Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 55.0s

Alternatives: 11

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 6.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 8.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), 0.5 \cdot \sqrt[3]{{\pi}^{3}}\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (/
   (fma 2.0 (asin (sqrt (- 0.5 (* 0.5 x)))) (* 0.5 (cbrt (pow PI 3.0))))
   (log
    (exp
     (-
      (* (pow PI 2.0) 0.25)
      (* 4.0 (expm1 (log1p (pow (asin (sqrt (+ 0.5 (* x -0.5)))) 2.0))))))))))

C

double code(double x) {
	return 1.0 / (fma(2.0, asin(sqrt((0.5 - (0.5 * x)))), (0.5 * cbrt(pow(((double) M_PI), 3.0)))) / log(exp(((pow(((double) M_PI), 2.0) * 0.25) - (4.0 * expm1(log1p(pow(asin(sqrt((0.5 + (x * -0.5)))), 2.0))))))));
}

Julia

function code(x)
	return Float64(1.0 / Float64(fma(2.0, asin(sqrt(Float64(0.5 - Float64(0.5 * x)))), Float64(0.5 * cbrt((pi ^ 3.0)))) / log(exp(Float64(Float64((pi ^ 2.0) * 0.25) - Float64(4.0 * expm1(log1p((asin(sqrt(Float64(0.5 + Float64(x * -0.5)))) ^ 2.0)))))))))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(2.0 * N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Log[N[Exp[N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25), $MachinePrecision] - N[(4.0 * N[(Exp[N[Log[1 + N[Power[N[ArcSin[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 6.5%

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

   2. clear-num 6.5%

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

4. Applied egg-rr 6.5%

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

   1. add-log-exp 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\color{blue}{\log \left(e^{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}\right)}}} \]

   2. unpow-prod-down 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{\color{blue}{{\pi}^{2} \cdot {0.5}^{2}} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}\right)}} \]

   3. metadata-eval 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot \color{blue}{0.25} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}\right)}} \]

   4. unpow-prod-down 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - \color{blue}{{2}^{2} \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}}\right)}} \]

   5. metadata-eval 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - \color{blue}{4} \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}\right)}} \]

   6. *-commutative 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot {\sin^{-1} \left(\sqrt{0.5 - \color{blue}{0.5 \cdot x}}\right)}^{2}}\right)}} \]

6. Applied egg-rr 6.5%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\color{blue}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot {\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}}\right)}}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}\right)\right)}}\right)}} \]

   2. expm1-undefine 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}\right)} - 1\right)}}\right)}} \]

   3. add-sqr-sqrt 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}} \cdot \sqrt{{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}}}\right)} - 1\right)}\right)}} \]

   4. add-sqr-sqrt 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}}\right)} - 1\right)}\right)}} \]

   5. cancel-sign-sub-inv 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)}^{2}\right)} - 1\right)}\right)}} \]

   6. metadata-eval 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)}^{2}\right)} - 1\right)}\right)}} \]

8. Applied egg-rr 8.3%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)}^{2}\right)} - 1\right)}}\right)}} \]
9. Step-by-step derivation

   1. expm1-define 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)}^{2}\right)\right)}}\right)}} \]

   2. *-commutative 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot -0.5}}\right)}^{2}\right)\right)}\right)}} \]

10. Simplified 8.3%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}}\right)}} \]
11. Step-by-step derivation

    1. add-cbrt-cube 8.3%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}\right)}} \]

    2. pow3 8.3%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}\right)}} \]

12. Applied egg-rr 8.3%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}\right)}} \]

13. Final simplification 8.3%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), 0.5 \cdot \sqrt[3]{{\pi}^{3}}\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}\right)}} \]
14. Add Preprocessing

Alternative 2: 8.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(2, 0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), 0.5 \cdot \pi\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (/
   (fma 2.0 (- (* 0.5 PI) (acos (sqrt (- 0.5 (* 0.5 x))))) (* 0.5 PI))
   (log
    (exp
     (-
      (* (pow PI 2.0) 0.25)
      (* 4.0 (expm1 (log1p (pow (asin (sqrt (+ 0.5 (* x -0.5)))) 2.0))))))))))

C

double code(double x) {
	return 1.0 / (fma(2.0, ((0.5 * ((double) M_PI)) - acos(sqrt((0.5 - (0.5 * x))))), (0.5 * ((double) M_PI))) / log(exp(((pow(((double) M_PI), 2.0) * 0.25) - (4.0 * expm1(log1p(pow(asin(sqrt((0.5 + (x * -0.5)))), 2.0))))))));
}

Julia

function code(x)
	return Float64(1.0 / Float64(fma(2.0, Float64(Float64(0.5 * pi) - acos(sqrt(Float64(0.5 - Float64(0.5 * x))))), Float64(0.5 * pi)) / log(exp(Float64(Float64((pi ^ 2.0) * 0.25) - Float64(4.0 * expm1(log1p((asin(sqrt(Float64(0.5 + Float64(x * -0.5)))) ^ 2.0)))))))))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(2.0 * N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] / N[Log[N[Exp[N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25), $MachinePrecision] - N[(4.0 * N[(Exp[N[Log[1 + N[Power[N[ArcSin[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 6.5%

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

   2. clear-num 6.5%

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

4. Applied egg-rr 6.5%

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

   1. add-log-exp 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\color{blue}{\log \left(e^{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}\right)}}} \]

   2. unpow-prod-down 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{\color{blue}{{\pi}^{2} \cdot {0.5}^{2}} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}\right)}} \]

   3. metadata-eval 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot \color{blue}{0.25} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}\right)}} \]

   4. unpow-prod-down 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - \color{blue}{{2}^{2} \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}}\right)}} \]

   5. metadata-eval 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - \color{blue}{4} \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}\right)}} \]

   6. *-commutative 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot {\sin^{-1} \left(\sqrt{0.5 - \color{blue}{0.5 \cdot x}}\right)}^{2}}\right)}} \]

6. Applied egg-rr 6.5%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\color{blue}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot {\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}}\right)}}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}\right)\right)}}\right)}} \]

   2. expm1-undefine 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}\right)} - 1\right)}}\right)}} \]

   3. add-sqr-sqrt 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}} \cdot \sqrt{{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}}}\right)} - 1\right)}\right)}} \]

   4. add-sqr-sqrt 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}}\right)} - 1\right)}\right)}} \]

   5. cancel-sign-sub-inv 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)}^{2}\right)} - 1\right)}\right)}} \]

   6. metadata-eval 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)}^{2}\right)} - 1\right)}\right)}} \]

8. Applied egg-rr 8.3%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)}^{2}\right)} - 1\right)}}\right)}} \]
9. Step-by-step derivation

   1. expm1-define 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)}^{2}\right)\right)}}\right)}} \]

   2. *-commutative 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot -0.5}}\right)}^{2}\right)\right)}\right)}} \]

10. Simplified 8.3%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}}\right)}} \]
11. Step-by-step derivation

    1. asin-acos 8.2%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}\right)}^{2}}} \]

    2. div-inv 8.2%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)\right)}^{2}}} \]

    3. metadata-eval 8.2%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)\right)}^{2}}} \]

    4. *-commutative 8.2%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - \color{blue}{0.5 \cdot x}}\right)\right)\right)}^{2}}} \]

12. Applied egg-rr 8.3%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \color{blue}{\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}, \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}\right)}} \]

13. Final simplification 8.3%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, 0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), 0.5 \cdot \pi\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}\right)}} \]
14. Add Preprocessing

Alternative 3: 8.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), 0.5 \cdot \pi\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (/
   (fma 2.0 (asin (sqrt (- 0.5 (* 0.5 x)))) (* 0.5 PI))
   (log
    (exp
     (-
      (* (pow PI 2.0) 0.25)
      (* 4.0 (expm1 (log1p (pow (asin (sqrt (+ 0.5 (* x -0.5)))) 2.0))))))))))

C

double code(double x) {
	return 1.0 / (fma(2.0, asin(sqrt((0.5 - (0.5 * x)))), (0.5 * ((double) M_PI))) / log(exp(((pow(((double) M_PI), 2.0) * 0.25) - (4.0 * expm1(log1p(pow(asin(sqrt((0.5 + (x * -0.5)))), 2.0))))))));
}

Julia

function code(x)
	return Float64(1.0 / Float64(fma(2.0, asin(sqrt(Float64(0.5 - Float64(0.5 * x)))), Float64(0.5 * pi)) / log(exp(Float64(Float64((pi ^ 2.0) * 0.25) - Float64(4.0 * expm1(log1p((asin(sqrt(Float64(0.5 + Float64(x * -0.5)))) ^ 2.0)))))))))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(2.0 * N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] / N[Log[N[Exp[N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25), $MachinePrecision] - N[(4.0 * N[(Exp[N[Log[1 + N[Power[N[ArcSin[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 6.5%

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

   2. clear-num 6.5%

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

4. Applied egg-rr 6.5%

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

   1. add-log-exp 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\color{blue}{\log \left(e^{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}\right)}}} \]

   2. unpow-prod-down 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{\color{blue}{{\pi}^{2} \cdot {0.5}^{2}} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}\right)}} \]

   3. metadata-eval 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot \color{blue}{0.25} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}\right)}} \]

   4. unpow-prod-down 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - \color{blue}{{2}^{2} \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}}\right)}} \]

   5. metadata-eval 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - \color{blue}{4} \cdot {\sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)}^{2}}\right)}} \]

   6. *-commutative 6.5%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot {\sin^{-1} \left(\sqrt{0.5 - \color{blue}{0.5 \cdot x}}\right)}^{2}}\right)}} \]

6. Applied egg-rr 6.5%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\color{blue}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot {\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}}\right)}}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}\right)\right)}}\right)}} \]

   2. expm1-undefine 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}\right)} - 1\right)}}\right)}} \]

   3. add-sqr-sqrt 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}} \cdot \sqrt{{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}}}\right)} - 1\right)}\right)}} \]

   4. add-sqr-sqrt 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}}\right)} - 1\right)}\right)}} \]

   5. cancel-sign-sub-inv 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)}^{2}\right)} - 1\right)}\right)}} \]

   6. metadata-eval 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)}^{2}\right)} - 1\right)}\right)}} \]

8. Applied egg-rr 8.3%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)}^{2}\right)} - 1\right)}}\right)}} \]
9. Step-by-step derivation

   1. expm1-define 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)}^{2}\right)\right)}}\right)}} \]

   2. *-commutative 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot -0.5}}\right)}^{2}\right)\right)}\right)}} \]

10. Simplified 8.3%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}}\right)}} \]

11. Final simplification 8.3%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), 0.5 \cdot \pi\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}\right)}} \]
12. Add Preprocessing

Alternative 4: 8.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (-
   (* (pow PI 2.0) 0.25)
   (* 4.0 (expm1 (log1p (pow (asin (sqrt (+ 0.5 (* x -0.5)))) 2.0)))))
  (+ (* 0.5 PI) (* 2.0 (asin (sqrt (- 0.5 (* 0.5 x))))))))

C

double code(double x) {
	return ((pow(((double) M_PI), 2.0) * 0.25) - (4.0 * expm1(log1p(pow(asin(sqrt((0.5 + (x * -0.5)))), 2.0))))) / ((0.5 * ((double) M_PI)) + (2.0 * asin(sqrt((0.5 - (0.5 * x))))));
}

Java

public static double code(double x) {
	return ((Math.pow(Math.PI, 2.0) * 0.25) - (4.0 * Math.expm1(Math.log1p(Math.pow(Math.asin(Math.sqrt((0.5 + (x * -0.5)))), 2.0))))) / ((0.5 * Math.PI) + (2.0 * Math.asin(Math.sqrt((0.5 - (0.5 * x))))));
}

Python

def code(x):
	return ((math.pow(math.pi, 2.0) * 0.25) - (4.0 * math.expm1(math.log1p(math.pow(math.asin(math.sqrt((0.5 + (x * -0.5)))), 2.0))))) / ((0.5 * math.pi) + (2.0 * math.asin(math.sqrt((0.5 - (0.5 * x))))))

Julia

function code(x)
	return Float64(Float64(Float64((pi ^ 2.0) * 0.25) - Float64(4.0 * expm1(log1p((asin(sqrt(Float64(0.5 + Float64(x * -0.5)))) ^ 2.0))))) / Float64(Float64(0.5 * pi) + Float64(2.0 * asin(sqrt(Float64(0.5 - Float64(0.5 * x)))))))
end

Wolfram

code[x_] := N[(N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25), $MachinePrecision] - N[(4.0 * N[(Exp[N[Log[1 + N[Power[N[ArcSin[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * Pi), $MachinePrecision] + N[(2.0 * N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 6.5%

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

   2. clear-num 6.5%

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

4. Applied egg-rr 6.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}}} \]

5. Taylor expanded in x around 0 6.5%

   \[\leadsto \color{blue}{\frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}\right)\right)}}\right)}} \]

   2. expm1-undefine 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}\right)} - 1\right)}}\right)}} \]

   3. add-sqr-sqrt 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}} \cdot \sqrt{{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}}}\right)} - 1\right)}\right)}} \]

   4. add-sqr-sqrt 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}}\right)} - 1\right)}\right)}} \]

   5. cancel-sign-sub-inv 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)}^{2}\right)} - 1\right)}\right)}} \]

   6. metadata-eval 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)}^{2}\right)} - 1\right)}\right)}} \]

7. Applied egg-rr 8.2%

   \[\leadsto \frac{0.25 \cdot {\pi}^{2} - 4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)}^{2}\right)} - 1\right)}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \]
8. Step-by-step derivation

   1. expm1-define 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)}^{2}\right)\right)}}\right)}} \]

   2. *-commutative 8.3%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{\log \left(e^{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot -0.5}}\right)}^{2}\right)\right)}\right)}} \]

9. Simplified 8.2%

   \[\leadsto \frac{0.25 \cdot {\pi}^{2} - 4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \]

10. Final simplification 8.2%

    \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - 4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)}^{2}\right)\right)}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \]
11. Add Preprocessing

Alternative 5: 8.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.5 - 0.5 \cdot x}\\ \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} t\_0, 0.5 \cdot \pi\right)}{{\left(0.5 \cdot \pi\right)}^{2} - {\left(2 \cdot \left(0.5 \cdot \pi - \cos^{-1} t\_0\right)\right)}^{2}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (- 0.5 (* 0.5 x)))))
   (/
    1.0
    (/
     (fma 2.0 (asin t_0) (* 0.5 PI))
     (- (pow (* 0.5 PI) 2.0) (pow (* 2.0 (- (* 0.5 PI) (acos t_0))) 2.0))))))

C

double code(double x) {
	double t_0 = sqrt((0.5 - (0.5 * x)));
	return 1.0 / (fma(2.0, asin(t_0), (0.5 * ((double) M_PI))) / (pow((0.5 * ((double) M_PI)), 2.0) - pow((2.0 * ((0.5 * ((double) M_PI)) - acos(t_0))), 2.0)));
}

Julia

function code(x)
	t_0 = sqrt(Float64(0.5 - Float64(0.5 * x)))
	return Float64(1.0 / Float64(fma(2.0, asin(t_0), Float64(0.5 * pi)) / Float64((Float64(0.5 * pi) ^ 2.0) - (Float64(2.0 * Float64(Float64(0.5 * pi) - acos(t_0))) ^ 2.0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(1.0 / N[(N[(2.0 * N[ArcSin[t$95$0], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(0.5 * Pi), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(2.0 * N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcCos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.6%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 6.5%

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

   2. clear-num 6.5%

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

4. Applied egg-rr 6.5%

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

   1. asin-acos 8.2%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}\right)}^{2}}} \]

   2. div-inv 8.2%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)\right)}^{2}}} \]

   3. metadata-eval 8.2%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)\right)}^{2}}} \]

   4. *-commutative 8.2%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - \color{blue}{0.5 \cdot x}}\right)\right)\right)}^{2}}} \]

6. Applied egg-rr 8.2%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}\right)}^{2}}} \]

7. Final simplification 8.2%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right), 0.5 \cdot \pi\right)}{{\left(0.5 \cdot \pi\right)}^{2} - {\left(2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}}} \]
8. Add Preprocessing

Alternative 6: 8.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\pi}^{2} \cdot 0.25 - 4 \cdot {\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (-
   (* (pow PI 2.0) 0.25)
   (* 4.0 (pow (- (* 0.5 PI) (acos (sqrt (+ 0.5 (* x -0.5))))) 2.0)))
  (+ (* 0.5 PI) (* 2.0 (asin (sqrt (- 0.5 (* 0.5 x))))))))

C

double code(double x) {
	return ((pow(((double) M_PI), 2.0) * 0.25) - (4.0 * pow(((0.5 * ((double) M_PI)) - acos(sqrt((0.5 + (x * -0.5))))), 2.0))) / ((0.5 * ((double) M_PI)) + (2.0 * asin(sqrt((0.5 - (0.5 * x))))));
}

Java

public static double code(double x) {
	return ((Math.pow(Math.PI, 2.0) * 0.25) - (4.0 * Math.pow(((0.5 * Math.PI) - Math.acos(Math.sqrt((0.5 + (x * -0.5))))), 2.0))) / ((0.5 * Math.PI) + (2.0 * Math.asin(Math.sqrt((0.5 - (0.5 * x))))));
}

Python

def code(x):
	return ((math.pow(math.pi, 2.0) * 0.25) - (4.0 * math.pow(((0.5 * math.pi) - math.acos(math.sqrt((0.5 + (x * -0.5))))), 2.0))) / ((0.5 * math.pi) + (2.0 * math.asin(math.sqrt((0.5 - (0.5 * x))))))

Julia

function code(x)
	return Float64(Float64(Float64((pi ^ 2.0) * 0.25) - Float64(4.0 * (Float64(Float64(0.5 * pi) - acos(sqrt(Float64(0.5 + Float64(x * -0.5))))) ^ 2.0))) / Float64(Float64(0.5 * pi) + Float64(2.0 * asin(sqrt(Float64(0.5 - Float64(0.5 * x)))))))
end

MATLAB

function tmp = code(x)
	tmp = (((pi ^ 2.0) * 0.25) - (4.0 * (((0.5 * pi) - acos(sqrt((0.5 + (x * -0.5))))) ^ 2.0))) / ((0.5 * pi) + (2.0 * asin(sqrt((0.5 - (0.5 * x))))));
end

Wolfram

code[x_] := N[(N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.25), $MachinePrecision] - N[(4.0 * N[Power[N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * Pi), $MachinePrecision] + N[(2.0 * N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 6.5%

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

   2. clear-num 6.5%

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

4. Applied egg-rr 6.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(2, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right), \pi \cdot 0.5\right)}{{\left(\pi \cdot 0.5\right)}^{2} - {\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)}^{2}}}} \]

5. Taylor expanded in x around 0 6.5%

   \[\leadsto \color{blue}{\frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}} \]
6. Step-by-step derivation

   1. asin-acos 8.2%

      \[\leadsto \frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \]

   2. div-inv 8.2%

      \[\leadsto \frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \]

   3. metadata-eval 8.2%

      \[\leadsto \frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \]

   4. sub-neg 8.2%

      \[\leadsto \frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\color{blue}{\left(\pi \cdot 0.5 + \left(-\cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \]

   5. cancel-sign-sub-inv 8.2%

      \[\leadsto \frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\left(\pi \cdot 0.5 + \left(-\cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right)\right)}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \]

   6. metadata-eval 8.2%

      \[\leadsto \frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\left(\pi \cdot 0.5 + \left(-\cos^{-1} \left(\sqrt{0.5 + \color{blue}{-0.5} \cdot x}\right)\right)\right)}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \]

7. Applied egg-rr 8.2%

   \[\leadsto \frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\color{blue}{\left(\pi \cdot 0.5 + \left(-\cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right)\right)}}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \]
8. Step-by-step derivation

   1. sub-neg 8.2%

      \[\leadsto \frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\right)}}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \]

   2. *-commutative 8.2%

      \[\leadsto \frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + \color{blue}{x \cdot -0.5}}\right)\right)}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \]

9. Simplified 8.2%

   \[\leadsto \frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \]

10. Final simplification 8.2%

    \[\leadsto \frac{{\pi}^{2} \cdot 0.25 - 4 \cdot {\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)\right)}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)} \]
11. Add Preprocessing

Alternative 7: 8.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} + 2 \cdot \left(\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) - \frac{\pi}{2}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ PI 2.0) (* 2.0 (- (acos (sqrt (+ 0.5 (* x -0.5)))) (/ PI 2.0)))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) + (2.0 * (acos(sqrt((0.5 + (x * -0.5)))) - (((double) M_PI) / 2.0)));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) + (2.0 * (Math.acos(Math.sqrt((0.5 + (x * -0.5)))) - (Math.PI / 2.0)));
}

Python

def code(x):
	return (math.pi / 2.0) + (2.0 * (math.acos(math.sqrt((0.5 + (x * -0.5)))) - (math.pi / 2.0)))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) + Float64(2.0 * Float64(acos(sqrt(Float64(0.5 + Float64(x * -0.5)))) - Float64(pi / 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) + (2.0 * (acos(sqrt((0.5 + (x * -0.5)))) - (pi / 2.0)));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] + N[(2.0 * N[(N[ArcCos[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. asin-acos 8.2%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)} \]

   2. add-cube-cbrt 6.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}}{2} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]

   3. associate-/l* 6.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \frac{\sqrt[3]{\pi}}{2}} - \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]

   4. fmm-def 6.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}, \frac{\sqrt[3]{\pi}}{2}, -\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)} \]

   5. pow2 6.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}}, \frac{\sqrt[3]{\pi}}{2}, -\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \]

   6. div-sub 6.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{fma}\left({\left(\sqrt[3]{\pi}\right)}^{2}, \frac{\sqrt[3]{\pi}}{2}, -\cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{x}{2}}}\right)\right) \]

   7. metadata-eval 6.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{fma}\left({\left(\sqrt[3]{\pi}\right)}^{2}, \frac{\sqrt[3]{\pi}}{2}, -\cos^{-1} \left(\sqrt{\color{blue}{0.5} - \frac{x}{2}}\right)\right) \]

   8. div-inv 6.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{fma}\left({\left(\sqrt[3]{\pi}\right)}^{2}, \frac{\sqrt[3]{\pi}}{2}, -\cos^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot \frac{1}{2}}}\right)\right) \]

   9. metadata-eval 6.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \mathsf{fma}\left({\left(\sqrt[3]{\pi}\right)}^{2}, \frac{\sqrt[3]{\pi}}{2}, -\cos^{-1} \left(\sqrt{0.5 - x \cdot \color{blue}{0.5}}\right)\right) \]

4. Applied egg-rr 6.3%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\pi}\right)}^{2}, \frac{\sqrt[3]{\pi}}{2}, -\cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)} \]
5. Step-by-step derivation

   1. fmm-undef 6.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \frac{\sqrt[3]{\pi}}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right)} \]

   2. associate-*r/ 6.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\color{blue}{\frac{{\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}}{2}} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \]

   3. unpow2 6.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)} \cdot \sqrt[3]{\pi}}{2} - \cos^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right)\right) \]

   4. rem-3cbrt-lft 8.2%

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

   5. sub-neg 8.2%

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

   6. distribute-rgt-neg-in 8.2%

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

   7. metadata-eval 8.2%

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

6. Simplified 8.2%

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

7. Final simplification 8.2%

   \[\leadsto \frac{\pi}{2} + 2 \cdot \left(\cos^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right) - \frac{\pi}{2}\right) \]
8. Add Preprocessing

Alternative 8: 6.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{\frac{2}{1 - x}}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (/ 1.0 (sqrt (/ 2.0 (- 1.0 x))))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin((1.0 / sqrt((2.0 / (1.0 - x))))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin((1.0 / Math.sqrt((2.0 / (1.0 - x))))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin((1.0 / math.sqrt((2.0 / (1.0 - x))))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(Float64(1.0 / sqrt(Float64(2.0 / Float64(1.0 - x)))))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin((1.0 / sqrt((2.0 / (1.0 - x))))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[(1.0 / N[Sqrt[N[(2.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num 6.5%

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

   2. sqrt-div 6.7%

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

   3. metadata-eval 6.7%

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

4. Applied egg-rr 6.7%

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

5. Final simplification 6.7%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\frac{1}{\sqrt{\frac{2}{1 - x}}}\right) \]
6. Add Preprocessing

Alternative 9: 6.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing

3. Final simplification 6.6%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
4. Add Preprocessing

Alternative 10: 3.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (* 0.5 PI) (* 2.0 (asin (sqrt 0.5)))))

C

double code(double x) {
	return (0.5 * ((double) M_PI)) + (2.0 * asin(sqrt(0.5)));
}

Java

public static double code(double x) {
	return (0.5 * Math.PI) + (2.0 * Math.asin(Math.sqrt(0.5)));
}

Python

def code(x):
	return (0.5 * math.pi) + (2.0 * math.asin(math.sqrt(0.5)))

Julia

function code(x)
	return Float64(Float64(0.5 * pi) + Float64(2.0 * asin(sqrt(0.5))))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * pi) + (2.0 * asin(sqrt(0.5)));
end

Wolfram

code[x_] := N[(N[(0.5 * Pi), $MachinePrecision] + N[(2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 6.6%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \cdot \sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right) \cdot \sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}} \]

   2. pow3 6.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}\right)}^{3}} \]

4. Applied egg-rr 6.6%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)}\right)}^{3}} \]
5. Step-by-step derivation

   1. rem-cube-cbrt 6.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\sqrt{0.5 - x \cdot 0.5}\right) \cdot -2\right)} \]

   2. fma-undefine 6.6%

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

   3. add-sqr-sqrt 0.0%

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

   4. sqrt-unprod 3.7%

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

   5. swap-sqr 3.7%

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

   6. pow2 3.7%

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

   7. *-commutative 3.7%

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

   8. metadata-eval 3.7%

      \[\leadsto \pi \cdot 0.5 + \sqrt{{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2} \cdot \color{blue}{4}} \]

   9. sqrt-prod 3.7%

      \[\leadsto \pi \cdot 0.5 + \color{blue}{\sqrt{{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{2}} \cdot \sqrt{4}} \]

6. Applied egg-rr 3.7%

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

7. Taylor expanded in x around 0 3.7%

   \[\leadsto \pi \cdot 0.5 + 2 \cdot \sin^{-1} \color{blue}{\left(\sqrt{0.5}\right)} \]

8. Final simplification 3.7%

   \[\leadsto 0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]
9. Add Preprocessing

Alternative 11: 4.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ PI 2.0) (* 2.0 (asin (sqrt 0.5)))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(0.5)));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(0.5)));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(0.5)))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(0.5))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(0.5)));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 3.9%

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

4. Final simplification 3.9%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]
5. Add Preprocessing

Developer target: 100.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asin x))

C

double code(double x) {
	return asin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = asin(x)
end function

Java

public static double code(double x) {
	return Math.asin(x);
}

Python

def code(x):
	return math.asin(x)

Julia

function code(x)
	return asin(x)
end

MATLAB

function tmp = code(x)
	tmp = asin(x);
end

Wolfram

code[x_] := N[ArcSin[x], $MachinePrecision]

Reproduce

herbie shell --seed 2024112 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :alt
  (asin x)

  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))