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

Percentage Accurate: 76.9% → 99.5%

Time: 17.5s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{t_0}{\sin x} \cdot \frac{t_0}{0.375} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (* (/ t_0 (sin x)) (/ t_0 0.375))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return (t_0 / sin(x)) * (t_0 / 0.375);
}

Fortran

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

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return (t_0 / Math.sin(x)) * (t_0 / 0.375);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return (t_0 / math.sin(x)) * (t_0 / 0.375)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(t_0 / sin(x)) * Float64(t_0 / 0.375))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = (t_0 / sin(x)) * (t_0 / 0.375);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 / N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / 0.375), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 73.7%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.2%

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

   4. metadata-eval 99.2%

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

   5. metadata-eval 99.2%

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

   6. metadata-eval 99.2%

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

   7. metadata-eval 99.2%

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

   8. times-frac 99.2%

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

   9. *-commutative 99.2%

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

   10. times-frac 99.2%

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

   11. associate-/l* 99.2%

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

   12. *-commutative 99.2%

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

   13. neg-mul-1 99.2%

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

   14. sin-neg 99.2%

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

   15. distribute-lft-neg-out 99.2%

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

   16. associate-*l/ 99.2%

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

3. Simplified 99.2%

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

   1. associate-/r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.3%

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

   4. associate-/r/ 99.2%

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

   5. associate-*l/ 73.6%

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

   6. div-inv 73.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-9)
   (/ x 1.5)
   (* 2.6666666666666665 (/ (pow (sin (* x 0.5)) 2.0) (sin x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2e-9) {
		tmp = x / 1.5;
	} else {
		tmp = 2.6666666666666665 * (pow(sin((x * 0.5)), 2.0) / sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2d-9) then
        tmp = x / 1.5d0
    else
        tmp = 2.6666666666666665d0 * ((sin((x * 0.5d0)) ** 2.0d0) / sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e-9) {
		tmp = x / 1.5;
	} else {
		tmp = 2.6666666666666665 * (Math.pow(Math.sin((x * 0.5)), 2.0) / Math.sin(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2e-9:
		tmp = x / 1.5
	else:
		tmp = 2.6666666666666665 * (math.pow(math.sin((x * 0.5)), 2.0) / math.sin(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e-9)
		tmp = Float64(x / 1.5);
	else
		tmp = Float64(2.6666666666666665 * Float64((sin(Float64(x * 0.5)) ^ 2.0) / sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2e-9)
		tmp = x / 1.5;
	else
		tmp = 2.6666666666666665 * ((sin((x * 0.5)) ^ 2.0) / sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e-9], N[(x / 1.5), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000012e-9

   1. Initial program 63.4%

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

      1. *-commutative 63.4%

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

      2. remove-double-neg 63.4%

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

      3. sin-neg 63.4%

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

      4. distribute-lft-neg-out 63.4%

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

      5. distribute-rgt-neg-in 63.4%

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

      6. associate-*l/ 99.3%

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

      7. *-commutative 99.3%

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

      8. distribute-rgt-neg-in 99.3%

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

      9. distribute-lft-neg-out 99.3%

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

      10. sin-neg 99.3%

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

      11. remove-double-neg 99.3%

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

      12. associate-*l* 99.3%

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

   3. 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 74.7%

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

      1. add-cube-cbrt 73.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{0.6666666666666666 \cdot x} \cdot \sqrt[3]{0.6666666666666666 \cdot x}\right) \cdot \sqrt[3]{0.6666666666666666 \cdot x}} \]

      2. pow3 73.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{0.6666666666666666 \cdot x}\right)}^{3}} \]

      3. *-commutative 73.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot 0.6666666666666666}}\right)}^{3} \]

   7. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot 0.6666666666666666}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 74.7%

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

      2. metadata-eval 74.7%

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

      3. div-inv 75.2%

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

   9. Applied egg-rr 75.2%

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

   if 2.00000000000000012e-9 < x

   1. Initial program 99.0%

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

      1. associate-/l* 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l/ 98.9%

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

      4. metadata-eval 98.9%

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

      5. metadata-eval 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. times-frac 99.0%

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

      9. *-commutative 99.0%

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

      10. times-frac 99.0%

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

      11. associate-/l* 99.0%

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

      12. *-commutative 99.0%

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

      13. neg-mul-1 99.0%

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

      14. sin-neg 99.0%

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

      15. distribute-lft-neg-out 99.0%

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

      16. associate-*l/ 99.0%

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

   3. Simplified 99.0%

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

      1. div-inv 99.0%

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

      2. clear-num 99.1%

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

      3. associate-*r* 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-*r/ 99.1%

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

      6. pow2 99.1%

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-18}:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e-18)
   (/ x 1.5)
   (* (pow (sin (* x 0.5)) 2.0) (/ 2.6666666666666665 (sin x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1e-18) {
		tmp = x / 1.5;
	} else {
		tmp = pow(sin((x * 0.5)), 2.0) * (2.6666666666666665 / sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d-18) then
        tmp = x / 1.5d0
    else
        tmp = (sin((x * 0.5d0)) ** 2.0d0) * (2.6666666666666665d0 / sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e-18) {
		tmp = x / 1.5;
	} else {
		tmp = Math.pow(Math.sin((x * 0.5)), 2.0) * (2.6666666666666665 / Math.sin(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e-18:
		tmp = x / 1.5
	else:
		tmp = math.pow(math.sin((x * 0.5)), 2.0) * (2.6666666666666665 / math.sin(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e-18)
		tmp = Float64(x / 1.5);
	else
		tmp = Float64((sin(Float64(x * 0.5)) ^ 2.0) * Float64(2.6666666666666665 / sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e-18)
		tmp = x / 1.5;
	else
		tmp = (sin((x * 0.5)) ^ 2.0) * (2.6666666666666665 / sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e-18], N[(x / 1.5), $MachinePrecision], N[(N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0000000000000001e-18

   1. Initial program 63.4%

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

      1. *-commutative 63.4%

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

      2. remove-double-neg 63.4%

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

      3. sin-neg 63.4%

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

      4. distribute-lft-neg-out 63.4%

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

      5. distribute-rgt-neg-in 63.4%

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

      6. associate-*l/ 99.3%

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

      7. *-commutative 99.3%

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

      8. distribute-rgt-neg-in 99.3%

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

      9. distribute-lft-neg-out 99.3%

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

      10. sin-neg 99.3%

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

      11. remove-double-neg 99.3%

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

      12. associate-*l* 99.3%

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

   3. 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 74.7%

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

      1. add-cube-cbrt 73.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{0.6666666666666666 \cdot x} \cdot \sqrt[3]{0.6666666666666666 \cdot x}\right) \cdot \sqrt[3]{0.6666666666666666 \cdot x}} \]

      2. pow3 73.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{0.6666666666666666 \cdot x}\right)}^{3}} \]

      3. *-commutative 73.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot 0.6666666666666666}}\right)}^{3} \]

   7. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot 0.6666666666666666}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 74.7%

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

      2. metadata-eval 74.7%

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

      3. div-inv 75.2%

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

   9. Applied egg-rr 75.2%

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

   if 1.0000000000000001e-18 < x

   1. Initial program 99.0%

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

      1. associate-/l* 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l/ 98.9%

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

      4. metadata-eval 98.9%

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

      5. metadata-eval 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. times-frac 99.0%

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

      9. *-commutative 99.0%

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

      10. times-frac 99.0%

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

      11. associate-/l* 99.0%

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

      12. *-commutative 99.0%

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

      13. neg-mul-1 99.0%

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

      14. sin-neg 99.0%

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

      15. distribute-lft-neg-out 99.0%

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

      16. associate-*l/ 99.0%

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

   3. Simplified 99.0%

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

      1. associate-/r/ 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l/ 99.1%

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

      4. associate-/r/ 99.0%

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

      5. associate-*l/ 98.8%

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

      6. div-inv 98.8%

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

      7. clear-num 99.0%

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

      8. pow2 99.0%

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

   6. Applied egg-rr 99.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-18}:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-18}:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e-18)
   (/ x 1.5)
   (/ 2.6666666666666665 (* (sin x) (pow (sin (* x 0.5)) -2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1e-18) {
		tmp = x / 1.5;
	} else {
		tmp = 2.6666666666666665 / (sin(x) * pow(sin((x * 0.5)), -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d-18) then
        tmp = x / 1.5d0
    else
        tmp = 2.6666666666666665d0 / (sin(x) * (sin((x * 0.5d0)) ** (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e-18) {
		tmp = x / 1.5;
	} else {
		tmp = 2.6666666666666665 / (Math.sin(x) * Math.pow(Math.sin((x * 0.5)), -2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e-18:
		tmp = x / 1.5
	else:
		tmp = 2.6666666666666665 / (math.sin(x) * math.pow(math.sin((x * 0.5)), -2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e-18)
		tmp = Float64(x / 1.5);
	else
		tmp = Float64(2.6666666666666665 / Float64(sin(x) * (sin(Float64(x * 0.5)) ^ -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e-18)
		tmp = x / 1.5;
	else
		tmp = 2.6666666666666665 / (sin(x) * (sin((x * 0.5)) ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e-18], N[(x / 1.5), $MachinePrecision], N[(2.6666666666666665 / N[(N[Sin[x], $MachinePrecision] * N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0000000000000001e-18

   1. Initial program 63.4%

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

      1. *-commutative 63.4%

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

      2. remove-double-neg 63.4%

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

      3. sin-neg 63.4%

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

      4. distribute-lft-neg-out 63.4%

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

      5. distribute-rgt-neg-in 63.4%

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

      6. associate-*l/ 99.3%

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

      7. *-commutative 99.3%

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

      8. distribute-rgt-neg-in 99.3%

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

      9. distribute-lft-neg-out 99.3%

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

      10. sin-neg 99.3%

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

      11. remove-double-neg 99.3%

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

      12. associate-*l* 99.3%

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

   3. 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 74.7%

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

      1. add-cube-cbrt 73.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{0.6666666666666666 \cdot x} \cdot \sqrt[3]{0.6666666666666666 \cdot x}\right) \cdot \sqrt[3]{0.6666666666666666 \cdot x}} \]

      2. pow3 73.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{0.6666666666666666 \cdot x}\right)}^{3}} \]

      3. *-commutative 73.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot 0.6666666666666666}}\right)}^{3} \]

   7. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot 0.6666666666666666}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 74.7%

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

      2. metadata-eval 74.7%

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

      3. div-inv 75.2%

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

   9. Applied egg-rr 75.2%

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

   if 1.0000000000000001e-18 < x

   1. Initial program 99.0%

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

      1. associate-/l* 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l/ 98.9%

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

      4. metadata-eval 98.9%

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

      5. metadata-eval 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. times-frac 99.0%

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

      9. *-commutative 99.0%

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

      10. times-frac 99.0%

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

      11. associate-/l* 99.0%

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

      12. *-commutative 99.0%

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

      13. neg-mul-1 99.0%

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

      14. sin-neg 99.0%

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

      15. distribute-lft-neg-out 99.0%

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

      16. associate-*l/ 99.0%

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

   3. Simplified 99.0%

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

      1. div-inv 99.0%

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

      2. clear-num 99.1%

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

      3. associate-*r* 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-*r/ 99.1%

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

      6. pow2 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. *-commutative 99.1%

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

      2. clear-num 98.9%

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

      3. un-div-inv 98.9%

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

      4. div-inv 98.9%

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

      5. pow-flip 99.0%

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

      6. metadata-eval 99.0%

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

   8. Applied egg-rr 99.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-18}:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x \cdot {\sin \left(x \cdot 0.5\right)}^{-2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

   1. *-commutative 73.7%

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

   2. remove-double-neg 73.7%

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

   3. sin-neg 73.7%

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

   4. distribute-lft-neg-out 73.7%

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

   5. distribute-rgt-neg-in 73.7%

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

   6. associate-*l/ 99.2%

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

   7. *-commutative 99.2%

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

   8. distribute-rgt-neg-in 99.2%

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

   9. distribute-lft-neg-out 99.2%

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

   10. sin-neg 99.2%

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

   11. remove-double-neg 99.2%

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

   12. associate-*l* 99.2%

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

3. Simplified 99.2%

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

5. 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 6: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000165:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - \frac{\cos x}{2}}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000165)
   (/ x 1.5)
   (* 2.6666666666666665 (/ (- 0.5 (/ (cos x) 2.0)) (sin x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.000165) {
		tmp = x / 1.5;
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (cos(x) / 2.0)) / sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.000165d0) then
        tmp = x / 1.5d0
    else
        tmp = 2.6666666666666665d0 * ((0.5d0 - (cos(x) / 2.0d0)) / sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.000165) {
		tmp = x / 1.5;
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (Math.cos(x) / 2.0)) / Math.sin(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000165:
		tmp = x / 1.5
	else:
		tmp = 2.6666666666666665 * ((0.5 - (math.cos(x) / 2.0)) / math.sin(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000165)
		tmp = Float64(x / 1.5);
	else
		tmp = Float64(2.6666666666666665 * Float64(Float64(0.5 - Float64(cos(x) / 2.0)) / sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.000165)
		tmp = x / 1.5;
	else
		tmp = 2.6666666666666665 * ((0.5 - (cos(x) / 2.0)) / sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000165], N[(x / 1.5), $MachinePrecision], N[(2.6666666666666665 * N[(N[(0.5 - N[(N[Cos[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65e-4

   1. Initial program 63.4%

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

      1. *-commutative 63.4%

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

      2. remove-double-neg 63.4%

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

      3. sin-neg 63.4%

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

      4. distribute-lft-neg-out 63.4%

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

      5. distribute-rgt-neg-in 63.4%

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

      6. associate-*l/ 99.3%

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

      7. *-commutative 99.3%

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

      8. distribute-rgt-neg-in 99.3%

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

      9. distribute-lft-neg-out 99.3%

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

      10. sin-neg 99.3%

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

      11. remove-double-neg 99.3%

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

      12. associate-*l* 99.3%

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

   3. 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 74.7%

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

      1. add-cube-cbrt 73.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{0.6666666666666666 \cdot x} \cdot \sqrt[3]{0.6666666666666666 \cdot x}\right) \cdot \sqrt[3]{0.6666666666666666 \cdot x}} \]

      2. pow3 73.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{0.6666666666666666 \cdot x}\right)}^{3}} \]

      3. *-commutative 73.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot 0.6666666666666666}}\right)}^{3} \]

   7. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot 0.6666666666666666}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 74.7%

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

      2. metadata-eval 74.7%

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

      3. div-inv 75.2%

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

   9. Applied egg-rr 75.2%

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

   if 1.65e-4 < x

   1. Initial program 99.0%

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

      1. associate-/l* 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l/ 98.9%

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

      4. metadata-eval 98.9%

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

      5. metadata-eval 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. times-frac 99.0%

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

      9. *-commutative 99.0%

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

      10. times-frac 99.0%

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

      11. associate-/l* 99.0%

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

      12. *-commutative 99.0%

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

      13. neg-mul-1 99.0%

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

      14. sin-neg 99.0%

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

      15. distribute-lft-neg-out 99.0%

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

      16. associate-*l/ 99.0%

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

   3. Simplified 99.0%

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

      1. div-inv 99.0%

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

      2. clear-num 99.1%

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

      3. associate-*r* 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-*r/ 99.1%

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

      6. pow2 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. unpow2 99.0%

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

      2. sin-mult 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. div-sub 98.4%

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

      2. +-inverses 98.4%

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

      3. cos-0 98.4%

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

      4. metadata-eval 98.4%

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

      5. distribute-lft-out 98.4%

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

      6. metadata-eval 98.4%

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

      7. *-rgt-identity 98.4%

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

   10. Simplified 98.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000165:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - \frac{\cos x}{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000165:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{\cos x}{2}}{\sin x \cdot 0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000165) (/ x 1.5) (/ (- 0.5 (/ (cos x) 2.0)) (* (sin x) 0.375))))

C

double code(double x) {
	double tmp;
	if (x <= 0.000165) {
		tmp = x / 1.5;
	} else {
		tmp = (0.5 - (cos(x) / 2.0)) / (sin(x) * 0.375);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.000165d0) then
        tmp = x / 1.5d0
    else
        tmp = (0.5d0 - (cos(x) / 2.0d0)) / (sin(x) * 0.375d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.000165) {
		tmp = x / 1.5;
	} else {
		tmp = (0.5 - (Math.cos(x) / 2.0)) / (Math.sin(x) * 0.375);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000165:
		tmp = x / 1.5
	else:
		tmp = (0.5 - (math.cos(x) / 2.0)) / (math.sin(x) * 0.375)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000165)
		tmp = Float64(x / 1.5);
	else
		tmp = Float64(Float64(0.5 - Float64(cos(x) / 2.0)) / Float64(sin(x) * 0.375));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.000165)
		tmp = x / 1.5;
	else
		tmp = (0.5 - (cos(x) / 2.0)) / (sin(x) * 0.375);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000165], N[(x / 1.5), $MachinePrecision], N[(N[(0.5 - N[(N[Cos[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65e-4

   1. Initial program 63.4%

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

      1. *-commutative 63.4%

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

      2. remove-double-neg 63.4%

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

      3. sin-neg 63.4%

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

      4. distribute-lft-neg-out 63.4%

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

      5. distribute-rgt-neg-in 63.4%

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

      6. associate-*l/ 99.3%

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

      7. *-commutative 99.3%

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

      8. distribute-rgt-neg-in 99.3%

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

      9. distribute-lft-neg-out 99.3%

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

      10. sin-neg 99.3%

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

      11. remove-double-neg 99.3%

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

      12. associate-*l* 99.3%

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

   3. 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 74.7%

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

      1. add-cube-cbrt 73.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{0.6666666666666666 \cdot x} \cdot \sqrt[3]{0.6666666666666666 \cdot x}\right) \cdot \sqrt[3]{0.6666666666666666 \cdot x}} \]

      2. pow3 73.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{0.6666666666666666 \cdot x}\right)}^{3}} \]

      3. *-commutative 73.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot 0.6666666666666666}}\right)}^{3} \]

   7. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot 0.6666666666666666}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 74.7%

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

      2. metadata-eval 74.7%

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

      3. div-inv 75.2%

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

   9. Applied egg-rr 75.2%

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

   if 1.65e-4 < x

   1. Initial program 99.0%

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

      1. *-commutative 99.0%

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

      2. remove-double-neg 99.0%

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

      3. sin-neg 99.0%

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

      4. distribute-lft-neg-out 99.0%

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

      5. distribute-rgt-neg-in 99.0%

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

      6. associate-*l/ 99.1%

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

      7. *-commutative 99.1%

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

      8. distribute-rgt-neg-in 99.1%

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

      9. distribute-lft-neg-out 99.1%

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

      10. sin-neg 99.1%

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

      11. remove-double-neg 99.1%

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

      12. associate-*l* 99.1%

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

   3. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*r/ 99.1%

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

      3. associate-/r/ 98.8%

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

      4. div-inv 99.0%

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

      5. pow2 99.0%

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

      6. metadata-eval 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. unpow2 99.0%

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

      2. sin-mult 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. div-sub 98.4%

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

      2. +-inverses 98.4%

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

      3. cos-0 98.4%

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

      4. metadata-eval 98.4%

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

      5. distribute-lft-out 98.4%

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

      6. metadata-eval 98.4%

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

      7. *-rgt-identity 98.4%

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

   10. Simplified 98.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000165:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{\cos x}{2}}{\sin x \cdot 0.375}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00021:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.3333333333333333 - \cos x \cdot 1.3333333333333333}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00021)
   (/ x 1.5)
   (/ (- 1.3333333333333333 (* (cos x) 1.3333333333333333)) (sin x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.00021) {
		tmp = x / 1.5;
	} else {
		tmp = (1.3333333333333333 - (cos(x) * 1.3333333333333333)) / sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00021d0) then
        tmp = x / 1.5d0
    else
        tmp = (1.3333333333333333d0 - (cos(x) * 1.3333333333333333d0)) / sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.00021) {
		tmp = x / 1.5;
	} else {
		tmp = (1.3333333333333333 - (Math.cos(x) * 1.3333333333333333)) / Math.sin(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00021:
		tmp = x / 1.5
	else:
		tmp = (1.3333333333333333 - (math.cos(x) * 1.3333333333333333)) / math.sin(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00021)
		tmp = Float64(x / 1.5);
	else
		tmp = Float64(Float64(1.3333333333333333 - Float64(cos(x) * 1.3333333333333333)) / sin(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00021)
		tmp = x / 1.5;
	else
		tmp = (1.3333333333333333 - (cos(x) * 1.3333333333333333)) / sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00021], N[(x / 1.5), $MachinePrecision], N[(N[(1.3333333333333333 - N[(N[Cos[x], $MachinePrecision] * 1.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1000000000000001e-4

   1. Initial program 63.4%

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

      1. *-commutative 63.4%

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

      2. remove-double-neg 63.4%

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

      3. sin-neg 63.4%

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

      4. distribute-lft-neg-out 63.4%

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

      5. distribute-rgt-neg-in 63.4%

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

      6. associate-*l/ 99.3%

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

      7. *-commutative 99.3%

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

      8. distribute-rgt-neg-in 99.3%

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

      9. distribute-lft-neg-out 99.3%

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

      10. sin-neg 99.3%

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

      11. remove-double-neg 99.3%

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

      12. associate-*l* 99.3%

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

   3. 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 74.7%

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

      1. add-cube-cbrt 73.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{0.6666666666666666 \cdot x} \cdot \sqrt[3]{0.6666666666666666 \cdot x}\right) \cdot \sqrt[3]{0.6666666666666666 \cdot x}} \]

      2. pow3 73.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{0.6666666666666666 \cdot x}\right)}^{3}} \]

      3. *-commutative 73.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot 0.6666666666666666}}\right)}^{3} \]

   7. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot 0.6666666666666666}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 74.7%

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

      2. metadata-eval 74.7%

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

      3. div-inv 75.2%

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

   9. Applied egg-rr 75.2%

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

   if 2.1000000000000001e-4 < x

   1. Initial program 99.0%

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

      1. associate-/l* 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l/ 98.9%

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

      4. metadata-eval 98.9%

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

      5. metadata-eval 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. times-frac 99.0%

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

      9. *-commutative 99.0%

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

      10. times-frac 99.0%

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

      11. associate-/l* 99.0%

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

      12. *-commutative 99.0%

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

      13. neg-mul-1 99.0%

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

      14. sin-neg 99.0%

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

      15. distribute-lft-neg-out 99.0%

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

      16. associate-*l/ 99.0%

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

   3. Simplified 99.0%

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

      1. associate-/r/ 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l/ 99.1%

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

      4. associate-/r/ 99.0%

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

      5. associate-*l/ 98.8%

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

      6. div-inv 99.0%

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

      7. times-frac 99.2%

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

      8. metadata-eval 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. frac-times 99.0%

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

      2. unpow2 99.0%

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

      3. clear-num 99.0%

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

   8. Applied egg-rr 99.0%

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

      1. unpow2 99.0%

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

      2. sin-mult 98.4%

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

   10. Applied egg-rr 98.4%

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

       1. div-sub 98.4%

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

       2. +-inverses 98.4%

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

       3. cos-0 98.4%

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

       4. metadata-eval 98.4%

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

       5. distribute-lft-out 98.4%

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

       6. metadata-eval 98.4%

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

       7. *-rgt-identity 98.4%

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

   12. Simplified 98.4%

       \[\leadsto \frac{1}{\frac{\sin x \cdot 0.375}{\color{blue}{0.5 - \frac{\cos x}{2}}}} \]
   13. Step-by-step derivation

       1. associate-/l* 98.2%

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

       2. associate-/r/ 98.3%

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

       3. div-sub 98.1%

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

       4. metadata-eval 98.1%

          \[\leadsto \frac{1}{\sin x} \cdot \left(\color{blue}{1.3333333333333333} - \frac{\frac{\cos x}{2}}{0.375}\right) \]

       5. clear-num 98.0%

          \[\leadsto \frac{1}{\sin x} \cdot \left(1.3333333333333333 - \frac{\color{blue}{\frac{1}{\frac{2}{\cos x}}}}{0.375}\right) \]

       6. associate-/r/ 98.1%

          \[\leadsto \frac{1}{\sin x} \cdot \left(1.3333333333333333 - \frac{\color{blue}{\frac{1}{2} \cdot \cos x}}{0.375}\right) \]

       7. metadata-eval 98.1%

          \[\leadsto \frac{1}{\sin x} \cdot \left(1.3333333333333333 - \frac{\color{blue}{0.5} \cdot \cos x}{0.375}\right) \]

   14. Applied egg-rr 98.1%

       \[\leadsto \color{blue}{\frac{1}{\sin x} \cdot \left(1.3333333333333333 - \frac{0.5 \cdot \cos x}{0.375}\right)} \]
   15. Step-by-step derivation

       1. associate-*l/ 98.1%

          \[\leadsto \color{blue}{\frac{1 \cdot \left(1.3333333333333333 - \frac{0.5 \cdot \cos x}{0.375}\right)}{\sin x}} \]

       2. *-lft-identity 98.1%

          \[\leadsto \frac{\color{blue}{1.3333333333333333 - \frac{0.5 \cdot \cos x}{0.375}}}{\sin x} \]

       3. associate-/l* 97.9%

          \[\leadsto \frac{1.3333333333333333 - \color{blue}{\frac{0.5}{\frac{0.375}{\cos x}}}}{\sin x} \]

       4. associate-/r/ 98.2%

          \[\leadsto \frac{1.3333333333333333 - \color{blue}{\frac{0.5}{0.375} \cdot \cos x}}{\sin x} \]

       5. metadata-eval 98.2%

          \[\leadsto \frac{1.3333333333333333 - \color{blue}{1.3333333333333333} \cdot \cos x}{\sin x} \]

   16. 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 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00021:\\ \;\;\;\;\frac{x}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.3333333333333333 - \cos x \cdot 1.3333333333333333}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

   1. *-commutative 73.7%

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

   2. remove-double-neg 73.7%

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

   3. sin-neg 73.7%

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

   4. distribute-lft-neg-out 73.7%

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

   5. distribute-rgt-neg-in 73.7%

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

   6. associate-*r/ 99.2%

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

   7. neg-mul-1 99.2%

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

   8. *-commutative 99.2%

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

   9. associate-/l* 99.2%

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

   10. associate-/r/ 99.2%

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

   11. *-commutative 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 58.2%

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

6. Final simplification 58.2%

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

Alternative 10: 55.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.2%

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

   4. metadata-eval 99.2%

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

   5. metadata-eval 99.2%

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

   6. metadata-eval 99.2%

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

   7. metadata-eval 99.2%

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

   8. times-frac 99.2%

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

   9. *-commutative 99.2%

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

   10. times-frac 99.2%

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

   11. associate-/l* 99.2%

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

   12. *-commutative 99.2%

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

   13. neg-mul-1 99.2%

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

   14. sin-neg 99.2%

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

   15. distribute-lft-neg-out 99.2%

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

   16. associate-*l/ 99.2%

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

3. Simplified 99.2%

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

   1. associate-/r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.3%

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

   4. associate-/r/ 99.2%

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

   5. associate-*l/ 73.6%

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

   6. div-inv 73.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

   1. frac-times 73.7%

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

   2. associate-/l* 99.4%

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

8. Applied egg-rr 99.4%

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

9. Taylor expanded in x around 0 58.5%

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

10. Final simplification 58.5%

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

Alternative 11: 51.7% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

   1. associate-/l* 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.2%

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

   4. metadata-eval 99.2%

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

   5. metadata-eval 99.2%

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

   6. metadata-eval 99.2%

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

   7. metadata-eval 99.2%

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

   8. times-frac 99.2%

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

   9. *-commutative 99.2%

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

   10. times-frac 99.2%

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

   11. associate-/l* 99.2%

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

   12. *-commutative 99.2%

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

   13. neg-mul-1 99.2%

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

   14. sin-neg 99.2%

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

   15. distribute-lft-neg-out 99.2%

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

   16. associate-*l/ 99.2%

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

3. Simplified 99.2%

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

   1. associate-/r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l/ 99.3%

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

   4. associate-/r/ 99.2%

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

   5. associate-*l/ 73.6%

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

   6. div-inv 73.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

   1. frac-times 73.7%

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

   2. unpow2 73.7%

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

   3. clear-num 73.6%

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

8. Applied egg-rr 73.6%

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

9. Taylor expanded in x around 0 54.6%

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

10. Final simplification 54.6%

    \[\leadsto \frac{1}{x \cdot -0.125 + 1.5 \cdot \frac{1}{x}} \]
11. Add Preprocessing

Alternative 12: 51.2% accurate, 104.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * 0.6666666666666666

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

   1. *-commutative 73.7%

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

   2. remove-double-neg 73.7%

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

   3. sin-neg 73.7%

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

   4. distribute-lft-neg-out 73.7%

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

   5. distribute-rgt-neg-in 73.7%

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

   6. associate-*l/ 99.2%

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

   7. *-commutative 99.2%

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

   8. distribute-rgt-neg-in 99.2%

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

   9. distribute-lft-neg-out 99.2%

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

   10. sin-neg 99.2%

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

   11. remove-double-neg 99.2%

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

   12. associate-*l* 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 54.0%

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

6. Final simplification 54.0%

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

Alternative 13: 51.4% accurate, 104.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x / 1.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

   1. *-commutative 73.7%

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

   2. remove-double-neg 73.7%

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

   3. sin-neg 73.7%

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

   4. distribute-lft-neg-out 73.7%

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

   5. distribute-rgt-neg-in 73.7%

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

   6. associate-*l/ 99.2%

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

   7. *-commutative 99.2%

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

   8. distribute-rgt-neg-in 99.2%

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

   9. distribute-lft-neg-out 99.2%

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

   10. sin-neg 99.2%

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

   11. remove-double-neg 99.2%

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

   12. associate-*l* 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 54.0%

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

   1. add-cube-cbrt 53.0%

      \[\leadsto \color{blue}{\left(\sqrt[3]{0.6666666666666666 \cdot x} \cdot \sqrt[3]{0.6666666666666666 \cdot x}\right) \cdot \sqrt[3]{0.6666666666666666 \cdot x}} \]

   2. pow3 53.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{0.6666666666666666 \cdot x}\right)}^{3}} \]

   3. *-commutative 53.0%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot 0.6666666666666666}}\right)}^{3} \]

7. Applied egg-rr 53.0%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot 0.6666666666666666}\right)}^{3}} \]
8. Step-by-step derivation

   1. rem-cube-cbrt 54.0%

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

   2. metadata-eval 54.0%

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

   3. div-inv 54.3%

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

9. Applied egg-rr 54.3%

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

10. Final simplification 54.3%

    \[\leadsto \frac{x}{1.5} \]
11. 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 2024020 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

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

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