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

Percentage Accurate: 76.9% → 99.5%

Time: 17.2s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{t\_0}{-0.375} \cdot \frac{t\_0}{-\sin x} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* 99.2%

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

   2. *-commutative 99.2%

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

   3. div-inv 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-/r/ 99.1%

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

   6. un-div-inv 99.2%

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

   7. *-un-lft-identity 99.2%

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

   8. times-frac 99.5%

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

   9. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{0.375 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]
7. 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 99.5%

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

8. Applied egg-rr 99.5%

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

   1. frac-2neg 99.5%

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

   2. associate-/r/ 99.6%

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

   3. metadata-eval 99.6%

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

   4. distribute-neg-frac2 99.6%

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

10. Applied egg-rr 99.6%

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

Alternative 2: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-163}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}}{0.375}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.1e-163)
   (/ (* 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 <= 2.1e-163) {
		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 <= 2.1d-163) 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 <= 2.1e-163) {
		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 <= 2.1e-163:
		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 <= 2.1e-163)
		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 <= 2.1e-163)
		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, 2.1e-163], 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 < 2.09999999999999998e-163

   1. Initial program 59.3%

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

      1. associate-/l* 99.4%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. clear-num 99.2%

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

      2. un-div-inv 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. metadata-eval 99.3%

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

      2. times-frac 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/r* 99.6%

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

      6. associate-/r/ 99.7%

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

      7. associate-*l/ 59.5%

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

      8. unpow2 59.5%

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

   8. Applied egg-rr 59.5%

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

   9. Taylor expanded in x around 0 64.7%

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

       1. *-commutative 64.7%

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

   11. Simplified 64.7%

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

   if 2.09999999999999998e-163 < x

   1. Initial program 97.8%

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

      1. associate-/l* 99.1%

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

      2. associate-*l* 99.1%

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

      3. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. clear-num 99.2%

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

      2. un-div-inv 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. metadata-eval 99.1%

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

      2. times-frac 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-/r* 99.3%

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

      6. associate-/r/ 99.3%

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

      7. associate-*l/ 98.1%

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

      8. unpow2 98.1%

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

   8. Applied egg-rr 98.1%

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

Alternative 3: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-163}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.1e-163)
   (/ (* 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 <= 2.1e-163) {
		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 <= 2.1d-163) 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 <= 2.1e-163) {
		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 <= 2.1e-163:
		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 <= 2.1e-163)
		tmp = Float64(Float64(x * 0.25) / 0.375);
	else
		tmp = Float64(2.6666666666666665 * Float64((sin(Float64(x * 0.5)) ^ 2.0) / sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.1e-163)
		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, 2.1e-163], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.09999999999999998e-163

   1. Initial program 59.3%

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

      1. associate-/l* 99.4%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. clear-num 99.2%

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

      2. un-div-inv 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. metadata-eval 99.3%

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

      2. times-frac 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/r* 99.6%

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

      6. associate-/r/ 99.7%

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

      7. associate-*l/ 59.5%

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

      8. unpow2 59.5%

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

   8. Applied egg-rr 59.5%

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

   9. Taylor expanded in x around 0 64.7%

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

       1. *-commutative 64.7%

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

   11. Simplified 64.7%

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

   if 2.09999999999999998e-163 < x

   1. Initial program 97.8%

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

      1. metadata-eval 97.8%

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

      2. associate-*r/ 99.1%

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

      3. associate-*r* 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-*r/ 97.9%

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

      6. pow2 97.9%

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

   4. Applied egg-rr 97.9%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-163}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(x \cdot 0.5\right)}^{2}}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{t\_0}{\frac{0.375}{\frac{t\_0}{\sin x}}} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* 99.2%

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

   2. *-commutative 99.2%

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

   3. div-inv 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-/r/ 99.1%

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

   6. un-div-inv 99.2%

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

   7. *-un-lft-identity 99.2%

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

   8. times-frac 99.5%

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

   9. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{0.375 \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right)}}} \]
7. 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 99.5%

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

8. Applied egg-rr 99.5%

   \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{0.375}{\frac{\sin \left(x \cdot 0.5\right)}{\sin x}}}} \]
9. 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 75.1%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* 99.2%

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

   2. *-commutative 99.2%

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

   3. div-inv 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-/r/ 99.1%

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

   6. un-div-inv 99.2%

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

   7. *-un-lft-identity 99.2%

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

   8. times-frac 99.5%

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

   9. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 6: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ 2.6666666666666665 \cdot \frac{t\_0}{\frac{\sin x}{t\_0}} \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return 2.6666666666666665 * (t_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 = 2.6666666666666665d0 * (t_0 / (sin(x) / t_0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. clear-num 99.2%

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

   2. un-div-inv 99.3%

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

6. Applied egg-rr 99.3%

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

Alternative 7: 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 75.1%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

Alternative 8: 75.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.85 \cdot 10^{-7}:\\ \;\;\;\;\frac{x \cdot 0.25}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x} \cdot \left(0.5 - 0.5 \cdot \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.85e-7)
   (/ (* x 0.25) 0.375)
   (* (/ 2.6666666666666665 (sin x)) (- 0.5 (* 0.5 (cos x))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.85e-7) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (2.6666666666666665 / sin(x)) * (0.5 - (0.5 * cos(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.85d-7) then
        tmp = (x * 0.25d0) / 0.375d0
    else
        tmp = (2.6666666666666665d0 / sin(x)) * (0.5d0 - (0.5d0 * cos(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.85e-7) {
		tmp = (x * 0.25) / 0.375;
	} else {
		tmp = (2.6666666666666665 / Math.sin(x)) * (0.5 - (0.5 * Math.cos(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.85e-7:
		tmp = (x * 0.25) / 0.375
	else:
		tmp = (2.6666666666666665 / math.sin(x)) * (0.5 - (0.5 * math.cos(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.85e-7)
		tmp = Float64(Float64(x * 0.25) / 0.375);
	else
		tmp = Float64(Float64(2.6666666666666665 / sin(x)) * Float64(0.5 - Float64(0.5 * cos(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.85e-7)
		tmp = (x * 0.25) / 0.375;
	else
		tmp = (2.6666666666666665 / sin(x)) * (0.5 - (0.5 * cos(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.85e-7], N[(N[(x * 0.25), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8500000000000002e-7

   1. Initial program 66.2%

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

      1. associate-/l* 99.4%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. clear-num 99.3%

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

      2. un-div-inv 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. metadata-eval 99.3%

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

      2. times-frac 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-/r* 99.7%

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

      6. associate-/r/ 99.7%

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

      7. associate-*l/ 66.4%

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

      8. unpow2 66.4%

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

   8. Applied egg-rr 66.4%

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

   9. Taylor expanded in x around 0 71.4%

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

       1. *-commutative 71.4%

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

   11. Simplified 71.4%

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

   if 2.8500000000000002e-7 < x

   1. Initial program 99.0%

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

      1. associate-/l* 98.9%

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

      2. associate-*l* 99.0%

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

      3. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

      1. associate-*r* 98.9%

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

      2. *-commutative 98.9%

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

      3. div-inv 98.9%

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

      4. associate-*l* 98.9%

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

      5. associate-/r/ 98.9%

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

      6. un-div-inv 98.9%

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

      7. *-un-lft-identity 98.9%

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

      8. times-frac 99.1%

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

      9. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. *-un-lft-identity 99.1%

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

      2. times-frac 99.0%

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

      3. metadata-eval 99.0%

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

      4. associate-*r/ 99.1%

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

      5. div-inv 98.9%

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

      6. times-frac 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. pow1 98.9%

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

      2. inv-pow 98.9%

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

      3. pow-div 99.0%

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

      4. metadata-eval 99.0%

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

      5. unpow2 99.0%

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

      6. sqr-sin-a 97.3%

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

   10. Applied egg-rr 97.3%

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

       1. *-commutative 97.3%

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

       2. associate-*r* 97.3%

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

       3. metadata-eval 97.3%

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

       4. *-lft-identity 97.3%

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

   12. Simplified 97.3%

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

Alternative 9: 75.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(0.25 + {x}^{2} \cdot 0.020833333333333332\right)}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.3333333333333333 + \cos x \cdot -1.3333333333333333}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.2e-5)
   (/ (* x (+ 0.25 (* (pow x 2.0) 0.020833333333333332))) 0.375)
   (/ (+ 1.3333333333333333 (* (cos x) -1.3333333333333333)) (sin x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.2e-5) {
		tmp = (x * (0.25 + (pow(x, 2.0) * 0.020833333333333332))) / 0.375;
	} else {
		tmp = (1.3333333333333333 + (cos(x) * -1.3333333333333333)) / sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.2d-5) then
        tmp = (x * (0.25d0 + ((x ** 2.0d0) * 0.020833333333333332d0))) / 0.375d0
    else
        tmp = (1.3333333333333333d0 + (cos(x) * (-1.3333333333333333d0))) / sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.2e-5) {
		tmp = (x * (0.25 + (Math.pow(x, 2.0) * 0.020833333333333332))) / 0.375;
	} else {
		tmp = (1.3333333333333333 + (Math.cos(x) * -1.3333333333333333)) / Math.sin(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.2e-5:
		tmp = (x * (0.25 + (math.pow(x, 2.0) * 0.020833333333333332))) / 0.375
	else:
		tmp = (1.3333333333333333 + (math.cos(x) * -1.3333333333333333)) / math.sin(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.2e-5)
		tmp = Float64(Float64(x * Float64(0.25 + Float64((x ^ 2.0) * 0.020833333333333332))) / 0.375);
	else
		tmp = Float64(Float64(1.3333333333333333 + Float64(cos(x) * -1.3333333333333333)) / sin(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.2e-5)
		tmp = (x * (0.25 + ((x ^ 2.0) * 0.020833333333333332))) / 0.375;
	else
		tmp = (1.3333333333333333 + (cos(x) * -1.3333333333333333)) / sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.2e-5], N[(N[(x * N[(0.25 + N[(N[Power[x, 2.0], $MachinePrecision] * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision], N[(N[(1.3333333333333333 + N[(N[Cos[x], $MachinePrecision] * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.19999999999999968e-5

   1. Initial program 66.2%

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

      1. associate-/l* 99.4%

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

      2. associate-*l* 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. clear-num 99.3%

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

      2. un-div-inv 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. metadata-eval 99.3%

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

      2. times-frac 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-/r* 99.7%

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

      6. associate-/r/ 99.7%

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

      7. associate-*l/ 66.4%

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

      8. unpow2 66.4%

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

   8. Applied egg-rr 66.4%

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

   9. Taylor expanded in x around 0 71.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(0.25 + 0.020833333333333332 \cdot {x}^{2}\right)}}{0.375} \]
   10. Step-by-step derivation

       1. *-commutative 71.2%

          \[\leadsto \frac{x \cdot \left(0.25 + \color{blue}{{x}^{2} \cdot 0.020833333333333332}\right)}{0.375} \]

   11. Simplified 71.2%

       \[\leadsto \frac{\color{blue}{x \cdot \left(0.25 + {x}^{2} \cdot 0.020833333333333332\right)}}{0.375} \]

   if 5.19999999999999968e-5 < x

   1. Initial program 99.0%

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

      1. associate-/l* 98.9%

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

      2. associate-*l* 99.0%

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

      3. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

      1. associate-*r* 98.9%

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

      2. *-commutative 98.9%

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

      3. div-inv 98.9%

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

      4. associate-*l* 98.9%

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

      5. associate-/r/ 98.9%

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

      6. un-div-inv 98.9%

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

      7. *-un-lft-identity 98.9%

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

      8. times-frac 99.1%

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

      9. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. clear-num 98.9%

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

      2. un-div-inv 99.0%

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

   8. Applied egg-rr 99.0%

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

      1. frac-2neg 99.0%

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

      2. associate-/r/ 99.0%

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

      3. metadata-eval 99.0%

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

      4. distribute-neg-frac2 99.0%

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

   10. Applied egg-rr 99.0%

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

       1. associate-*r/ 99.0%

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

       2. frac-2neg 99.0%

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

       3. associate-*l/ 99.0%

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

       4. unpow2 99.0%

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

       5. remove-double-neg 99.0%

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

   12. Applied egg-rr 99.0%

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

       1. unpow2 99.0%

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

       2. sin-mult 97.3%

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

   14. Applied egg-rr 97.3%

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

       1. div-sub 97.3%

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

       2. +-inverses 97.3%

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

       3. cos-0 97.3%

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

       4. metadata-eval 97.3%

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

       5. distribute-lft-out 97.3%

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

       6. metadata-eval 97.3%

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

       7. *-rgt-identity 97.3%

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

   16. Simplified 97.3%

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

       1. clear-num 97.4%

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

       2. associate-/r/ 97.3%

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

       3. neg-sub0 97.3%

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

       4. div-sub 96.7%

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

       5. associate--r- 96.7%

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

       6. metadata-eval 96.7%

          \[\leadsto \frac{1}{\sin x} \cdot \left(\left(0 - \color{blue}{-1.3333333333333333}\right) + \frac{\frac{\cos x}{2}}{-0.375}\right) \]

       7. metadata-eval 96.7%

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

       8. div-inv 96.7%

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

       9. metadata-eval 96.7%

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

       10. associate-/l* 96.8%

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

       11. metadata-eval 96.8%

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

   18. Applied egg-rr 96.8%

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

   19. Taylor expanded in x around inf 96.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(0.25 + {x}^{2} \cdot 0.020833333333333332\right)}{0.375}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.3333333333333333 + \cos x \cdot -1.3333333333333333}{\sin x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 32.2% 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 75.1%

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

   1. *-commutative 75.1%

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

   2. associate-/l* 99.2%

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

   3. remove-double-neg 99.2%

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

   4. sin-neg 99.2%

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

   5. distribute-lft-neg-out 99.2%

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

   6. distribute-rgt-neg-in 99.2%

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

   7. distribute-frac-neg 99.2%

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

   8. distribute-frac-neg2 99.2%

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

   9. neg-mul-1 99.2%

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

   10. associate-/r* 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 56.3%

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

   1. add-sqr-sqrt 29.4%

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

   2. sqrt-unprod 20.4%

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

   3. swap-sqr 20.4%

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

   4. unpow2 20.4%

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

   5. metadata-eval 20.4%

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

7. Applied egg-rr 20.4%

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

   1. unpow2 20.4%

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

   2. metadata-eval 20.4%

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

   3. swap-sqr 20.4%

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

   4. rem-sqrt-square 33.4%

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

   5. *-commutative 33.4%

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

9. Simplified 33.4%

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

10. Final simplification 33.4%

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

Alternative 11: 56.0% 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 75.1%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. associate-*r* 99.2%

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

   2. *-commutative 99.2%

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

   3. div-inv 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-/r/ 99.1%

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

   6. un-div-inv 99.2%

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

   7. *-un-lft-identity 99.2%

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

   8. times-frac 99.5%

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

   9. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0 56.5%

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

Alternative 12: 55.8% 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 75.1%

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

   1. *-commutative 75.1%

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

   2. associate-/l* 99.2%

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

   3. remove-double-neg 99.2%

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

   4. sin-neg 99.2%

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

   5. distribute-lft-neg-out 99.2%

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

   6. distribute-rgt-neg-in 99.2%

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

   7. distribute-frac-neg 99.2%

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

   8. distribute-frac-neg2 99.2%

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

   9. neg-mul-1 99.2%

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

   10. associate-/r* 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 56.3%

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

Alternative 13: 51.8% 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 75.1%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. clear-num 99.2%

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

   2. un-div-inv 99.3%

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

6. Applied egg-rr 99.3%

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

   1. metadata-eval 99.3%

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

   2. times-frac 99.5%

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

   3. *-un-lft-identity 99.5%

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

   4. *-commutative 99.5%

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

   5. associate-/r* 99.5%

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

   6. associate-/r/ 99.5%

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

   7. associate-*l/ 75.3%

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

   8. unpow2 75.3%

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

8. Applied egg-rr 75.3%

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

9. Taylor expanded in x around 0 53.1%

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

    1. *-commutative 53.1%

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

11. Simplified 53.1%

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

Alternative 14: 51.6% 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 75.1%

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

   1. associate-/l* 99.2%

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

   2. associate-*l* 99.2%

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

   3. metadata-eval 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 52.8%

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

6. Final simplification 52.8%

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

Developer target: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024105 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

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

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