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

Percentage Accurate: 77.0% → 99.6%

Time: 11.8s

Alternatives: 5

Speedup: 3.0×

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: 77.0% 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: 99.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{\frac{\tan \left(x \cdot 0.5\right)}{0.75}}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ 1.0 (/ (tan (* x 0.5)) 0.75))))

C

double code(double x) {
	return 1.0 / (1.0 / (tan((x * 0.5)) / 0.75));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 / (tan((x * 0.5d0)) / 0.75d0))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 / (Math.tan((x * 0.5)) / 0.75));
}

Python

def code(x):
	return 1.0 / (1.0 / (math.tan((x * 0.5)) / 0.75))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 / Float64(tan(Float64(x * 0.5)) / 0.75)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 / (tan((x * 0.5)) / 0.75));
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 / N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.4%

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

   1. *-commutative 75.4%

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

   2. *-commutative 75.4%

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

   3. associate-*r* 75.3%

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

   4. sqr-neg 75.3%

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

   5. sin-neg 75.3%

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

   6. distribute-lft-neg-out 75.3%

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

   7. sin-neg 75.3%

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

   8. distribute-lft-neg-out 75.3%

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

   9. associate-/l* 75.4%

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

3. Simplified 75.4%

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

   1. *-commutative 75.4%

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

   2. associate-*l/ 75.3%

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

   3. associate-*l* 75.4%

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

   4. metadata-eval 75.4%

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

   5. add-log-exp 51.4%

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

   6. *-un-lft-identity 51.4%

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

   7. log-prod 51.4%

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

   8. metadata-eval 51.4%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}}\right) \]

   9. add-log-exp 75.4%

      \[\leadsto 0 + \color{blue}{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]

   10. metadata-eval 75.4%

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

   11. associate-*l* 75.3%

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

   12. associate-*l/ 75.4%

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

   13. clear-num 75.4%

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

6. Applied egg-rr 50.5%

   \[\leadsto \color{blue}{0 + \frac{1 - \cos x}{\left(\sin x \cdot 0.375\right) \cdot 2}} \]
7. Step-by-step derivation

   1. +-lft-identity 50.5%

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

   2. *-rgt-identity 50.5%

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

   3. associate-*l* 50.5%

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

   4. times-frac 50.4%

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

   5. metadata-eval 50.4%

      \[\leadsto \frac{1 - \cos x}{\sin x} \cdot \frac{1}{\color{blue}{0.75}} \]

   6. metadata-eval 50.4%

      \[\leadsto \frac{1 - \cos x}{\sin x} \cdot \color{blue}{1.3333333333333333} \]

   7. metadata-eval 50.4%

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

   8. times-frac 50.5%

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

   9. *-commutative 50.5%

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

   10. *-commutative 50.5%

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

   11. times-frac 50.4%

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

   12. metadata-eval 50.4%

       \[\leadsto \color{blue}{1.3333333333333333} \cdot \frac{1 - \cos x}{\sin x} \]

   13. hang-p0-tan 99.4%

       \[\leadsto 1.3333333333333333 \cdot \color{blue}{\tan \left(\frac{x}{2}\right)} \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \tan \left(\frac{x}{2}\right)} \]
9. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right) \cdot 1.3333333333333333} \]

   2. tan-quot 99.3%

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

   3. div-inv 99.3%

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

   4. metadata-eval 99.3%

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

   5. associate-*l/ 99.4%

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

   6. div-inv 99.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot 1.3333333333333333}{\cos \color{blue}{\left(x \cdot \frac{1}{2}\right)}} \]

   7. metadata-eval 99.4%

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

10. Applied egg-rr 99.4%

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

    1. clear-num 99.2%

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

    2. clear-num 99.3%

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

    3. *-commutative 99.3%

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

    4. associate-*r/ 99.2%

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

    5. tan-quot 99.3%

       \[\leadsto \frac{1}{\frac{1}{1.3333333333333333 \cdot \color{blue}{\tan \left(x \cdot 0.5\right)}}} \]

12. Applied egg-rr 99.3%

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

    1. associate-/r* 99.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{1.3333333333333333}}{\tan \left(x \cdot 0.5\right)}}} \]

    2. clear-num 99.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\tan \left(x \cdot 0.5\right)}{\frac{1}{1.3333333333333333}}}}} \]

    3. metadata-eval 99.6%

       \[\leadsto \frac{1}{\frac{1}{\frac{\tan \left(x \cdot 0.5\right)}{\color{blue}{0.75}}}} \]

14. Applied egg-rr 99.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\tan \left(x \cdot 0.5\right)}{0.75}}}} \]
15. Add Preprocessing

Alternative 2: 99.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{0.75}{\tan \left(x \cdot 0.5\right)}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ 0.75 (tan (* x 0.5)))))

C

double code(double x) {
	return 1.0 / (0.75 / tan((x * 0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (0.75d0 / tan((x * 0.5d0)))
end function

Java

public static double code(double x) {
	return 1.0 / (0.75 / Math.tan((x * 0.5)));
}

Python

def code(x):
	return 1.0 / (0.75 / math.tan((x * 0.5)))

Julia

function code(x)
	return Float64(1.0 / Float64(0.75 / tan(Float64(x * 0.5))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (0.75 / tan((x * 0.5)));
end

Wolfram

code[x_] := N[(1.0 / N[(0.75 / N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.4%

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

   1. *-commutative 75.4%

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

   2. *-commutative 75.4%

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

   3. associate-*r* 75.3%

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

   4. sqr-neg 75.3%

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

   5. sin-neg 75.3%

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

   6. distribute-lft-neg-out 75.3%

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

   7. sin-neg 75.3%

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

   8. distribute-lft-neg-out 75.3%

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

   9. associate-/l* 75.4%

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

3. Simplified 75.4%

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

   1. *-commutative 75.4%

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

   2. associate-*l/ 75.3%

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

   3. associate-*l* 75.4%

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

   4. metadata-eval 75.4%

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

   5. add-log-exp 51.4%

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

   6. *-un-lft-identity 51.4%

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

   7. log-prod 51.4%

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

   8. metadata-eval 51.4%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}}\right) \]

   9. add-log-exp 75.4%

      \[\leadsto 0 + \color{blue}{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]

   10. metadata-eval 75.4%

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

   11. associate-*l* 75.3%

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

   12. associate-*l/ 75.4%

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

   13. clear-num 75.4%

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

6. Applied egg-rr 50.5%

   \[\leadsto \color{blue}{0 + \frac{1 - \cos x}{\left(\sin x \cdot 0.375\right) \cdot 2}} \]
7. Step-by-step derivation

   1. +-lft-identity 50.5%

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

   2. *-rgt-identity 50.5%

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

   3. associate-*l* 50.5%

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

   4. times-frac 50.4%

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

   5. metadata-eval 50.4%

      \[\leadsto \frac{1 - \cos x}{\sin x} \cdot \frac{1}{\color{blue}{0.75}} \]

   6. metadata-eval 50.4%

      \[\leadsto \frac{1 - \cos x}{\sin x} \cdot \color{blue}{1.3333333333333333} \]

   7. metadata-eval 50.4%

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

   8. times-frac 50.5%

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

   9. *-commutative 50.5%

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

   10. *-commutative 50.5%

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

   11. times-frac 50.4%

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

   12. metadata-eval 50.4%

       \[\leadsto \color{blue}{1.3333333333333333} \cdot \frac{1 - \cos x}{\sin x} \]

   13. hang-p0-tan 99.4%

       \[\leadsto 1.3333333333333333 \cdot \color{blue}{\tan \left(\frac{x}{2}\right)} \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \tan \left(\frac{x}{2}\right)} \]
9. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right) \cdot 1.3333333333333333} \]

   2. tan-quot 99.3%

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

   3. div-inv 99.3%

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

   4. metadata-eval 99.3%

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

   5. associate-*l/ 99.4%

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

   6. div-inv 99.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot 1.3333333333333333}{\cos \color{blue}{\left(x \cdot \frac{1}{2}\right)}} \]

   7. metadata-eval 99.4%

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

10. Applied egg-rr 99.4%

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

    1. clear-num 99.2%

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

    2. clear-num 99.3%

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

    3. *-commutative 99.3%

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

    4. associate-*r/ 99.2%

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

    5. tan-quot 99.3%

       \[\leadsto \frac{1}{\frac{1}{1.3333333333333333 \cdot \color{blue}{\tan \left(x \cdot 0.5\right)}}} \]

12. Applied egg-rr 99.3%

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

    1. *-un-lft-identity 99.3%

       \[\leadsto \frac{1}{\color{blue}{1 \cdot \frac{1}{1.3333333333333333 \cdot \tan \left(x \cdot 0.5\right)}}} \]

    2. associate-/r* 99.5%

       \[\leadsto \frac{1}{1 \cdot \color{blue}{\frac{\frac{1}{1.3333333333333333}}{\tan \left(x \cdot 0.5\right)}}} \]

    3. metadata-eval 99.5%

       \[\leadsto \frac{1}{1 \cdot \frac{\color{blue}{0.75}}{\tan \left(x \cdot 0.5\right)}} \]

14. Applied egg-rr 99.5%

    \[\leadsto \frac{1}{\color{blue}{1 \cdot \frac{0.75}{\tan \left(x \cdot 0.5\right)}}} \]
15. Step-by-step derivation

    1. *-lft-identity 99.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{0.75}{\tan \left(x \cdot 0.5\right)}}} \]

16. Simplified 99.5%

    \[\leadsto \frac{1}{\color{blue}{\frac{0.75}{\tan \left(x \cdot 0.5\right)}}} \]
17. Add Preprocessing

Alternative 3: 99.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 1.3333333333333333 \cdot \tan \left(\frac{x}{2}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 1.3333333333333333 (tan (/ x 2.0))))

C

double code(double x) {
	return 1.3333333333333333 * tan((x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.3333333333333333d0 * tan((x / 2.0d0))
end function

Java

public static double code(double x) {
	return 1.3333333333333333 * Math.tan((x / 2.0));
}

Python

def code(x):
	return 1.3333333333333333 * math.tan((x / 2.0))

Julia

function code(x)
	return Float64(1.3333333333333333 * tan(Float64(x / 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.3333333333333333 * tan((x / 2.0));
end

Wolfram

code[x_] := N[(1.3333333333333333 * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.4%

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

   1. *-commutative 75.4%

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

   2. *-commutative 75.4%

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

   3. associate-*r* 75.3%

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

   4. sqr-neg 75.3%

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

   5. sin-neg 75.3%

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

   6. distribute-lft-neg-out 75.3%

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

   7. sin-neg 75.3%

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

   8. distribute-lft-neg-out 75.3%

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

   9. associate-/l* 75.4%

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

3. Simplified 75.4%

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

   1. *-commutative 75.4%

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

   2. associate-*l/ 75.3%

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

   3. associate-*l* 75.4%

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

   4. metadata-eval 75.4%

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

   5. add-log-exp 51.4%

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

   6. *-un-lft-identity 51.4%

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

   7. log-prod 51.4%

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

   8. metadata-eval 51.4%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}}\right) \]

   9. add-log-exp 75.4%

      \[\leadsto 0 + \color{blue}{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]

   10. metadata-eval 75.4%

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

   11. associate-*l* 75.3%

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

   12. associate-*l/ 75.4%

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

   13. clear-num 75.4%

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

6. Applied egg-rr 50.5%

   \[\leadsto \color{blue}{0 + \frac{1 - \cos x}{\left(\sin x \cdot 0.375\right) \cdot 2}} \]
7. Step-by-step derivation

   1. +-lft-identity 50.5%

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

   2. *-rgt-identity 50.5%

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

   3. associate-*l* 50.5%

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

   4. times-frac 50.4%

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

   5. metadata-eval 50.4%

      \[\leadsto \frac{1 - \cos x}{\sin x} \cdot \frac{1}{\color{blue}{0.75}} \]

   6. metadata-eval 50.4%

      \[\leadsto \frac{1 - \cos x}{\sin x} \cdot \color{blue}{1.3333333333333333} \]

   7. metadata-eval 50.4%

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

   8. times-frac 50.5%

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

   9. *-commutative 50.5%

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

   10. *-commutative 50.5%

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

   11. times-frac 50.4%

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

   12. metadata-eval 50.4%

       \[\leadsto \color{blue}{1.3333333333333333} \cdot \frac{1 - \cos x}{\sin x} \]

   13. hang-p0-tan 99.4%

       \[\leadsto 1.3333333333333333 \cdot \color{blue}{\tan \left(\frac{x}{2}\right)} \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \tan \left(\frac{x}{2}\right)} \]
9. Add Preprocessing

Alternative 4: 50.2% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1.5}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ 1.5 x)))

C

double code(double x) {
	return 1.0 / (1.5 / x);
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 / (1.5 / x);
}

Python

def code(x):
	return 1.0 / (1.5 / x)

Julia

function code(x)
	return Float64(1.0 / Float64(1.5 / x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.5 / x);
end

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. *-commutative 75.4%

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

   2. *-commutative 75.4%

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

   3. associate-*r* 75.3%

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

   4. sqr-neg 75.3%

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

   5. sin-neg 75.3%

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

   6. distribute-lft-neg-out 75.3%

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

   7. sin-neg 75.3%

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

   8. distribute-lft-neg-out 75.3%

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

   9. associate-/l* 75.4%

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

3. Simplified 75.4%

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

   1. *-commutative 75.4%

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

   2. associate-*l/ 75.3%

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

   3. associate-*l* 75.4%

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

   4. metadata-eval 75.4%

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

   5. add-log-exp 51.4%

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

   6. *-un-lft-identity 51.4%

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

   7. log-prod 51.4%

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

   8. metadata-eval 51.4%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}}\right) \]

   9. add-log-exp 75.4%

      \[\leadsto 0 + \color{blue}{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}} \]

   10. metadata-eval 75.4%

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

   11. associate-*l* 75.3%

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

   12. associate-*l/ 75.4%

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

   13. clear-num 75.4%

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

6. Applied egg-rr 50.5%

   \[\leadsto \color{blue}{0 + \frac{1 - \cos x}{\left(\sin x \cdot 0.375\right) \cdot 2}} \]
7. Step-by-step derivation

   1. +-lft-identity 50.5%

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

   2. *-rgt-identity 50.5%

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

   3. associate-*l* 50.5%

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

   4. times-frac 50.4%

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

   5. metadata-eval 50.4%

      \[\leadsto \frac{1 - \cos x}{\sin x} \cdot \frac{1}{\color{blue}{0.75}} \]

   6. metadata-eval 50.4%

      \[\leadsto \frac{1 - \cos x}{\sin x} \cdot \color{blue}{1.3333333333333333} \]

   7. metadata-eval 50.4%

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

   8. times-frac 50.5%

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

   9. *-commutative 50.5%

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

   10. *-commutative 50.5%

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

   11. times-frac 50.4%

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

   12. metadata-eval 50.4%

       \[\leadsto \color{blue}{1.3333333333333333} \cdot \frac{1 - \cos x}{\sin x} \]

   13. hang-p0-tan 99.4%

       \[\leadsto 1.3333333333333333 \cdot \color{blue}{\tan \left(\frac{x}{2}\right)} \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \tan \left(\frac{x}{2}\right)} \]
9. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right) \cdot 1.3333333333333333} \]

   2. tan-quot 99.3%

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

   3. div-inv 99.3%

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

   4. metadata-eval 99.3%

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

   5. associate-*l/ 99.4%

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

   6. div-inv 99.4%

      \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot 1.3333333333333333}{\cos \color{blue}{\left(x \cdot \frac{1}{2}\right)}} \]

   7. metadata-eval 99.4%

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

10. Applied egg-rr 99.4%

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

    1. clear-num 99.2%

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

    2. clear-num 99.3%

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

    3. *-commutative 99.3%

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

    4. associate-*r/ 99.2%

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

    5. tan-quot 99.3%

       \[\leadsto \frac{1}{\frac{1}{1.3333333333333333 \cdot \color{blue}{\tan \left(x \cdot 0.5\right)}}} \]

12. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\frac{1}{\frac{1}{1.3333333333333333 \cdot \tan \left(x \cdot 0.5\right)}}} \]

13. Taylor expanded in x around 0 53.3%

    \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
14. Add Preprocessing

Alternative 5: 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 75.4%

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

   1. *-commutative 75.4%

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

   2. *-commutative 75.4%

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

   3. associate-*r* 75.3%

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

   4. sqr-neg 75.3%

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

   5. sin-neg 75.3%

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

   6. distribute-lft-neg-out 75.3%

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

   7. sin-neg 75.3%

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

   8. distribute-lft-neg-out 75.3%

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

   9. associate-/l* 75.4%

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

3. Simplified 75.4%

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

5. Taylor expanded in x around 0 53.2%

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

6. Final simplification 53.2%

   \[\leadsto x \cdot 0.6666666666666666 \]
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 2024096 
(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)))