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

Percentage Accurate: 75.7% → 99.5%

Time: 14.8s

Alternatives: 13

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.7% 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

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x_m \cdot 0.5\right)}^{2} \cdot \frac{1}{\sin x_m \cdot 0.375}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5e-12)
    (/ (* x_m 0.25) 0.375)
    (* (pow (sin (* x_m 0.5)) 2.0) (/ 1.0 (* (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 <= 5e-12) {
		tmp = (x_m * 0.25) / 0.375;
	} else {
		tmp = pow(sin((x_m * 0.5)), 2.0) * (1.0 / (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 <= 5d-12) then
        tmp = (x_m * 0.25d0) / 0.375d0
    else
        tmp = (sin((x_m * 0.5d0)) ** 2.0d0) * (1.0d0 / (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 <= 5e-12) {
		tmp = (x_m * 0.25) / 0.375;
	} else {
		tmp = Math.pow(Math.sin((x_m * 0.5)), 2.0) * (1.0 / (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 <= 5e-12:
		tmp = (x_m * 0.25) / 0.375
	else:
		tmp = math.pow(math.sin((x_m * 0.5)), 2.0) * (1.0 / (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 <= 5e-12)
		tmp = Float64(Float64(x_m * 0.25) / 0.375);
	else
		tmp = Float64((sin(Float64(x_m * 0.5)) ^ 2.0) * Float64(1.0 / Float64(sin(x_m) * 0.375)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 5e-12)
		tmp = (x_m * 0.25) / 0.375;
	else
		tmp = (sin((x_m * 0.5)) ^ 2.0) * (1.0 / (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, 5e-12], N[(N[(x$95$m * 0.25), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[Power[N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / N[(N[Sin[x$95$m], $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999997e-12

   1. Initial program 69.4%

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

      1. *-commutative 69.4%

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

      2. remove-double-neg 69.4%

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

      3. sin-neg 69.4%

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

      4. distribute-lft-neg-out 69.4%

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

      5. distribute-rgt-neg-in 69.4%

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

      6. associate-*l/ 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-rgt-neg-in 99.5%

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

      9. distribute-lft-neg-out 99.5%

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

      10. sin-neg 99.5%

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

      11. remove-double-neg 99.5%

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

      12. associate-*l* 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*r/ 69.3%

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

      3. associate-/r/ 69.2%

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

      4. div-inv 69.3%

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

      5. associate-/r* 69.4%

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

      6. pow2 69.4%

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

      7. metadata-eval 69.4%

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

   6. Applied egg-rr 69.4%

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

   7. Taylor expanded in x around 0 65.7%

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

   if 4.9999999999999997e-12 < x

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. remove-double-neg 99.2%

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

      3. sin-neg 99.2%

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

      4. distribute-lft-neg-out 99.2%

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

      5. distribute-rgt-neg-in 99.2%

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

      6. associate-*l/ 99.3%

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

      7. *-commutative 99.3%

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

      8. distribute-rgt-neg-in 99.3%

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

      9. distribute-lft-neg-out 99.3%

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

      10. sin-neg 99.3%

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

      11. remove-double-neg 99.3%

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

      12. associate-*l* 99.2%

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

   3. 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%

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

      2. associate-*r/ 99.1%

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

      3. associate-/l* 99.3%

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

      4. pow2 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.1%

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

      3. clear-num 99.2%

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

      4. unpow2 99.2%

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

      5. associate-*l/ 99.2%

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

      6. metadata-eval 99.2%

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

      7. div-inv 99.2%

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

      8. associate-*r/ 99.3%

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

      9. frac-times 99.0%

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

      10. unpow2 99.0%

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

      11. div-inv 99.3%

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

   8. Applied egg-rr 99.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{1}{\sin x \cdot 0.375}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1e-9)
    (/ (* x_m 0.25) 0.375)
    (* 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 <= 1e-9) {
		tmp = (x_m * 0.25) / 0.375;
	} else {
		tmp = 2.6666666666666665 * (pow(sin((x_m * 0.5)), 2.0) / sin(x_m));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1d-9) then
        tmp = (x_m * 0.25d0) / 0.375d0
    else
        tmp = 2.6666666666666665d0 * ((sin((x_m * 0.5d0)) ** 2.0d0) / sin(x_m))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1e-9) {
		tmp = (x_m * 0.25) / 0.375;
	} else {
		tmp = 2.6666666666666665 * (Math.pow(Math.sin((x_m * 0.5)), 2.0) / Math.sin(x_m));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1e-9:
		tmp = (x_m * 0.25) / 0.375
	else:
		tmp = 2.6666666666666665 * (math.pow(math.sin((x_m * 0.5)), 2.0) / math.sin(x_m))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1e-9)
		tmp = Float64(Float64(x_m * 0.25) / 0.375);
	else
		tmp = Float64(2.6666666666666665 * Float64((sin(Float64(x_m * 0.5)) ^ 2.0) / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1e-9)
		tmp = (x_m * 0.25) / 0.375;
	else
		tmp = 2.6666666666666665 * ((sin((x_m * 0.5)) ^ 2.0) / sin(x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1e-9], N[(N[(x$95$m * 0.25), $MachinePrecision] / 0.375), $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 < 1.00000000000000006e-9

   1. Initial program 69.5%

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

      1. *-commutative 69.5%

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

      2. remove-double-neg 69.5%

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

      3. sin-neg 69.5%

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

      4. distribute-lft-neg-out 69.5%

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

      5. distribute-rgt-neg-in 69.5%

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

      6. associate-*l/ 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-rgt-neg-in 99.5%

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

      9. distribute-lft-neg-out 99.5%

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

      10. sin-neg 99.5%

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

      11. remove-double-neg 99.5%

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

      12. associate-*l* 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*r/ 69.4%

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

      3. associate-/r/ 69.4%

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

      4. div-inv 69.5%

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

      5. associate-/r* 69.6%

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

      6. pow2 69.6%

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

      7. metadata-eval 69.6%

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

   6. Applied egg-rr 69.6%

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

   7. Taylor expanded in x around 0 65.9%

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

   if 1.00000000000000006e-9 < x

   1. Initial program 99.2%

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

      1. associate-/l* 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l/ 99.1%

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

      4. metadata-eval 99.1%

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

      5. metadata-eval 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. times-frac 99.3%

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

      9. *-commutative 99.3%

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

      10. times-frac 99.2%

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

      11. associate-/l* 99.2%

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

      12. *-commutative 99.2%

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

      13. neg-mul-1 99.2%

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

      14. sin-neg 99.2%

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

      15. distribute-lft-neg-out 99.2%

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

      16. associate-*l/ 99.3%

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

   3. Simplified 99.3%

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

      1. div-inv 99.2%

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

      2. clear-num 99.3%

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

      3. associate-*r* 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. *-commutative 99.2%

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

      5. associate-*r/ 99.3%

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

      6. pow2 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-9}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 6 \cdot 10^{-12}:\\ \;\;\;\;\frac{x_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x_m \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 6e-12)
    (/ (* x_m 0.25) 0.375)
    (* (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 <= 6e-12) {
		tmp = (x_m * 0.25) / 0.375;
	} else {
		tmp = pow(sin((x_m * 0.5)), 2.0) * (2.6666666666666665 / sin(x_m));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 6d-12) then
        tmp = (x_m * 0.25d0) / 0.375d0
    else
        tmp = (sin((x_m * 0.5d0)) ** 2.0d0) * (2.6666666666666665d0 / sin(x_m))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 6e-12) {
		tmp = (x_m * 0.25) / 0.375;
	} else {
		tmp = Math.pow(Math.sin((x_m * 0.5)), 2.0) * (2.6666666666666665 / Math.sin(x_m));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 6e-12:
		tmp = (x_m * 0.25) / 0.375
	else:
		tmp = math.pow(math.sin((x_m * 0.5)), 2.0) * (2.6666666666666665 / math.sin(x_m))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 6e-12)
		tmp = Float64(Float64(x_m * 0.25) / 0.375);
	else
		tmp = Float64((sin(Float64(x_m * 0.5)) ^ 2.0) * Float64(2.6666666666666665 / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 6e-12)
		tmp = (x_m * 0.25) / 0.375;
	else
		tmp = (sin((x_m * 0.5)) ^ 2.0) * (2.6666666666666665 / sin(x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 6e-12], N[(N[(x$95$m * 0.25), $MachinePrecision] / 0.375), $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 < 6.0000000000000003e-12

   1. Initial program 69.4%

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

      1. *-commutative 69.4%

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

      2. remove-double-neg 69.4%

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

      3. sin-neg 69.4%

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

      4. distribute-lft-neg-out 69.4%

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

      5. distribute-rgt-neg-in 69.4%

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

      6. associate-*l/ 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-rgt-neg-in 99.5%

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

      9. distribute-lft-neg-out 99.5%

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

      10. sin-neg 99.5%

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

      11. remove-double-neg 99.5%

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

      12. associate-*l* 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*r/ 69.3%

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

      3. associate-/r/ 69.2%

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

      4. div-inv 69.3%

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

      5. associate-/r* 69.4%

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

      6. pow2 69.4%

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

      7. metadata-eval 69.4%

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

   6. Applied egg-rr 69.4%

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

   7. Taylor expanded in x around 0 65.7%

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

   if 6.0000000000000003e-12 < x

   1. Initial program 99.2%

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

      1. associate-/l* 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l/ 99.1%

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

      4. metadata-eval 99.1%

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

      5. metadata-eval 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. times-frac 99.3%

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

      9. *-commutative 99.3%

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

      10. times-frac 99.2%

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

      11. associate-/l* 99.2%

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

      12. *-commutative 99.2%

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

      13. neg-mul-1 99.2%

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

      14. sin-neg 99.2%

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

      15. distribute-lft-neg-out 99.2%

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

      16. associate-*l/ 99.3%

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

   3. Simplified 99.3%

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

      1. associate-/r/ 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-/l* 99.2%

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

      4. associate-/r/ 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*r* 99.2%

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

      7. pow2 99.2%

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

   6. Applied egg-rr 99.2%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. associate-/l* 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*l/ 99.4%

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

   4. metadata-eval 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. times-frac 99.4%

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

   9. *-commutative 99.4%

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

   10. times-frac 99.4%

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

   11. associate-/l* 99.4%

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

   12. *-commutative 99.4%

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

   13. neg-mul-1 99.4%

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

   14. sin-neg 99.4%

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

   15. distribute-lft-neg-out 99.4%

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

   16. associate-*l/ 99.4%

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

3. Simplified 99.4%

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

   1. associate-/r/ 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*l/ 99.4%

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

   4. associate-/r/ 99.4%

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

   5. associate-*l/ 75.4%

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

   6. div-inv 75.5%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. *-commutative 75.5%

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

   2. remove-double-neg 75.5%

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

   3. sin-neg 75.5%

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

   4. distribute-lft-neg-out 75.5%

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

   5. distribute-rgt-neg-in 75.5%

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

   6. associate-*l/ 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-rgt-neg-in 99.4%

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

   9. distribute-lft-neg-out 99.4%

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

   10. sin-neg 99.4%

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

   11. remove-double-neg 99.4%

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

   12. associate-*l* 99.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 6: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.000165:\\ \;\;\;\;\frac{x_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x_m}{0.5 - \frac{\cos x_m}{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.000165)
    (/ (* x_m 0.25) 0.375)
    (/ 2.6666666666666665 (/ (sin x_m) (- 0.5 (/ (cos x_m) 2.0)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.000165) {
		tmp = (x_m * 0.25) / 0.375;
	} else {
		tmp = 2.6666666666666665 / (sin(x_m) / (0.5 - (cos(x_m) / 2.0)));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.000165d0) then
        tmp = (x_m * 0.25d0) / 0.375d0
    else
        tmp = 2.6666666666666665d0 / (sin(x_m) / (0.5d0 - (cos(x_m) / 2.0d0)))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.000165) {
		tmp = (x_m * 0.25) / 0.375;
	} else {
		tmp = 2.6666666666666665 / (Math.sin(x_m) / (0.5 - (Math.cos(x_m) / 2.0)));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.000165:
		tmp = (x_m * 0.25) / 0.375
	else:
		tmp = 2.6666666666666665 / (math.sin(x_m) / (0.5 - (math.cos(x_m) / 2.0)))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.000165)
		tmp = Float64(Float64(x_m * 0.25) / 0.375);
	else
		tmp = Float64(2.6666666666666665 / Float64(sin(x_m) / Float64(0.5 - Float64(cos(x_m) / 2.0))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.000165)
		tmp = (x_m * 0.25) / 0.375;
	else
		tmp = 2.6666666666666665 / (sin(x_m) / (0.5 - (cos(x_m) / 2.0)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.000165], N[(N[(x$95$m * 0.25), $MachinePrecision] / 0.375), $MachinePrecision], N[(2.6666666666666665 / N[(N[Sin[x$95$m], $MachinePrecision] / N[(0.5 - N[(N[Cos[x$95$m], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.65e-4

   1. Initial program 69.5%

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

      1. *-commutative 69.5%

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

      2. remove-double-neg 69.5%

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

      3. sin-neg 69.5%

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

      4. distribute-lft-neg-out 69.5%

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

      5. distribute-rgt-neg-in 69.5%

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

      6. associate-*l/ 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-rgt-neg-in 99.5%

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

      9. distribute-lft-neg-out 99.5%

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

      10. sin-neg 99.5%

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

      11. remove-double-neg 99.5%

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

      12. associate-*l* 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*r/ 69.4%

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

      3. associate-/r/ 69.4%

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

      4. div-inv 69.5%

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

      5. associate-/r* 69.6%

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

      6. pow2 69.6%

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

      7. metadata-eval 69.6%

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

   6. Applied egg-rr 69.6%

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

   7. Taylor expanded in x around 0 65.9%

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

   if 1.65e-4 < x

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. remove-double-neg 99.2%

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

      3. sin-neg 99.2%

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

      4. distribute-lft-neg-out 99.2%

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

      5. distribute-rgt-neg-in 99.2%

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

      6. associate-*l/ 99.3%

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

      7. *-commutative 99.3%

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

      8. distribute-rgt-neg-in 99.3%

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

      9. distribute-lft-neg-out 99.3%

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

      10. sin-neg 99.3%

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

      11. remove-double-neg 99.3%

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

      12. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

      1. associate-*r/ 99.3%

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

      2. associate-*r/ 99.1%

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

      3. associate-/l* 99.3%

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

      4. pow2 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. unpow2 99.3%

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

      2. sin-mult 97.9%

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

   8. Applied egg-rr 97.9%

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

      1. div-sub 97.9%

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

      2. +-inverses 97.9%

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

      3. cos-0 97.9%

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

      4. metadata-eval 97.9%

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

      5. distribute-lft-out 97.9%

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

      6. metadata-eval 97.9%

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

      7. *-rgt-identity 97.9%

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

   10. Simplified 97.9%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000165:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\frac{\sin x}{0.5 - \frac{\cos x}{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.000165:\\ \;\;\;\;\frac{x_m \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 - \frac{\cos x_m}{2}}{\sin x_m}}{0.375}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.000165)
    (/ (* x_m 0.25) 0.375)
    (/ (/ (- 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 <= 0.000165) {
		tmp = (x_m * 0.25) / 0.375;
	} else {
		tmp = ((0.5 - (cos(x_m) / 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 <= 0.000165d0) then
        tmp = (x_m * 0.25d0) / 0.375d0
    else
        tmp = ((0.5d0 - (cos(x_m) / 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 <= 0.000165) {
		tmp = (x_m * 0.25) / 0.375;
	} else {
		tmp = ((0.5 - (Math.cos(x_m) / 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 <= 0.000165:
		tmp = (x_m * 0.25) / 0.375
	else:
		tmp = ((0.5 - (math.cos(x_m) / 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 <= 0.000165)
		tmp = Float64(Float64(x_m * 0.25) / 0.375);
	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

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.000165)
		tmp = (x_m * 0.25) / 0.375;
	else
		tmp = ((0.5 - (cos(x_m) / 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, 0.000165], N[(N[(x$95$m * 0.25), $MachinePrecision] / 0.375), $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 < 1.65e-4

   1. Initial program 69.5%

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

      1. *-commutative 69.5%

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

      2. remove-double-neg 69.5%

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

      3. sin-neg 69.5%

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

      4. distribute-lft-neg-out 69.5%

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

      5. distribute-rgt-neg-in 69.5%

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

      6. associate-*l/ 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-rgt-neg-in 99.5%

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

      9. distribute-lft-neg-out 99.5%

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

      10. sin-neg 99.5%

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

      11. remove-double-neg 99.5%

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

      12. associate-*l* 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*r/ 69.4%

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

      3. associate-/r/ 69.4%

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

      4. div-inv 69.5%

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

      5. associate-/r* 69.6%

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

      6. pow2 69.6%

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

      7. metadata-eval 69.6%

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

   6. Applied egg-rr 69.6%

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

   7. Taylor expanded in x around 0 65.9%

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

   if 1.65e-4 < x

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. remove-double-neg 99.2%

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

      3. sin-neg 99.2%

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

      4. distribute-lft-neg-out 99.2%

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

      5. distribute-rgt-neg-in 99.2%

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

      6. associate-*l/ 99.3%

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

      7. *-commutative 99.3%

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

      8. distribute-rgt-neg-in 99.3%

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

      9. distribute-lft-neg-out 99.3%

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

      10. sin-neg 99.3%

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

      11. remove-double-neg 99.3%

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

      12. associate-*l* 99.2%

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

   3. 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. *-commutative 99.2%

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

      2. associate-*r/ 99.3%

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

      3. associate-/r/ 99.1%

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

      4. div-inv 99.0%

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

      5. associate-/r* 99.1%

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

      6. pow2 99.1%

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

      7. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. unpow2 99.3%

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

      2. sin-mult 97.9%

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

   8. Applied egg-rr 98.1%

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

      1. div-sub 97.9%

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

      2. +-inverses 97.9%

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

      3. cos-0 97.9%

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

      4. metadata-eval 97.9%

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

      5. distribute-lft-out 97.9%

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

      6. metadata-eval 97.9%

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

      7. *-rgt-identity 97.9%

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

   10. Simplified 98.1%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000165:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 - \frac{\cos x}{2}}{\sin x}}{0.375}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. associate-/l* 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*l/ 99.4%

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

   4. metadata-eval 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. times-frac 99.4%

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

   9. *-commutative 99.4%

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

   10. times-frac 99.4%

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

   11. associate-/l* 99.4%

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

   12. *-commutative 99.4%

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

   13. neg-mul-1 99.4%

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

   14. sin-neg 99.4%

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

   15. distribute-lft-neg-out 99.4%

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

   16. associate-*l/ 99.4%

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

3. Simplified 99.4%

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

   1. associate-/r/ 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*l/ 99.4%

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

   4. associate-/r/ 99.4%

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

   5. associate-*l/ 75.4%

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

   6. div-inv 75.5%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in x around 0 56.7%

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

8. Final simplification 56.7%

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

Alternative 9: 56.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. *-commutative 75.5%

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

   2. remove-double-neg 75.5%

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

   3. sin-neg 75.5%

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

   4. distribute-lft-neg-out 75.5%

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

   5. distribute-rgt-neg-in 75.5%

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

   6. associate-*r/ 99.4%

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

   7. neg-mul-1 99.4%

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

   8. *-commutative 99.4%

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

   9. associate-/l* 99.4%

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

   10. associate-/r/ 99.4%

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

   11. *-commutative 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 56.6%

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

6. Final simplification 56.6%

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

Alternative 10: 52.4% accurate, 34.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \frac{2.6666666666666665}{x_m \cdot -0.3333333333333333 + \frac{4}{x_m}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ 2.6666666666666665 (+ (* x_m -0.3333333333333333) (/ 4.0 x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (2.6666666666666665 / ((x_m * -0.3333333333333333) + (4.0 / 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 * (2.6666666666666665d0 / ((x_m * (-0.3333333333333333d0)) + (4.0d0 / 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 * (2.6666666666666665 / ((x_m * -0.3333333333333333) + (4.0 / x_m)));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (2.6666666666666665 / ((x_m * -0.3333333333333333) + (4.0 / x_m)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(2.6666666666666665 / Float64(Float64(x_m * -0.3333333333333333) + Float64(4.0 / 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 * (2.6666666666666665 / ((x_m * -0.3333333333333333) + (4.0 / 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[(2.6666666666666665 / N[(N[(x$95$m * -0.3333333333333333), $MachinePrecision] + N[(4.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.5%

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

   1. *-commutative 75.5%

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

   2. remove-double-neg 75.5%

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

   3. sin-neg 75.5%

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

   4. distribute-lft-neg-out 75.5%

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

   5. distribute-rgt-neg-in 75.5%

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

   6. associate-*l/ 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-rgt-neg-in 99.4%

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

   9. distribute-lft-neg-out 99.4%

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

   10. sin-neg 99.4%

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

   11. remove-double-neg 99.4%

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

   12. associate-*l* 99.4%

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

3. Simplified 99.4%

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

   1. associate-*r/ 75.5%

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

   2. associate-*r/ 75.5%

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

   3. associate-/l* 75.5%

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

   4. pow2 75.5%

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

6. Applied egg-rr 75.5%

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

7. Taylor expanded in x around 0 53.4%

   \[\leadsto \frac{2.6666666666666665}{\color{blue}{-0.3333333333333333 \cdot x + \left(-0.005555555555555556 \cdot {x}^{3} + 4 \cdot \frac{1}{x}\right)}} \]

8. Taylor expanded in x around 0 53.5%

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

9. Final simplification 53.5%

   \[\leadsto \frac{2.6666666666666665}{x \cdot -0.3333333333333333 + \frac{4}{x}} \]
10. Add Preprocessing

Alternative 11: 52.1% accurate, 62.6× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (0.5 / (0.75 / 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 * (0.5d0 / (0.75d0 / 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 * (0.5 / (0.75 / x_m));
}

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(0.5 / Float64(0.75 / 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 * (0.5 / (0.75 / 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[(0.5 / N[(0.75 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.5%

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

   1. associate-/l* 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*l/ 99.4%

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

   4. metadata-eval 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. times-frac 99.4%

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

   9. *-commutative 99.4%

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

   10. times-frac 99.4%

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

   11. associate-/l* 99.4%

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

   12. *-commutative 99.4%

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

   13. neg-mul-1 99.4%

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

   14. sin-neg 99.4%

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

   15. distribute-lft-neg-out 99.4%

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

   16. associate-*l/ 99.4%

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

3. Simplified 99.4%

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

   1. associate-/r/ 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*l/ 99.4%

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

   4. associate-/r/ 99.4%

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

   5. associate-*l/ 75.4%

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

   6. div-inv 75.5%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

   1. clear-num 99.4%

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

   2. un-div-inv 99.4%

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

8. Applied egg-rr 99.4%

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

9. Taylor expanded in x around 0 56.6%

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

10. Taylor expanded in x around 0 53.2%

    \[\leadsto \frac{0.5}{\color{blue}{\frac{0.75}{x}}} \]

11. Final simplification 53.2%

    \[\leadsto \frac{0.5}{\frac{0.75}{x}} \]
12. Add Preprocessing

Alternative 12: 52.3% 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 1 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 75.5%

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

   1. *-commutative 75.5%

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

   2. remove-double-neg 75.5%

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

   3. sin-neg 75.5%

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

   4. distribute-lft-neg-out 75.5%

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

   5. distribute-rgt-neg-in 75.5%

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

   6. associate-*l/ 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-rgt-neg-in 99.4%

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

   9. distribute-lft-neg-out 99.4%

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

   10. sin-neg 99.4%

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

   11. remove-double-neg 99.4%

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

   12. associate-*l* 99.4%

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

3. Simplified 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*r/ 75.5%

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

   3. associate-/r/ 75.4%

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

   4. div-inv 75.5%

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

   5. associate-/r* 75.6%

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

   6. pow2 75.6%

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

   7. metadata-eval 75.6%

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

6. Applied egg-rr 75.6%

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

7. Taylor expanded in x around 0 53.4%

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

8. Final simplification 53.4%

   \[\leadsto \frac{x \cdot 0.25}{0.375} \]
9. Add Preprocessing

Alternative 13: 52.0% accurate, 104.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. associate-/l* 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*l/ 99.4%

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

   4. metadata-eval 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. times-frac 99.4%

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

   9. *-commutative 99.4%

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

   10. times-frac 99.4%

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

   11. associate-/l* 99.4%

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

   12. *-commutative 99.4%

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

   13. neg-mul-1 99.4%

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

   14. sin-neg 99.4%

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

   15. distribute-lft-neg-out 99.4%

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

   16. associate-*l/ 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 53.2%

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

   1. *-commutative 53.2%

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

7. Simplified 53.2%

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

8. Final simplification 53.2%

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

Developer target: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024017 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

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

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