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

Percentage Accurate: 77.7% → 99.5%

Time: 13.8s

Alternatives: 14

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.7% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x\_m \cdot 0.5\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-9}:\\ \;\;\;\;\frac{t\_0}{0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.375 \cdot \frac{\sin x\_m}{{t\_0}^{2}}}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5))))
   (*
    x_s
    (if (<= x_m 1e-9)
      (/ t_0 0.75)
      (/ 1.0 (* 0.375 (/ (sin x_m) (pow t_0 2.0))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((x_m * 0.5));
	double tmp;
	if (x_m <= 1e-9) {
		tmp = t_0 / 0.75;
	} else {
		tmp = 1.0 / (0.375 * (sin(x_m) / pow(t_0, 2.0)));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x_m * 0.5d0))
    if (x_m <= 1d-9) then
        tmp = t_0 / 0.75d0
    else
        tmp = 1.0d0 / (0.375d0 * (sin(x_m) / (t_0 ** 2.0d0)))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin((x_m * 0.5));
	double tmp;
	if (x_m <= 1e-9) {
		tmp = t_0 / 0.75;
	} else {
		tmp = 1.0 / (0.375 * (Math.sin(x_m) / Math.pow(t_0, 2.0)));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin((x_m * 0.5))
	tmp = 0
	if x_m <= 1e-9:
		tmp = t_0 / 0.75
	else:
		tmp = 1.0 / (0.375 * (math.sin(x_m) / math.pow(t_0, 2.0)))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = sin(Float64(x_m * 0.5))
	tmp = 0.0
	if (x_m <= 1e-9)
		tmp = Float64(t_0 / 0.75);
	else
		tmp = Float64(1.0 / Float64(0.375 * Float64(sin(x_m) / (t_0 ^ 2.0))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = sin((x_m * 0.5));
	tmp = 0.0;
	if (x_m <= 1e-9)
		tmp = t_0 / 0.75;
	else
		tmp = 1.0 / (0.375 * (sin(x_m) / (t_0 ^ 2.0)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 1e-9], N[(t$95$0 / 0.75), $MachinePrecision], N[(1.0 / N[(0.375 * N[(N[Sin[x$95$m], $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000006e-9

   1. Initial program 62.8%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.3%

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

      2. *-commutative 99.3%

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

      3. div-inv 99.1%

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

      4. associate-*l* 99.1%

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

      5. associate-/r/ 99.1%

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

      6. un-div-inv 99.3%

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

      7. *-un-lft-identity 99.3%

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

      8. times-frac 99.7%

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

      9. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 76.8%

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

   if 1.00000000000000006e-9 < x

   1. Initial program 98.9%

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

      1. associate-/l* 99.0%

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

      2. associate-*l* 98.8%

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

      3. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

      1. associate-*r* 99.0%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. clear-num 98.7%

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

      5. *-un-lft-identity 98.7%

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

      6. metadata-eval 98.7%

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

      7. associate-*l* 98.9%

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

      8. times-frac 99.0%

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

      9. metadata-eval 99.0%

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

      10. pow2 99.0%

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

   6. Applied egg-rr 99.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-9}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.375 \cdot \frac{\sin x}{{\sin \left(x \cdot 0.5\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x\_m \cdot 0.5\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_0}{0.75 + {x\_m}^{2} \cdot -0.09375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \left({t\_0}^{2} \cdot \frac{1}{\sin x\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5))))
   (*
    x_s
    (if (<= x_m 2e-7)
      (/ t_0 (+ 0.75 (* (pow x_m 2.0) -0.09375)))
      (* 2.6666666666666665 (* (pow t_0 2.0) (/ 1.0 (sin x_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((x_m * 0.5));
	double tmp;
	if (x_m <= 2e-7) {
		tmp = t_0 / (0.75 + (pow(x_m, 2.0) * -0.09375));
	} else {
		tmp = 2.6666666666666665 * (pow(t_0, 2.0) * (1.0 / sin(x_m)));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x_m * 0.5d0))
    if (x_m <= 2d-7) then
        tmp = t_0 / (0.75d0 + ((x_m ** 2.0d0) * (-0.09375d0)))
    else
        tmp = 2.6666666666666665d0 * ((t_0 ** 2.0d0) * (1.0d0 / sin(x_m)))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin((x_m * 0.5));
	double tmp;
	if (x_m <= 2e-7) {
		tmp = t_0 / (0.75 + (Math.pow(x_m, 2.0) * -0.09375));
	} else {
		tmp = 2.6666666666666665 * (Math.pow(t_0, 2.0) * (1.0 / Math.sin(x_m)));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin((x_m * 0.5))
	tmp = 0
	if x_m <= 2e-7:
		tmp = t_0 / (0.75 + (math.pow(x_m, 2.0) * -0.09375))
	else:
		tmp = 2.6666666666666665 * (math.pow(t_0, 2.0) * (1.0 / math.sin(x_m)))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = sin(Float64(x_m * 0.5))
	tmp = 0.0
	if (x_m <= 2e-7)
		tmp = Float64(t_0 / Float64(0.75 + Float64((x_m ^ 2.0) * -0.09375)));
	else
		tmp = Float64(2.6666666666666665 * Float64((t_0 ^ 2.0) * Float64(1.0 / sin(x_m))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = sin((x_m * 0.5));
	tmp = 0.0;
	if (x_m <= 2e-7)
		tmp = t_0 / (0.75 + ((x_m ^ 2.0) * -0.09375));
	else
		tmp = 2.6666666666666665 * ((t_0 ^ 2.0) * (1.0 / sin(x_m)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 2e-7], N[(t$95$0 / N[(0.75 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.09375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(1.0 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e-7

   1. Initial program 62.8%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.3%

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

      2. *-commutative 99.3%

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

      3. div-inv 99.1%

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

      4. associate-*l* 99.1%

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

      5. associate-/r/ 99.1%

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

      6. un-div-inv 99.3%

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

      7. *-un-lft-identity 99.3%

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

      8. times-frac 99.7%

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

      9. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 74.5%

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

      1. *-commutative 74.5%

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

   9. Simplified 74.5%

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

   if 1.9999999999999999e-7 < x

   1. Initial program 98.9%

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

      1. metadata-eval 98.9%

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

      2. add-sqr-sqrt 98.6%

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

      3. pow2 98.6%

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

      4. associate-*l* 98.7%

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

      5. sqrt-prod 98.8%

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

      6. sqrt-unprod 60.6%

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

      7. add-sqr-sqrt 98.8%

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

   4. Applied egg-rr 98.8%

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

      1. div-inv 98.7%

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

      2. *-commutative 98.7%

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

      3. *-commutative 98.7%

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

      4. unpow-prod-down 99.0%

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

      5. pow2 99.0%

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

      6. rem-square-sqrt 99.0%

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

      7. associate-*r* 99.0%

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

   6. Applied egg-rr 99.0%

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

4. Final simplification 82.4%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x\_m \cdot 0.5\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_0}{0.75 + {x\_m}^{2} \cdot -0.09375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{t\_0}^{2}}{\sin x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5))))
   (*
    x_s
    (if (<= x_m 2e-7)
      (/ t_0 (+ 0.75 (* (pow x_m 2.0) -0.09375)))
      (* 2.6666666666666665 (/ (pow t_0 2.0) (sin x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((x_m * 0.5));
	double tmp;
	if (x_m <= 2e-7) {
		tmp = t_0 / (0.75 + (pow(x_m, 2.0) * -0.09375));
	} else {
		tmp = 2.6666666666666665 * (pow(t_0, 2.0) / sin(x_m));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x_m * 0.5d0))
    if (x_m <= 2d-7) then
        tmp = t_0 / (0.75d0 + ((x_m ** 2.0d0) * (-0.09375d0)))
    else
        tmp = 2.6666666666666665d0 * ((t_0 ** 2.0d0) / sin(x_m))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin((x_m * 0.5));
	double tmp;
	if (x_m <= 2e-7) {
		tmp = t_0 / (0.75 + (Math.pow(x_m, 2.0) * -0.09375));
	} else {
		tmp = 2.6666666666666665 * (Math.pow(t_0, 2.0) / Math.sin(x_m));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin((x_m * 0.5))
	tmp = 0
	if x_m <= 2e-7:
		tmp = t_0 / (0.75 + (math.pow(x_m, 2.0) * -0.09375))
	else:
		tmp = 2.6666666666666665 * (math.pow(t_0, 2.0) / math.sin(x_m))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = sin(Float64(x_m * 0.5))
	tmp = 0.0
	if (x_m <= 2e-7)
		tmp = Float64(t_0 / Float64(0.75 + Float64((x_m ^ 2.0) * -0.09375)));
	else
		tmp = Float64(2.6666666666666665 * Float64((t_0 ^ 2.0) / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = sin((x_m * 0.5));
	tmp = 0.0;
	if (x_m <= 2e-7)
		tmp = t_0 / (0.75 + ((x_m ^ 2.0) * -0.09375));
	else
		tmp = 2.6666666666666665 * ((t_0 ^ 2.0) / sin(x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 2e-7], N[(t$95$0 / N[(0.75 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.09375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e-7

   1. Initial program 62.8%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.3%

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

      2. *-commutative 99.3%

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

      3. div-inv 99.1%

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

      4. associate-*l* 99.1%

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

      5. associate-/r/ 99.1%

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

      6. un-div-inv 99.3%

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

      7. *-un-lft-identity 99.3%

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

      8. times-frac 99.7%

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

      9. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 74.5%

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

      1. *-commutative 74.5%

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

   9. Simplified 74.5%

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

   if 1.9999999999999999e-7 < x

   1. Initial program 98.9%

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

      1. metadata-eval 98.9%

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

      2. associate-*r/ 99.0%

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

      3. associate-*r* 98.8%

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

      4. *-commutative 98.8%

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

      5. associate-*r/ 99.0%

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

      6. pow2 99.0%

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

   4. Applied egg-rr 99.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + {x}^{2} \cdot -0.09375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0044:\\ \;\;\;\;\frac{\sin \left(x\_m \cdot 0.5\right)}{0.75 + {x\_m}^{2} \cdot -0.09375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x\_m} \cdot \mathsf{fma}\left(\cos x\_m, -0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0044)
    (/ (sin (* x_m 0.5)) (+ 0.75 (* (pow x_m 2.0) -0.09375)))
    (* (/ 2.6666666666666665 (sin x_m)) (fma (cos x_m) -0.5 0.5)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0044) {
		tmp = sin((x_m * 0.5)) / (0.75 + (pow(x_m, 2.0) * -0.09375));
	} else {
		tmp = (2.6666666666666665 / sin(x_m)) * fma(cos(x_m), -0.5, 0.5);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0044)
		tmp = Float64(sin(Float64(x_m * 0.5)) / Float64(0.75 + Float64((x_m ^ 2.0) * -0.09375)));
	else
		tmp = Float64(Float64(2.6666666666666665 / sin(x_m)) * fma(cos(x_m), -0.5, 0.5));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0044], N[(N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] / N[(0.75 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.09375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.6666666666666665 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x$95$m], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.00440000000000000027

   1. Initial program 63.2%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.3%

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

      2. *-commutative 99.3%

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

      3. div-inv 99.1%

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

      4. associate-*l* 99.1%

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

      5. associate-/r/ 99.1%

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

      6. un-div-inv 99.3%

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

      7. *-un-lft-identity 99.3%

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

      8. times-frac 99.7%

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

      9. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 74.8%

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

      1. *-commutative 74.8%

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

   9. Simplified 74.8%

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

   if 0.00440000000000000027 < x

   1. Initial program 98.9%

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

      1. clear-num 98.7%

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

      2. associate-/r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. associate-*l* 99.0%

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

      5. pow2 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. unpow2 99.0%

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

      2. sin-mult 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. div-sub 98.3%

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

      2. +-inverses 98.3%

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

      3. cos-0 98.3%

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

      4. metadata-eval 98.3%

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

      5. distribute-lft-out 98.3%

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

      6. metadata-eval 98.3%

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

      7. *-rgt-identity 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in x around inf 98.4%

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

       1. clear-num 98.2%

          \[\leadsto 2.6666666666666665 \cdot \color{blue}{\frac{1}{\frac{\sin x}{0.5 - 0.5 \cdot \cos x}}} \]

       2. un-div-inv 98.3%

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

       3. sub-neg 98.3%

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

       4. *-commutative 98.3%

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

       5. distribute-rgt-neg-in 98.3%

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

       6. metadata-eval 98.3%

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

   11. Applied egg-rr 98.3%

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

       1. associate-/r/ 98.4%

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

       2. +-commutative 98.4%

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

       3. fma-define 98.4%

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

   13. Simplified 98.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + {x}^{2} \cdot -0.09375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x} \cdot \mathsf{fma}\left(\cos x, -0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* 99.2%

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

   2. *-commutative 99.2%

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

   3. div-inv 99.1%

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

   4. associate-*l* 99.0%

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

   5. associate-/r/ 99.0%

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

   6. un-div-inv 99.2%

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

   7. *-un-lft-identity 99.2%

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

   8. times-frac 99.5%

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

   9. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 6: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 7: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. *-commutative 74.5%

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

   2. associate-/l* 99.2%

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

   3. remove-double-neg 99.2%

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

   4. sin-neg 99.2%

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

   5. distribute-lft-neg-out 99.2%

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

   6. distribute-rgt-neg-in 99.2%

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

   7. distribute-frac-neg 99.2%

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

   8. distribute-frac-neg2 99.2%

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

   9. neg-mul-1 99.2%

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

   10. associate-/r* 99.2%

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

3. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 8: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. /-rgt-identity 99.2%

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

   4. metadata-eval 99.2%

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

   5. distribute-neg-frac2 99.2%

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

   6. distribute-frac-neg 99.2%

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

   7. sin-neg 99.2%

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

   8. distribute-lft-neg-out 99.2%

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

   9. associate-*l/ 99.2%

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

   10. *-commutative 99.2%

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

3. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 9: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0044:\\ \;\;\;\;\frac{\sin \left(x\_m \cdot 0.5\right)}{0.75 + {x\_m}^{2} \cdot -0.09375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x\_m}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0044)
    (/ (sin (* x_m 0.5)) (+ 0.75 (* (pow x_m 2.0) -0.09375)))
    (* 2.6666666666666665 (/ (- 0.5 (* 0.5 (cos x_m))) (sin x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0044) {
		tmp = sin((x_m * 0.5)) / (0.75 + (pow(x_m, 2.0) * -0.09375));
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x_m))) / sin(x_m));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0044d0) then
        tmp = sin((x_m * 0.5d0)) / (0.75d0 + ((x_m ** 2.0d0) * (-0.09375d0)))
    else
        tmp = 2.6666666666666665d0 * ((0.5d0 - (0.5d0 * cos(x_m))) / sin(x_m))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0044) {
		tmp = Math.sin((x_m * 0.5)) / (0.75 + (Math.pow(x_m, 2.0) * -0.09375));
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * Math.cos(x_m))) / Math.sin(x_m));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.0044:
		tmp = math.sin((x_m * 0.5)) / (0.75 + (math.pow(x_m, 2.0) * -0.09375))
	else:
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * math.cos(x_m))) / math.sin(x_m))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0044)
		tmp = Float64(sin(Float64(x_m * 0.5)) / Float64(0.75 + Float64((x_m ^ 2.0) * -0.09375)));
	else
		tmp = Float64(2.6666666666666665 * Float64(Float64(0.5 - Float64(0.5 * cos(x_m))) / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.0044)
		tmp = sin((x_m * 0.5)) / (0.75 + ((x_m ^ 2.0) * -0.09375));
	else
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x_m))) / sin(x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0044], N[(N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision] / N[(0.75 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.09375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[(0.5 - N[(0.5 * N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.00440000000000000027

   1. Initial program 63.2%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.3%

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

      2. *-commutative 99.3%

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

      3. div-inv 99.1%

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

      4. associate-*l* 99.1%

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

      5. associate-/r/ 99.1%

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

      6. un-div-inv 99.3%

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

      7. *-un-lft-identity 99.3%

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

      8. times-frac 99.7%

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

      9. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 74.8%

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

      1. *-commutative 74.8%

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

   9. Simplified 74.8%

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

   if 0.00440000000000000027 < x

   1. Initial program 98.9%

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

      1. clear-num 98.7%

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

      2. associate-/r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. associate-*l* 99.0%

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

      5. pow2 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. unpow2 99.0%

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

      2. sin-mult 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. div-sub 98.3%

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

      2. +-inverses 98.3%

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

      3. cos-0 98.3%

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

      4. metadata-eval 98.3%

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

      5. distribute-lft-out 98.3%

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

      6. metadata-eval 98.3%

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

      7. *-rgt-identity 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in x around inf 98.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + {x}^{2} \cdot -0.09375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0044:\\ \;\;\;\;x\_m \cdot \left(0.6666666666666666 + {x\_m}^{2} \cdot 0.05555555555555555\right)\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x\_m}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.0044)
    (* x_m (+ 0.6666666666666666 (* (pow x_m 2.0) 0.05555555555555555)))
    (* 2.6666666666666665 (/ (- 0.5 (* 0.5 (cos x_m))) (sin x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0044) {
		tmp = x_m * (0.6666666666666666 + (pow(x_m, 2.0) * 0.05555555555555555));
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x_m))) / sin(x_m));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0044d0) then
        tmp = x_m * (0.6666666666666666d0 + ((x_m ** 2.0d0) * 0.05555555555555555d0))
    else
        tmp = 2.6666666666666665d0 * ((0.5d0 - (0.5d0 * cos(x_m))) / sin(x_m))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.0044) {
		tmp = x_m * (0.6666666666666666 + (Math.pow(x_m, 2.0) * 0.05555555555555555));
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * Math.cos(x_m))) / Math.sin(x_m));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.0044:
		tmp = x_m * (0.6666666666666666 + (math.pow(x_m, 2.0) * 0.05555555555555555))
	else:
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * math.cos(x_m))) / math.sin(x_m))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.0044)
		tmp = Float64(x_m * Float64(0.6666666666666666 + Float64((x_m ^ 2.0) * 0.05555555555555555)));
	else
		tmp = Float64(2.6666666666666665 * Float64(Float64(0.5 - Float64(0.5 * cos(x_m))) / sin(x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.0044)
		tmp = x_m * (0.6666666666666666 + ((x_m ^ 2.0) * 0.05555555555555555));
	else
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x_m))) / sin(x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.0044], N[(x$95$m * N[(0.6666666666666666 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.05555555555555555), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[(0.5 - N[(0.5 * N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.00440000000000000027

   1. Initial program 63.2%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 73.9%

      \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 + 0.05555555555555555 \cdot {x}^{2}\right)} \]

   if 0.00440000000000000027 < x

   1. Initial program 98.9%

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

      1. clear-num 98.7%

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

      2. associate-/r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. associate-*l* 99.0%

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

      5. pow2 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. unpow2 99.0%

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

      2. sin-mult 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. div-sub 98.3%

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

      2. +-inverses 98.3%

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

      3. cos-0 98.3%

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

      4. metadata-eval 98.3%

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

      5. distribute-lft-out 98.3%

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

      6. metadata-eval 98.3%

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

      7. *-rgt-identity 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in x around inf 98.4%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;x \cdot \left(0.6666666666666666 + {x}^{2} \cdot 0.05555555555555555\right)\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.005:\\ \;\;\;\;x\_m \cdot \left(0.6666666666666666 + {x\_m}^{2} \cdot 0.05555555555555555\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1.3333333333333333 + \cos x\_m \cdot -1.3333333333333333}{\sin x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.005)
    (* x_m (+ 0.6666666666666666 (* (pow x_m 2.0) 0.05555555555555555)))
    (/ (+ 1.3333333333333333 (* (cos x_m) -1.3333333333333333)) (sin x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.005) {
		tmp = x_m * (0.6666666666666666 + (pow(x_m, 2.0) * 0.05555555555555555));
	} else {
		tmp = (1.3333333333333333 + (cos(x_m) * -1.3333333333333333)) / sin(x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.005d0) then
        tmp = x_m * (0.6666666666666666d0 + ((x_m ** 2.0d0) * 0.05555555555555555d0))
    else
        tmp = (1.3333333333333333d0 + (cos(x_m) * (-1.3333333333333333d0))) / sin(x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.005) {
		tmp = x_m * (0.6666666666666666 + (Math.pow(x_m, 2.0) * 0.05555555555555555));
	} else {
		tmp = (1.3333333333333333 + (Math.cos(x_m) * -1.3333333333333333)) / Math.sin(x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.005:
		tmp = x_m * (0.6666666666666666 + (math.pow(x_m, 2.0) * 0.05555555555555555))
	else:
		tmp = (1.3333333333333333 + (math.cos(x_m) * -1.3333333333333333)) / math.sin(x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.005)
		tmp = Float64(x_m * Float64(0.6666666666666666 + Float64((x_m ^ 2.0) * 0.05555555555555555)));
	else
		tmp = Float64(Float64(1.3333333333333333 + Float64(cos(x_m) * -1.3333333333333333)) / sin(x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.005)
		tmp = x_m * (0.6666666666666666 + ((x_m ^ 2.0) * 0.05555555555555555));
	else
		tmp = (1.3333333333333333 + (cos(x_m) * -1.3333333333333333)) / sin(x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.005], N[(x$95$m * N[(0.6666666666666666 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.05555555555555555), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.3333333333333333 + N[(N[Cos[x$95$m], $MachinePrecision] * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.0050000000000000001

   1. Initial program 63.2%

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

      1. associate-/l* 99.3%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 73.9%

      \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 + 0.05555555555555555 \cdot {x}^{2}\right)} \]

   if 0.0050000000000000001 < x

   1. Initial program 98.9%

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

      1. clear-num 98.7%

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

      2. associate-/r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. associate-*l* 99.0%

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

      5. pow2 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. unpow2 99.0%

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

      2. sin-mult 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. div-sub 98.3%

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

      2. +-inverses 98.3%

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

      3. cos-0 98.3%

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

      4. metadata-eval 98.3%

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

      5. distribute-lft-out 98.3%

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

      6. metadata-eval 98.3%

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

      7. *-rgt-identity 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in x around inf 98.4%

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

       1. clear-num 98.2%

          \[\leadsto 2.6666666666666665 \cdot \color{blue}{\frac{1}{\frac{\sin x}{0.5 - 0.5 \cdot \cos x}}} \]

       2. un-div-inv 98.3%

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

       3. sub-neg 98.3%

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

       4. *-commutative 98.3%

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

       5. distribute-rgt-neg-in 98.3%

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

       6. metadata-eval 98.3%

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

   11. Applied egg-rr 98.3%

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

       1. associate-/r/ 98.4%

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

       2. associate-*l/ 98.5%

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

       3. associate-*r/ 98.4%

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

       4. remove-double-neg 98.4%

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

       5. distribute-neg-frac2 98.4%

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

       6. distribute-neg-frac 98.4%

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

       7. distribute-neg-in 98.4%

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

       8. metadata-eval 98.4%

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

       9. associate-/l* 98.5%

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

       10. distribute-lft-in 98.2%

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

       11. metadata-eval 98.2%

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

       12. distribute-rgt-neg-in 98.2%

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

       13. metadata-eval 98.2%

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

       14. distribute-neg-in 98.2%

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

       15. distribute-frac-neg 98.2%

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

       16. distribute-neg-frac2 98.2%

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

   13. Simplified 98.2%

       \[\leadsto \color{blue}{\frac{1.3333333333333333 + -1.3333333333333333 \cdot \cos x}{\sin x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;x \cdot \left(0.6666666666666666 + {x}^{2} \cdot 0.05555555555555555\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1.3333333333333333 + \cos x \cdot -1.3333333333333333}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. *-commutative 74.5%

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

   2. associate-/l* 99.2%

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

   3. remove-double-neg 99.2%

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

   4. sin-neg 99.2%

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

   5. distribute-lft-neg-out 99.2%

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

   6. distribute-rgt-neg-in 99.2%

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

   7. distribute-frac-neg 99.2%

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

   8. distribute-frac-neg2 99.2%

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

   9. neg-mul-1 99.2%

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

   10. associate-/r* 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 55.3%

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

6. Final simplification 55.3%

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

Alternative 13: 54.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* 99.2%

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

   2. *-commutative 99.2%

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

   3. div-inv 99.1%

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

   4. associate-*l* 99.0%

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

   5. associate-/r/ 99.0%

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

   6. un-div-inv 99.2%

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

   7. *-un-lft-identity 99.2%

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

   8. times-frac 99.5%

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

   9. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0 55.6%

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

8. Final simplification 55.6%

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

Alternative 14: 50.0% 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 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* x_m 0.6666666666666666)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 51.4%

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

6. Final simplification 51.4%

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

Developer target: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024115 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

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

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