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

Percentage Accurate: 77.2% → 99.5%

Time: 17.6s

Alternatives: 19

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: 77.2% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{t_0}{\frac{0.375}{\frac{1}{\frac{\sin x}{t_0}}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ t_0 (/ 0.375 (/ 1.0 (/ (sin x) t_0))))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return t_0 / (0.375 / (1.0 / (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 = t_0 / (0.375d0 / (1.0d0 / (sin(x) / t_0)))
end function

Java

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

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return t_0 / (0.375 / (1.0 / (math.sin(x) / t_0)))

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(t_0 / Float64(0.375 / Float64(1.0 / Float64(sin(x) / t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = t_0 / (0.375 / (1.0 / (sin(x) / t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 / N[(0.375 / N[(1.0 / N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.5%

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

   1. associate-/l* 99.2%

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

   2. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

   1. clear-num 99.1%

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

   2. associate-*l/ 99.2%

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

   3. *-un-lft-identity 99.2%

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

   4. *-un-lft-identity 99.2%

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

   5. times-frac 99.6%

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

   6. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

   1. clear-num 99.5%

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

   2. un-div-inv 99.6%

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

8. Applied egg-rr 99.6%

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

   1. clear-num 99.6%

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

   2. inv-pow 99.6%

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

10. Applied egg-rr 99.6%

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

    1. unpow-1 99.6%

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

12. Applied egg-rr 99.6%

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

13. Final simplification 99.6%

    \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\frac{0.375}{\frac{1}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}}} \]
14. Add Preprocessing

Alternative 2: 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 76.5%

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

   1. *-commutative 76.5%

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

   2. remove-double-neg 76.5%

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

   3. sin-neg 76.5%

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

   4. distribute-lft-neg-out 76.5%

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

   5. distribute-rgt-neg-in 76.5%

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

   6. associate-*l/ 99.2%

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

   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 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. associate-/l* 99.2%

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

   2. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*l/ 99.2%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

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

Alternative 4: 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 76.5%

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

   1. associate-/l* 99.2%

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

   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. *-lft-identity 99.2%

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

   4. metadata-eval 99.2%

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

   5. times-frac 99.3%

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

   6. associate-/l* 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. sin-neg 99.3%

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

   10. distribute-lft-neg-out 99.3%

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

   11. times-frac 99.2%

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

   12. *-commutative 99.2%

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

   13. associate-/l/ 99.2%

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

   14. associate-/r* 99.2%

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

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/ 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*l/ 99.2%

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

   4. associate-/r/ 99.1%

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

   5. associate-*l/ 76.5%

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

   6. div-inv 76.6%

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

   7. times-frac 99.5%

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

   8. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return t_0 / (0.375 * (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 = t_0 / (0.375d0 * (sin(x) / t_0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. associate-/l* 99.2%

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

   2. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

   1. clear-num 99.1%

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

   2. associate-*l/ 99.2%

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

   3. *-un-lft-identity 99.2%

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

   4. *-un-lft-identity 99.2%

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

   5. times-frac 99.6%

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

   6. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 6: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-16)
   (/ (* x 0.25) 0.375)
   (* (pow (sin (* x 0.5)) 2.0) (/ 2.6666666666666665 (sin x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2e-16) {
		tmp = (x * 0.25) / 0.375;
	} 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 <= 2d-16) then
        tmp = (x * 0.25d0) / 0.375d0
    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 <= 2e-16) {
		tmp = (x * 0.25) / 0.375;
	} 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 <= 2e-16:
		tmp = (x * 0.25) / 0.375
	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 <= 2e-16)
		tmp = Float64(Float64(x * 0.25) / 0.375);
	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 <= 2e-16)
		tmp = (x * 0.25) / 0.375;
	else
		tmp = (sin((x * 0.5)) ^ 2.0) * (2.6666666666666665 / sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e-16], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $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 < 2e-16

   1. Initial program 70.2%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l/ 99.3%

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

      3. associate-/r/ 99.2%

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

      4. associate-*l/ 70.2%

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

      5. div-inv 70.3%

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

      6. associate-/r* 70.4%

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

      7. pow2 70.4%

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

      8. metadata-eval 70.4%

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

   6. Applied egg-rr 70.4%

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

   7. Taylor expanded in x around 0 64.2%

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

      1. *-commutative 64.2%

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

   9. Simplified 64.2%

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

   if 2e-16 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. *-commutative 99.1%

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

      3. *-lft-identity 99.1%

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

      4. metadata-eval 99.1%

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

      5. times-frac 99.1%

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

      6. associate-/l* 99.1%

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

      7. *-commutative 99.1%

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

      8. neg-mul-1 99.1%

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

      9. sin-neg 99.1%

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

      10. distribute-lft-neg-out 99.1%

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

      11. times-frac 99.1%

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

      12. *-commutative 99.1%

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

      13. associate-/l/ 99.1%

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

      14. associate-/r* 99.1%

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

   3. Simplified 99.1%

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

      1. associate-/r/ 99.1%

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

      2. *-commutative 99.1%

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

      3. div-inv 98.9%

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

      4. *-commutative 98.9%

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

      5. associate-*l* 99.0%

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

      6. div-inv 99.1%

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

      7. associate-*r* 99.0%

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

      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 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;{\sin \left(x \cdot 0.5\right)}^{2} \cdot \frac{2.6666666666666665}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-10)
   (/ (* x 0.25) 0.375)
   (/ 2.6666666666666665 (/ (sin x) (pow (sin (* x 0.5)) 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 2e-10) {
		tmp = (x * 0.25) / 0.375;
	} 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 <= 2d-10) then
        tmp = (x * 0.25d0) / 0.375d0
    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 <= 2e-10) {
		tmp = (x * 0.25) / 0.375;
	} 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 <= 2e-10:
		tmp = (x * 0.25) / 0.375
	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 <= 2e-10)
		tmp = Float64(Float64(x * 0.25) / 0.375);
	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 <= 2e-10)
		tmp = (x * 0.25) / 0.375;
	else
		tmp = 2.6666666666666665 / (sin(x) / (sin((x * 0.5)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e-10], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $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 < 2.00000000000000007e-10

   1. Initial program 70.4%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l/ 99.3%

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

      3. associate-/r/ 99.2%

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

      4. associate-*l/ 70.3%

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

      5. div-inv 70.5%

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

      6. associate-/r* 70.6%

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

      7. pow2 70.6%

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

      8. metadata-eval 70.6%

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

   6. Applied egg-rr 70.6%

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

   7. Taylor expanded in x around 0 64.4%

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

      1. *-commutative 64.4%

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

   9. Simplified 64.4%

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

   if 2.00000000000000007e-10 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.0%

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

      2. associate-/r/ 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*l/ 99.1%

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

      2. associate-*l* 99.1%

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

      3. associate-/l* 99.0%

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

      4. pow2 99.1%

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 71.9%

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

Alternative 8: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665 \cdot {\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e-14)
   (/ (* x 0.25) 0.375)
   (/ (* 2.6666666666666665 (pow (sin (* x 0.5)) 2.0)) (sin x))))

C

double code(double x) {
	double tmp;
	if (x <= 5e-14) {
		tmp = (x * 0.25) / 0.375;
	} 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 <= 5d-14) then
        tmp = (x * 0.25d0) / 0.375d0
    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 <= 5e-14) {
		tmp = (x * 0.25) / 0.375;
	} 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 <= 5e-14:
		tmp = (x * 0.25) / 0.375
	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 <= 5e-14)
		tmp = Float64(Float64(x * 0.25) / 0.375);
	else
		tmp = Float64(Float64(2.6666666666666665 * (sin(Float64(x * 0.5)) ^ 2.0)) / sin(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e-14)
		tmp = (x * 0.25) / 0.375;
	else
		tmp = (2.6666666666666665 * (sin((x * 0.5)) ^ 2.0)) / sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e-14], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(2.6666666666666665 * N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e-14

   1. Initial program 70.2%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l/ 99.3%

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

      3. associate-/r/ 99.2%

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

      4. associate-*l/ 70.2%

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

      5. div-inv 70.3%

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

      6. associate-/r* 70.4%

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

      7. pow2 70.4%

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

      8. metadata-eval 70.4%

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

   6. Applied egg-rr 70.4%

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

   7. Taylor expanded in x around 0 64.2%

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

      1. *-commutative 64.2%

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

   9. Simplified 64.2%

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

   if 5.0000000000000002e-14 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf 99.2%

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

      1. *-commutative 99.2%

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

   5. Simplified 99.2%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665 \cdot {\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-16}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375}}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e-16)
   (/ (* x 0.25) 0.375)
   (/ (/ (pow (sin (* x 0.5)) 2.0) 0.375) (sin x))))

C

double code(double x) {
	double tmp;
	if (x <= 1e-16) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (pow(sin((x * 0.5)), 2.0) / 0.375) / sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d-16) then
        tmp = (x * 0.25d0) / 0.375d0
    else
        tmp = ((sin((x * 0.5d0)) ** 2.0d0) / 0.375d0) / sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e-16) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (Math.pow(Math.sin((x * 0.5)), 2.0) / 0.375) / Math.sin(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e-16:
		tmp = (x * 0.25) / 0.375
	else:
		tmp = (math.pow(math.sin((x * 0.5)), 2.0) / 0.375) / math.sin(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e-16)
		tmp = Float64(Float64(x * 0.25) / 0.375);
	else
		tmp = Float64(Float64((sin(Float64(x * 0.5)) ^ 2.0) / 0.375) / sin(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e-16)
		tmp = (x * 0.25) / 0.375;
	else
		tmp = ((sin((x * 0.5)) ^ 2.0) / 0.375) / sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e-16], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / 0.375), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999998e-17

   1. Initial program 70.2%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l/ 99.3%

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

      3. associate-/r/ 99.2%

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

      4. associate-*l/ 70.2%

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

      5. div-inv 70.3%

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

      6. associate-/r* 70.4%

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

      7. pow2 70.4%

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

      8. metadata-eval 70.4%

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

   6. Applied egg-rr 70.4%

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

   7. Taylor expanded in x around 0 64.2%

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

      1. *-commutative 64.2%

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

   9. Simplified 64.2%

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

   if 9.9999999999999998e-17 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf 99.2%

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

      1. *-commutative 99.2%

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

   5. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. unpow2 99.1%

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

      3. associate-*r* 99.1%

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

      4. metadata-eval 99.1%

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

      5. div-inv 99.1%

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

      6. associate-*r/ 99.1%

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

      7. unpow2 99.2%

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

   7. Applied egg-rr 99.2%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-16}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{0.375}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-20}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e-20)
   (/ (* x 0.25) 0.375)
   (/ (/ (pow (sin (* x 0.5)) 2.0) (sin x)) 0.375)))

C

double code(double x) {
	double tmp;
	if (x <= 1e-20) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (pow(sin((x * 0.5)), 2.0) / sin(x)) / 0.375;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1d-20) then
        tmp = (x * 0.25d0) / 0.375d0
    else
        tmp = ((sin((x * 0.5d0)) ** 2.0d0) / sin(x)) / 0.375d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e-20) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (Math.pow(Math.sin((x * 0.5)), 2.0) / Math.sin(x)) / 0.375;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e-20:
		tmp = (x * 0.25) / 0.375
	else:
		tmp = (math.pow(math.sin((x * 0.5)), 2.0) / math.sin(x)) / 0.375
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e-20)
		tmp = Float64(Float64(x * 0.25) / 0.375);
	else
		tmp = Float64(Float64((sin(Float64(x * 0.5)) ^ 2.0) / sin(x)) / 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e-20)
		tmp = (x * 0.25) / 0.375;
	else
		tmp = ((sin((x * 0.5)) ^ 2.0) / sin(x)) / 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e-20], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999945e-21

   1. Initial program 70.2%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l/ 99.3%

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

      3. associate-/r/ 99.2%

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

      4. associate-*l/ 70.2%

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

      5. div-inv 70.3%

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

      6. associate-/r* 70.4%

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

      7. pow2 70.4%

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

      8. metadata-eval 70.4%

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

   6. Applied egg-rr 70.4%

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

   7. Taylor expanded in x around 0 64.2%

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

      1. *-commutative 64.2%

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

   9. Simplified 64.2%

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

   if 9.99999999999999945e-21 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. associate-/r/ 99.1%

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

   3. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*l/ 99.0%

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

      3. associate-/r/ 99.0%

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

      4. associate-*l/ 99.0%

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

      5. div-inv 99.0%

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

      6. associate-/r* 99.1%

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

      7. pow2 99.2%

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

      8. metadata-eval 99.2%

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

   6. Applied egg-rr 99.2%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-20}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 - \frac{\cos x}{2}}{\sin x}}{0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.005)
   (/ (sin (* x 0.5)) (+ 0.75 (* -0.09375 (pow x 2.0))))
   (/ (/ (- 0.5 (/ (cos x) 2.0)) (sin x)) 0.375)))

C

double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = sin((x * 0.5)) / (0.75 + (-0.09375 * pow(x, 2.0)));
	} 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 <= 0.005d0) then
        tmp = sin((x * 0.5d0)) / (0.75d0 + ((-0.09375d0) * (x ** 2.0d0)))
    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 <= 0.005) {
		tmp = Math.sin((x * 0.5)) / (0.75 + (-0.09375 * Math.pow(x, 2.0)));
	} else {
		tmp = ((0.5 - (Math.cos(x) / 2.0)) / Math.sin(x)) / 0.375;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.005:
		tmp = math.sin((x * 0.5)) / (0.75 + (-0.09375 * math.pow(x, 2.0)))
	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 <= 0.005)
		tmp = Float64(sin(Float64(x * 0.5)) / Float64(0.75 + Float64(-0.09375 * (x ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(0.5 - Float64(cos(x) / 2.0)) / sin(x)) / 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.005)
		tmp = sin((x * 0.5)) / (0.75 + (-0.09375 * (x ^ 2.0)));
	else
		tmp = ((0.5 - (cos(x) / 2.0)) / sin(x)) / 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.005], N[(N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[(0.75 + N[(-0.09375 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(N[Cos[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0050000000000000001

   1. Initial program 70.6%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

      1. clear-num 99.1%

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

      2. associate-*l/ 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. *-un-lft-identity 99.3%

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

      5. times-frac 99.7%

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

      6. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 64.8%

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

   if 0.0050000000000000001 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. associate-/r/ 99.1%

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

   3. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*l/ 99.0%

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

      3. associate-/r/ 99.0%

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

      4. associate-*l/ 99.0%

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

      5. div-inv 99.0%

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

      6. associate-/r* 99.1%

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

      7. pow2 99.1%

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

      8. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. unpow2 99.2%

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

      2. sin-mult 98.0%

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

   8. Applied egg-rr 98.1%

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

      1. div-sub 98.0%

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

      2. +-inverses 98.0%

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

      3. cos-0 98.0%

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

      4. metadata-eval 98.0%

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

      5. distribute-lft-out 98.0%

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

      6. metadata-eval 98.0%

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

      7. *-rgt-identity 98.0%

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

   10. Simplified 98.1%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;\frac{\sin \left(x \cdot 0.5\right)}{0.75 + -0.09375 \cdot {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 - \frac{\cos x}{2}}{\sin x}}{0.375}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (/ (+ (* 0.020833333333333332 (pow x 3.0)) (* x 0.25)) 0.375)
   (* 2.6666666666666665 (/ (- 0.5 (* 0.5 (cos x))) (sin x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = ((0.020833333333333332 * pow(x, 3.0)) + (x * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x))) / sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0042d0) then
        tmp = ((0.020833333333333332d0 * (x ** 3.0d0)) + (x * 0.25d0)) / 0.375d0
    else
        tmp = 2.6666666666666665d0 * ((0.5d0 - (0.5d0 * cos(x))) / sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = ((0.020833333333333332 * Math.pow(x, 3.0)) + (x * 0.25)) / 0.375;
	} else {
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * Math.cos(x))) / Math.sin(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0042:
		tmp = ((0.020833333333333332 * math.pow(x, 3.0)) + (x * 0.25)) / 0.375
	else:
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * math.cos(x))) / math.sin(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x ^ 3.0)) + Float64(x * 0.25)) / 0.375);
	else
		tmp = Float64(2.6666666666666665 * Float64(Float64(0.5 - Float64(0.5 * cos(x))) / sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0042)
		tmp = ((0.020833333333333332 * (x ^ 3.0)) + (x * 0.25)) / 0.375;
	else
		tmp = 2.6666666666666665 * ((0.5 - (0.5 * cos(x))) / sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(N[(N[(0.020833333333333332 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(2.6666666666666665 * N[(N[(0.5 - N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 70.6%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l/ 99.3%

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

      3. associate-/r/ 99.2%

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

      4. associate-*l/ 70.6%

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

      5. div-inv 70.7%

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

      6. associate-/r* 70.8%

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

      7. pow2 70.8%

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

      8. metadata-eval 70.8%

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

   6. Applied egg-rr 70.8%

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

   7. Taylor expanded in x around 0 64.3%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]

   if 0.00419999999999999974 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf 99.2%

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

      1. *-commutative 99.2%

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

   5. Simplified 99.2%

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

      1. unpow2 99.2%

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

      2. sin-mult 98.0%

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

   7. Applied egg-rr 98.0%

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

      1. div-sub 98.0%

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

      2. +-inverses 98.0%

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

      3. cos-0 98.0%

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

      4. metadata-eval 98.0%

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

      5. distribute-lft-out 98.0%

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

      6. metadata-eval 98.0%

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

      7. *-rgt-identity 98.0%

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

   9. Simplified 98.0%

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

       1. *-un-lft-identity 98.0%

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

       2. times-frac 98.0%

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

       3. metadata-eval 98.0%

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

       4. div-inv 98.0%

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

       5. metadata-eval 98.0%

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

   11. Applied egg-rr 98.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{0.5 - 0.5 \cdot \cos x}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 - \frac{\cos x}{2}}{\sin x}}{0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (/ (+ (* 0.020833333333333332 (pow x 3.0)) (* x 0.25)) 0.375)
   (/ (/ (- 0.5 (/ (cos x) 2.0)) (sin x)) 0.375)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = ((0.020833333333333332 * pow(x, 3.0)) + (x * 0.25)) / 0.375;
	} 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 <= 0.0042d0) then
        tmp = ((0.020833333333333332d0 * (x ** 3.0d0)) + (x * 0.25d0)) / 0.375d0
    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 <= 0.0042) {
		tmp = ((0.020833333333333332 * Math.pow(x, 3.0)) + (x * 0.25)) / 0.375;
	} else {
		tmp = ((0.5 - (Math.cos(x) / 2.0)) / Math.sin(x)) / 0.375;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0042:
		tmp = ((0.020833333333333332 * math.pow(x, 3.0)) + (x * 0.25)) / 0.375
	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 <= 0.0042)
		tmp = Float64(Float64(Float64(0.020833333333333332 * (x ^ 3.0)) + Float64(x * 0.25)) / 0.375);
	else
		tmp = Float64(Float64(Float64(0.5 - Float64(cos(x) / 2.0)) / sin(x)) / 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0042)
		tmp = ((0.020833333333333332 * (x ^ 3.0)) + (x * 0.25)) / 0.375;
	else
		tmp = ((0.5 - (cos(x) / 2.0)) / sin(x)) / 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(N[(N[(0.020833333333333332 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 0.25), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(N[(0.5 - N[(N[Cos[x], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 70.6%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l/ 99.3%

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

      3. associate-/r/ 99.2%

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

      4. associate-*l/ 70.6%

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

      5. div-inv 70.7%

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

      6. associate-/r* 70.8%

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

      7. pow2 70.8%

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

      8. metadata-eval 70.8%

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

   6. Applied egg-rr 70.8%

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

   7. Taylor expanded in x around 0 64.3%

      \[\leadsto \frac{\color{blue}{0.020833333333333332 \cdot {x}^{3} + 0.25 \cdot x}}{0.375} \]

   if 0.00419999999999999974 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. associate-/r/ 99.1%

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

   3. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*l/ 99.0%

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

      3. associate-/r/ 99.0%

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

      4. associate-*l/ 99.0%

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

      5. div-inv 99.0%

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

      6. associate-/r* 99.1%

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

      7. pow2 99.1%

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

      8. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. unpow2 99.2%

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

      2. sin-mult 98.0%

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

   8. Applied egg-rr 98.1%

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

      1. div-sub 98.0%

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

      2. +-inverses 98.0%

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

      3. cos-0 98.0%

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

      4. metadata-eval 98.0%

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

      5. distribute-lft-out 98.0%

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

      6. metadata-eval 98.0%

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

      7. *-rgt-identity 98.0%

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

   10. Simplified 98.1%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;\frac{0.020833333333333332 \cdot {x}^{3} + x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 - \frac{\cos x}{2}}{\sin x}}{0.375}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. associate-/l* 99.2%

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

   2. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 55.6%

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

   1. add-sqr-sqrt 28.8%

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

   2. sqrt-unprod 20.1%

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

   3. *-commutative 20.1%

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

   4. *-commutative 20.1%

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

   5. swap-sqr 20.1%

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

   6. unpow2 20.1%

      \[\leadsto \sqrt{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}} \cdot \left(1.3333333333333333 \cdot 1.3333333333333333\right)} \]

   7. metadata-eval 20.1%

      \[\leadsto \sqrt{{\sin \left(x \cdot 0.5\right)}^{2} \cdot \color{blue}{1.7777777777777777}} \]

7. Applied egg-rr 20.1%

   \[\leadsto \color{blue}{\sqrt{{\sin \left(x \cdot 0.5\right)}^{2} \cdot 1.7777777777777777}} \]
8. Step-by-step derivation

   1. unpow2 20.1%

      \[\leadsto \sqrt{\color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \sin \left(x \cdot 0.5\right)\right)} \cdot 1.7777777777777777} \]

   2. metadata-eval 20.1%

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

   3. swap-sqr 20.1%

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

   4. rem-sqrt-square 32.7%

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

   5. *-commutative 32.7%

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

9. Simplified 32.7%

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

10. Final simplification 32.7%

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

Alternative 15: 54.4% 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 76.5%

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

   1. associate-/l* 99.2%

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

   2. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 55.6%

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

6. Final simplification 55.6%

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

Alternative 16: 54.7% 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 76.5%

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

   1. associate-/l* 99.2%

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

   2. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

   1. clear-num 99.1%

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

   2. associate-*l/ 99.2%

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

   3. *-un-lft-identity 99.2%

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

   4. *-un-lft-identity 99.2%

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

   5. times-frac 99.6%

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

   6. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in x around 0 55.9%

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

8. Final simplification 55.9%

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

Alternative 17: 53.0% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 1.3333333333333333}{1 + \left(1 + x \cdot 0.6666666666666666\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* x 1.3333333333333333) (+ 1.0 (+ 1.0 (* x 0.6666666666666666)))))

C

double code(double x) {
	return (x * 1.3333333333333333) / (1.0 + (1.0 + (x * 0.6666666666666666)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 1.3333333333333333d0) / (1.0d0 + (1.0d0 + (x * 0.6666666666666666d0)))
end function

Java

public static double code(double x) {
	return (x * 1.3333333333333333) / (1.0 + (1.0 + (x * 0.6666666666666666)));
}

Python

def code(x):
	return (x * 1.3333333333333333) / (1.0 + (1.0 + (x * 0.6666666666666666)))

Julia

function code(x)
	return Float64(Float64(x * 1.3333333333333333) / Float64(1.0 + Float64(1.0 + Float64(x * 0.6666666666666666))))
end

MATLAB

function tmp = code(x)
	tmp = (x * 1.3333333333333333) / (1.0 + (1.0 + (x * 0.6666666666666666)));
end

Wolfram

code[x_] := N[(N[(x * 1.3333333333333333), $MachinePrecision] / N[(1.0 + N[(1.0 + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

   1. associate-/l* 99.2%

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

   2. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 51.6%

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

   1. *-commutative 51.6%

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

7. Simplified 51.6%

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

   1. expm1-log1p-u 50.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)\right)} \]

9. Applied egg-rr 50.6%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)\right)} \]
10. Step-by-step derivation

    1. expm1-udef 4.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)} - 1} \]

    2. flip-- 4.5%

       \[\leadsto \color{blue}{\frac{e^{\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)} \cdot e^{\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)} + 1}} \]

    3. log1p-udef 4.5%

       \[\leadsto \frac{e^{\color{blue}{\log \left(1 + x \cdot 0.6666666666666666\right)}} \cdot e^{\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)} + 1} \]

    4. rem-exp-log 4.5%

       \[\leadsto \frac{\color{blue}{\left(1 + x \cdot 0.6666666666666666\right)} \cdot e^{\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)} + 1} \]

    5. log1p-udef 4.5%

       \[\leadsto \frac{\left(1 + x \cdot 0.6666666666666666\right) \cdot e^{\color{blue}{\log \left(1 + x \cdot 0.6666666666666666\right)}} - 1 \cdot 1}{e^{\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)} + 1} \]

    6. rem-exp-log 4.5%

       \[\leadsto \frac{\left(1 + x \cdot 0.6666666666666666\right) \cdot \color{blue}{\left(1 + x \cdot 0.6666666666666666\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)} + 1} \]

    7. +-commutative 4.5%

       \[\leadsto \frac{\color{blue}{\left(x \cdot 0.6666666666666666 + 1\right)} \cdot \left(1 + x \cdot 0.6666666666666666\right) - 1 \cdot 1}{e^{\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)} + 1} \]

    8. +-commutative 4.5%

       \[\leadsto \frac{\left(x \cdot 0.6666666666666666 + 1\right) \cdot \color{blue}{\left(x \cdot 0.6666666666666666 + 1\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)} + 1} \]

    9. metadata-eval 4.5%

       \[\leadsto \frac{\left(x \cdot 0.6666666666666666 + 1\right) \cdot \left(x \cdot 0.6666666666666666 + 1\right) - \color{blue}{1}}{e^{\mathsf{log1p}\left(x \cdot 0.6666666666666666\right)} + 1} \]

    10. log1p-udef 4.5%

        \[\leadsto \frac{\left(x \cdot 0.6666666666666666 + 1\right) \cdot \left(x \cdot 0.6666666666666666 + 1\right) - 1}{e^{\color{blue}{\log \left(1 + x \cdot 0.6666666666666666\right)}} + 1} \]

    11. rem-exp-log 5.4%

        \[\leadsto \frac{\left(x \cdot 0.6666666666666666 + 1\right) \cdot \left(x \cdot 0.6666666666666666 + 1\right) - 1}{\color{blue}{\left(1 + x \cdot 0.6666666666666666\right)} + 1} \]

    12. +-commutative 5.4%

        \[\leadsto \frac{\left(x \cdot 0.6666666666666666 + 1\right) \cdot \left(x \cdot 0.6666666666666666 + 1\right) - 1}{\color{blue}{\left(x \cdot 0.6666666666666666 + 1\right)} + 1} \]

11. Applied egg-rr 5.4%

    \[\leadsto \color{blue}{\frac{\left(x \cdot 0.6666666666666666 + 1\right) \cdot \left(x \cdot 0.6666666666666666 + 1\right) - 1}{\left(x \cdot 0.6666666666666666 + 1\right) + 1}} \]

12. Taylor expanded in x around 0 54.7%

    \[\leadsto \frac{\color{blue}{1.3333333333333333 \cdot x}}{\left(x \cdot 0.6666666666666666 + 1\right) + 1} \]
13. Step-by-step derivation

    1. *-commutative 54.7%

       \[\leadsto \frac{\color{blue}{x \cdot 1.3333333333333333}}{\left(x \cdot 0.6666666666666666 + 1\right) + 1} \]

14. Simplified 54.7%

    \[\leadsto \frac{\color{blue}{x \cdot 1.3333333333333333}}{\left(x \cdot 0.6666666666666666 + 1\right) + 1} \]

15. Final simplification 54.7%

    \[\leadsto \frac{x \cdot 1.3333333333333333}{1 + \left(1 + x \cdot 0.6666666666666666\right)} \]
16. Add Preprocessing

Alternative 18: 50.5% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 0.25}{0.375} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* x 0.25) 0.375))

C

double code(double x) {
	return (x * 0.25) / 0.375;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 0.25d0) / 0.375d0
end function

Java

public static double code(double x) {
	return (x * 0.25) / 0.375;
}

Python

def code(x):
	return (x * 0.25) / 0.375

Julia

function code(x)
	return Float64(Float64(x * 0.25) / 0.375)
end

MATLAB

function tmp = code(x)
	tmp = (x * 0.25) / 0.375;
end

Wolfram

code[x_] := N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

   1. associate-/l* 99.2%

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

   2. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*l/ 99.2%

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

   3. associate-/r/ 99.1%

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

   4. associate-*l/ 76.5%

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

   5. div-inv 76.6%

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

   6. associate-/r* 76.7%

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

   7. pow2 76.7%

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

   8. metadata-eval 76.7%

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

6. Applied egg-rr 76.7%

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

7. Taylor expanded in x around 0 51.8%

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

   1. *-commutative 51.8%

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

9. Simplified 51.8%

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

10. Final simplification 51.8%

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

Alternative 19: 50.2% 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 76.5%

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

   1. associate-/l* 99.2%

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

   2. associate-/r/ 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 51.6%

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

   1. *-commutative 51.6%

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

7. Simplified 51.6%

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

8. Final simplification 51.6%

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

Developer target: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024011 
(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)))