3%

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

      8. add-sqr-sqrt 67.8%

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

      9. clear-num 67.8%

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

      10. div-inv 67.9%

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

      11. metadata-eval 67.9%

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

      12. associate-/r* 67.8%

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

      13. *-commutative 67.8%

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

      14. associate-/r* 67.9%

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

      15. clear-num 68.0%

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

   8. Applied egg-rr 68.0%

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

   9. Taylor expanded in x around 0 63.7%

      \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]
   10. Step-by-step derivation

       1. *-commutative 63.7%

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

   11. Simplified 63.7%

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

   if 4.00000000000000025e-9 < x

   1. Initial program 98.9%

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

      1. associate-/l* 99.1%

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

      2. associate-*l* 99.0%

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

      3. metadata-eval 99.0%

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

   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. add-sqr-sqrt 52.0%

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

      2. pow2 52.0%

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

      3. *-commutative 52.0%

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

      4. sqrt-prod 52.0%

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

      5. associate-*r/ 51.9%

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

      6. sqrt-div 51.9%

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

      7. sqrt-unprod 27.5%

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

      8. add-sqr-sqrt 51.9%

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

   6. Applied egg-rr 51.9%

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

      1. *-commutative 51.9%

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

      2. unpow-prod-down 51.8%

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

      3. pow2 51.8%

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

      4. rem-square-sqrt 51.8%

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

      5. pow2 51.8%

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

      6. frac-times 51.9%

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

      7. unpow2 51.9%

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

      8. add-sqr-sqrt 98.9%

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

      9. clear-num 98.9%

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

      10. div-inv 99.0%

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

      11. div-inv 99.1%

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

      12. pow-flip 99.2%

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

      13. metadata-eval 99.2%

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

   8. Applied egg-rr 99.2%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0005:\\ \;\;\;\;\frac{x\_m}{0.375 \cdot \mathsf{fma}\left({x\_m}^{2}, -0.3333333333333333, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x\_m \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0005)
    (/ x_m (* 0.375 (fma (pow x_m 2.0) -0.3333333333333333 4.0)))
    (* (pow (sin (* x_m 0.5)) 2.0) (/ 2.6666666666666665 (sin x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0005) {
		tmp = x_m / (0.375 * fma(pow(x_m, 2.0), -0.3333333333333333, 4.0));
	} else {
		tmp = pow(sin((x_m * 0.5)), 2.0) * (2.6666666666666665 / 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 <= 0.0005)
		tmp = Float64(x_m / Float64(0.375 * fma((x_m ^ 2.0), -0.3333333333333333, 4.0)));
	else
		tmp = Float64((sin(Float64(x_m * 0.5)) ^ 2.0) * Float64(2.6666666666666665 / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0005], N[(x$95$m / N[(0.375 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.3333333333333333 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(2.6666666666666665 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-4

   1. Initial program 68.0%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.3%

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

      2. associate-*r/ 68.0%

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

      3. metadata-eval 68.0%

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

      4. clear-num 68.0%

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

      5. *-un-lft-identity 68.0%

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

      6. metadata-eval 68.0%

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

      7. associate-*l* 68.0%

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

      8. times-frac 67.9%

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

      9. metadata-eval 67.9%

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

      10. pow2 67.9%

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

   6. Applied egg-rr 67.9%

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

   7. Taylor expanded in x around 0 63.9%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + -0.3333333333333333 \cdot {x}^{2}}{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 63.9%

         \[\leadsto \frac{1}{0.375 \cdot \frac{4 + \color{blue}{{x}^{2} \cdot -0.3333333333333333}}{x}} \]

   9. Simplified 63.9%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]
   10. Step-by-step derivation

       1. associate-/r* 64.0%

          \[\leadsto \color{blue}{\frac{\frac{1}{0.375}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       2. metadata-eval 64.0%

          \[\leadsto \frac{\color{blue}{2.6666666666666665}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}} \]

       3. div-inv 64.0%

          \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{1}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       4. clear-num 64.0%

          \[\leadsto 2.6666666666666665 \cdot \color{blue}{\frac{x}{4 + {x}^{2} \cdot -0.3333333333333333}} \]

       5. +-commutative 64.0%

          \[\leadsto 2.6666666666666665 \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.3333333333333333 + 4}} \]

       6. fma-define 64.0%

          \[\leadsto 2.6666666666666665 \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

   11. Applied egg-rr 64.0%

       \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]
   12. Step-by-step derivation

       1. metadata-eval 64.0%

          \[\leadsto \color{blue}{\frac{1}{0.375}} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

       2. times-frac 64.4%

          \[\leadsto \color{blue}{\frac{1 \cdot x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

       3. *-lft-identity 64.4%

          \[\leadsto \frac{\color{blue}{x}}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

   13. Simplified 64.4%

       \[\leadsto \color{blue}{\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

   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* 99.0%

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

      2. associate-*l* 99.0%

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

      3. metadata-eval 99.0%

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

   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. associate-*r* 99.0%

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

      2. *-commutative 99.0%

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

      3. div-inv 98.9%

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

      4. associate-*l* 99.0%

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

      5. associate-/r/ 99.0%

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

      6. un-div-inv 99.0%

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

      7. *-un-lft-identity 99.0%

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

      8. times-frac 99.0%

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

      9. metadata-eval 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in x around inf 98.9%

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

      1. associate-*r/ 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l/ 98.9%

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

      4. *-commutative 98.9%

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

   9. Simplified 98.9%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0002:\\ \;\;\;\;\frac{x\_m}{0.375 \cdot \mathsf{fma}\left({x\_m}^{2}, -0.3333333333333333, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x\_m \cdot 0.5\right)}^{2}}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0002)
    (/ x_m (* 0.375 (fma (pow x_m 2.0) -0.3333333333333333 4.0)))
    (* 2.6666666666666665 (/ (pow (sin (* x_m 0.5)) 2.0) (sin x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0002) {
		tmp = x_m / (0.375 * fma(pow(x_m, 2.0), -0.3333333333333333, 4.0));
	} else {
		tmp = 2.6666666666666665 * (pow(sin((x_m * 0.5)), 2.0) / 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 <= 0.0002)
		tmp = Float64(x_m / Float64(0.375 * fma((x_m ^ 2.0), -0.3333333333333333, 4.0)));
	else
		tmp = Float64(2.6666666666666665 * Float64((sin(Float64(x_m * 0.5)) ^ 2.0) / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0002], N[(x$95$m / N[(0.375 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.3333333333333333 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-4

   1. Initial program 68.0%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.3%

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

      2. associate-*r/ 68.0%

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

      3. metadata-eval 68.0%

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

      4. clear-num 68.0%

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

      5. *-un-lft-identity 68.0%

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

      6. metadata-eval 68.0%

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

      7. associate-*l* 68.0%

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

      8. times-frac 67.9%

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

      9. metadata-eval 67.9%

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

      10. pow2 67.9%

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

   6. Applied egg-rr 67.9%

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

   7. Taylor expanded in x around 0 63.9%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + -0.3333333333333333 \cdot {x}^{2}}{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 63.9%

         \[\leadsto \frac{1}{0.375 \cdot \frac{4 + \color{blue}{{x}^{2} \cdot -0.3333333333333333}}{x}} \]

   9. Simplified 63.9%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]
   10. Step-by-step derivation

       1. associate-/r* 64.0%

          \[\leadsto \color{blue}{\frac{\frac{1}{0.375}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       2. metadata-eval 64.0%

          \[\leadsto \frac{\color{blue}{2.6666666666666665}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}} \]

       3. div-inv 64.0%

          \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{1}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       4. clear-num 64.0%

          \[\leadsto 2.6666666666666665 \cdot \color{blue}{\frac{x}{4 + {x}^{2} \cdot -0.3333333333333333}} \]

       5. +-commutative 64.0%

          \[\leadsto 2.6666666666666665 \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.3333333333333333 + 4}} \]

       6. fma-define 64.0%

          \[\leadsto 2.6666666666666665 \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

   11. Applied egg-rr 64.0%

       \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]
   12. Step-by-step derivation

       1. metadata-eval 64.0%

          \[\leadsto \color{blue}{\frac{1}{0.375}} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

       2. times-frac 64.4%

          \[\leadsto \color{blue}{\frac{1 \cdot x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

       3. *-lft-identity 64.4%

          \[\leadsto \frac{\color{blue}{x}}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

   13. Simplified 64.4%

       \[\leadsto \color{blue}{\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

   if 2.0000000000000001e-4 < x

   1. Initial program 98.9%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. 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/ 99.0%

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

      3. associate-*r* 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-*r/ 98.9%

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

      6. pow2 98.9%

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

   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 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0002:\\ \;\;\;\;\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* 99.2%

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

   2. *-commutative 99.2%

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

   3. div-inv 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-/r/ 99.1%

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

   6. un-div-inv 99.2%

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

   7. *-un-lft-identity 99.2%

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

   8. times-frac 99.5%

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

   9. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 7: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. *-commutative 74.7%

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

   2. associate-/l* 99.3%

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

   3. remove-double-neg 99.3%

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

   4. sin-neg 99.3%

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

   5. distribute-lft-neg-out 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

   7. distribute-frac-neg 99.3%

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

   8. distribute-frac-neg2 99.3%

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

   9. neg-mul-1 99.3%

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

   10. associate-/r* 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 8: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. *-commutative 74.7%

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

   2. associate-/l* 99.3%

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

   3. remove-double-neg 99.3%

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

   4. sin-neg 99.3%

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

   5. distribute-lft-neg-out 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

   7. distribute-frac-neg 99.3%

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

   8. distribute-frac-neg2 99.3%

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

   9. neg-mul-1 99.3%

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

   10. associate-/r* 99.3%

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

3. Simplified 99.3%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

   3. *-un-lft-identity 99.1%

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

   4. times-frac 99.2%

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

   5. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

   1. unpow-1 99.2%

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

   2. associate-/r* 99.2%

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

   3. metadata-eval 99.2%

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

8. Simplified 99.2%

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

Alternative 9: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

Alternative 10: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0062:\\ \;\;\;\;\frac{x\_m}{0.375 \cdot \mathsf{fma}\left({x\_m}^{2}, -0.3333333333333333, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.375}{\frac{0.5 - \frac{\cos x\_m}{2}}{\sin x\_m}}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0062)
    (/ x_m (* 0.375 (fma (pow x_m 2.0) -0.3333333333333333 4.0)))
    (/ 1.0 (/ 0.375 (/ (- 0.5 (/ (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 <= 0.0062) {
		tmp = x_m / (0.375 * fma(pow(x_m, 2.0), -0.3333333333333333, 4.0));
	} else {
		tmp = 1.0 / (0.375 / ((0.5 - (cos(x_m) / 2.0)) / 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 <= 0.0062)
		tmp = Float64(x_m / Float64(0.375 * fma((x_m ^ 2.0), -0.3333333333333333, 4.0)));
	else
		tmp = Float64(1.0 / Float64(0.375 / Float64(Float64(0.5 - Float64(cos(x_m) / 2.0)) / sin(x_m))));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0062], N[(x$95$m / N[(0.375 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.3333333333333333 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.375 / N[(N[(0.5 - N[(N[Cos[x$95$m], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.00619999999999999978

   1. Initial program 68.1%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.3%

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

      2. associate-*r/ 68.1%

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

      3. metadata-eval 68.1%

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

      4. clear-num 68.1%

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

      5. *-un-lft-identity 68.1%

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

      6. metadata-eval 68.1%

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

      7. associate-*l* 68.1%

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

      8. times-frac 68.1%

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

      9. metadata-eval 68.1%

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

      10. pow2 68.1%

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

   6. Applied egg-rr 68.1%

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

   7. Taylor expanded in x around 0 64.1%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + -0.3333333333333333 \cdot {x}^{2}}{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 64.1%

         \[\leadsto \frac{1}{0.375 \cdot \frac{4 + \color{blue}{{x}^{2} \cdot -0.3333333333333333}}{x}} \]

   9. Simplified 64.1%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]
   10. Step-by-step derivation

       1. associate-/r* 64.2%

          \[\leadsto \color{blue}{\frac{\frac{1}{0.375}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       2. metadata-eval 64.2%

          \[\leadsto \frac{\color{blue}{2.6666666666666665}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}} \]

       3. div-inv 64.1%

          \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{1}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       4. clear-num 64.2%

          \[\leadsto 2.6666666666666665 \cdot \color{blue}{\frac{x}{4 + {x}^{2} \cdot -0.3333333333333333}} \]

       5. +-commutative 64.2%

          \[\leadsto 2.6666666666666665 \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.3333333333333333 + 4}} \]

       6. fma-define 64.2%

          \[\leadsto 2.6666666666666665 \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

   11. Applied egg-rr 64.2%

       \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]
   12. Step-by-step derivation

       1. metadata-eval 64.2%

          \[\leadsto \color{blue}{\frac{1}{0.375}} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

       2. times-frac 64.5%

          \[\leadsto \color{blue}{\frac{1 \cdot x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

       3. *-lft-identity 64.5%

          \[\leadsto \frac{\color{blue}{x}}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

   13. Simplified 64.5%

       \[\leadsto \color{blue}{\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

   if 0.00619999999999999978 < x

   1. Initial program 98.9%

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

      1. associate-/l* 99.0%

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

      2. associate-*l* 99.0%

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

      3. metadata-eval 99.0%

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

   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. associate-*r* 99.0%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. clear-num 98.9%

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

      5. *-un-lft-identity 98.9%

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

      6. metadata-eval 98.9%

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

      7. associate-*l* 99.0%

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

      8. times-frac 99.2%

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

      9. metadata-eval 99.2%

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

      10. pow2 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. clear-num 99.1%

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

      2. unpow2 99.1%

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

      3. add-sqr-sqrt 50.4%

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

      4. frac-times 50.2%

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

      5. pow2 50.2%

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

      6. un-div-inv 50.2%

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

      7. pow2 50.2%

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

      8. frac-times 50.4%

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

      9. unpow2 50.4%

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

      10. add-sqr-sqrt 99.2%

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

   8. Applied egg-rr 99.2%

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

      1. unpow2 99.2%

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

      2. sin-mult 98.9%

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

   10. Applied egg-rr 98.9%

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

       1. div-sub 98.9%

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

       2. +-inverses 98.9%

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

       3. cos-0 98.9%

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

       4. metadata-eval 98.9%

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

       5. distribute-lft-out 98.9%

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

       6. metadata-eval 98.9%

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

       7. *-rgt-identity 98.9%

          \[\leadsto \frac{1}{\frac{0.375}{\frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x}}} \]

   12. Simplified 98.9%

       \[\leadsto \frac{1}{\frac{0.375}{\frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{\sin x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0062:\\ \;\;\;\;\frac{x\_m}{0.375 \cdot \mathsf{fma}\left({x\_m}^{2}, -0.3333333333333333, 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 - \frac{\cos x\_m}{2}}{\sin x\_m}}{0.375}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0062)
    (/ x_m (* 0.375 (fma (pow x_m 2.0) -0.3333333333333333 4.0)))
    (/ (/ (- 0.5 (/ (cos x_m) 2.0)) (sin x_m)) 0.375))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0062) {
		tmp = x_m / (0.375 * fma(pow(x_m, 2.0), -0.3333333333333333, 4.0));
	} else {
		tmp = ((0.5 - (cos(x_m) / 2.0)) / sin(x_m)) / 0.375;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0062)
		tmp = Float64(x_m / Float64(0.375 * fma((x_m ^ 2.0), -0.3333333333333333, 4.0)));
	else
		tmp = Float64(Float64(Float64(0.5 - Float64(cos(x_m) / 2.0)) / sin(x_m)) / 0.375);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0062], N[(x$95$m / N[(0.375 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.3333333333333333 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(N[Cos[x$95$m], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.00619999999999999978

   1. Initial program 68.1%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.3%

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

      2. associate-*r/ 68.1%

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

      3. metadata-eval 68.1%

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

      4. clear-num 68.1%

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

      5. *-un-lft-identity 68.1%

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

      6. metadata-eval 68.1%

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

      7. associate-*l* 68.1%

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

      8. times-frac 68.1%

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

      9. metadata-eval 68.1%

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

      10. pow2 68.1%

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

   6. Applied egg-rr 68.1%

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

   7. Taylor expanded in x around 0 64.1%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + -0.3333333333333333 \cdot {x}^{2}}{x}}} \]
   8. Step-by-step derivation

      1. *-commutative 64.1%

         \[\leadsto \frac{1}{0.375 \cdot \frac{4 + \color{blue}{{x}^{2} \cdot -0.3333333333333333}}{x}} \]

   9. Simplified 64.1%

      \[\leadsto \frac{1}{0.375 \cdot \color{blue}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]
   10. Step-by-step derivation

       1. associate-/r* 64.2%

          \[\leadsto \color{blue}{\frac{\frac{1}{0.375}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       2. metadata-eval 64.2%

          \[\leadsto \frac{\color{blue}{2.6666666666666665}}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}} \]

       3. div-inv 64.1%

          \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{1}{\frac{4 + {x}^{2} \cdot -0.3333333333333333}{x}}} \]

       4. clear-num 64.2%

          \[\leadsto 2.6666666666666665 \cdot \color{blue}{\frac{x}{4 + {x}^{2} \cdot -0.3333333333333333}} \]

       5. +-commutative 64.2%

          \[\leadsto 2.6666666666666665 \cdot \frac{x}{\color{blue}{{x}^{2} \cdot -0.3333333333333333 + 4}} \]

       6. fma-define 64.2%

          \[\leadsto 2.6666666666666665 \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

   11. Applied egg-rr 64.2%

       \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]
   12. Step-by-step derivation

       1. metadata-eval 64.2%

          \[\leadsto \color{blue}{\frac{1}{0.375}} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

       2. times-frac 64.5%

          \[\leadsto \color{blue}{\frac{1 \cdot x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

       3. *-lft-identity 64.5%

          \[\leadsto \frac{\color{blue}{x}}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)} \]

   13. Simplified 64.5%

       \[\leadsto \color{blue}{\frac{x}{0.375 \cdot \mathsf{fma}\left({x}^{2}, -0.3333333333333333, 4\right)}} \]

   if 0.00619999999999999978 < x

   1. Initial program 98.9%

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

      1. associate-/l* 99.0%

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

      2. associate-*l* 99.0%

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

      3. metadata-eval 99.0%

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

   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. add-sqr-sqrt 50.3%

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

      2. pow2 50.3%

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

      3. *-commutative 50.3%

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

      4. sqrt-prod 50.2%

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

      5. associate-*r/ 50.2%

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

      6. sqrt-div 50.2%

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

      7. sqrt-unprod 25.0%

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

      8. add-sqr-sqrt 50.2%

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

   6. Applied egg-rr 50.2%

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

      1. *-commutative 50.2%

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

      2. unpow-prod-down 50.1%

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

      3. pow2 50.1%

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

      4. rem-square-sqrt 50.1%

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

      5. pow2 50.1%

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

      6. frac-times 50.2%

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

      7. unpow2 50.2%

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

      8. add-sqr-sqrt 99.0%

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

      9. clear-num 98.9%

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

      10. div-inv 99.0%

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

      11. metadata-eval 99.0%

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

      12. associate-/r* 99.2%

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

      13. *-commutative 99.2%

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

      14. associate-/r* 99.1%

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

      15. clear-num 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. unpow2 99.2%

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

      2. sin-mult 98.9%

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

   10. Applied egg-rr 98.8%

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

       1. div-sub 98.9%

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

       2. +-inverses 98.9%

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

       3. cos-0 98.9%

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

       4. metadata-eval 98.9%

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

       5. distribute-lft-out 98.9%

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

       6. metadata-eval 98.9%

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

       7. *-rgt-identity 98.9%

          \[\leadsto \frac{1}{\frac{0.375}{\frac{0.5 - \frac{\cos \color{blue}{x}}{2}}{\sin x}}} \]

   12. Simplified 98.8%

       \[\leadsto \frac{\frac{\color{blue}{0.5 - \frac{\cos x}{2}}}{\sin x}}{0.375} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 54.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* 99.2%

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

   2. *-commutative 99.2%

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

   3. div-inv 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-/r/ 99.1%

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

   6. un-div-inv 99.2%

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

   7. *-un-lft-identity 99.2%

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

   8. times-frac 99.5%

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

   9. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0 54.4%

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

Alternative 13: 54.5% accurate, 3.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* (sin (* x_m 0.5)) 1.3333333333333333)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. *-commutative 74.7%

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

   2. associate-/l* 99.3%

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

   3. remove-double-neg 99.3%

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

   4. sin-neg 99.3%

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

   5. distribute-lft-neg-out 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

   7. distribute-frac-neg 99.3%

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

   8. distribute-frac-neg2 99.3%

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

   9. neg-mul-1 99.3%

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

   10. associate-/r* 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 54.2%

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

Alternative 14: 50.7% accurate, 62.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m \cdot 0.25}{0.375} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ (* x_m 0.25) 0.375)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. add-sqr-sqrt 47.9%

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

   2. pow2 47.9%

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

   3. *-commutative 47.9%

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

   4. sqrt-prod 47.9%

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

   5. associate-*r/ 36.9%

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

   6. sqrt-div 35.8%

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

   7. sqrt-unprod 35.3%

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

   8. add-sqr-sqrt 47.8%

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

6. Applied egg-rr 47.8%

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

   1. *-commutative 47.8%

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

   2. unpow-prod-down 47.8%

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

   3. pow2 47.8%

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

   4. rem-square-sqrt 47.8%

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

   5. pow2 47.8%

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

   6. frac-times 35.9%

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

   7. unpow2 35.9%

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

   8. add-sqr-sqrt 74.7%

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

   9. clear-num 74.7%

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

   10. div-inv 74.8%

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

   11. metadata-eval 74.8%

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

   12. associate-/r* 74.8%

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

   13. *-commutative 74.8%

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

   14. associate-/r* 74.9%

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

   15. clear-num 74.9%

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

8. Applied egg-rr 74.9%

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

9. Taylor expanded in x around 0 50.7%

   \[\leadsto \frac{\color{blue}{0.25 \cdot x}}{0.375} \]
10. Step-by-step derivation

    1. *-commutative 50.7%

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

11. Simplified 50.7%

    \[\leadsto \frac{\color{blue}{x \cdot 0.25}}{0.375} \]
12. Add Preprocessing

Alternative 15: 50.5% accurate, 62.6× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ 1.0 (/ 1.5 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (1.0 / (1.5 / 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
    code = x_s * (1.0d0 / (1.5d0 / 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) {
	return x_s * (1.0 / (1.5 / x_m));
}

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (1.0 / (1.5 / 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_] := N[(x$95$s * N[(1.0 / N[(1.5 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.7%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* 99.2%

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

   2. associate-*r/ 74.7%

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

   3. metadata-eval 74.7%

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

   4. clear-num 74.8%

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

   5. *-un-lft-identity 74.8%

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

   6. metadata-eval 74.8%

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

   7. associate-*l* 74.8%

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

   8. times-frac 74.8%

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

   9. metadata-eval 74.8%

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

   10. pow2 74.8%

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

6. Applied egg-rr 74.8%

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

7. Taylor expanded in x around 0 50.5%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
8. Add Preprocessing

Alternative 16: 50.5% accurate, 104.3× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* x_m 0.6666666666666666)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 50.4%

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

6. Final simplification 50.4%

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

Developer Target 1: 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 2024180 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (* 8 (sin (* x 1/2))) 3) (/ (sin x) (sin (* x 1/2)))))

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