Ian Simplification

Percentage Accurate: 6.7% → 8.2%

Time: 31.4s

Alternatives: 8

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: 6.7% 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: 8.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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. div-inv 6.5%

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

4. Applied egg-rr 6.5%

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

   1. asin-acos 8.0%

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

   2. div-inv 8.0%

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

   3. metadata-eval 8.0%

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

   4. sub-neg 8.0%

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

   5. distribute-rgt-neg-in 8.0%

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

   6. metadata-eval 8.0%

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

6. Applied egg-rr 8.0%

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

   1. metadata-eval 8.0%

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

   2. div-inv 8.0%

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

   3. asin-acos 6.5%

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

   4. pow2 6.5%

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

   5. add-log-exp 6.5%

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

   6. pow2 6.5%

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

8. Applied egg-rr 6.5%

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

   1. rem-log-exp 6.5%

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

   2. add-cube-cbrt 8.0%

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

   3. log-prod 8.0%

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

10. Applied egg-rr 8.0%

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

    1. log-pow 8.0%

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

    2. distribute-lft1-in 8.0%

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

    3. metadata-eval 8.0%

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

    4. fma-def 8.0%

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

    5. metadata-eval 8.0%

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

    6. distribute-lft-neg-in 8.0%

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

    7. +-commutative 8.0%

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

12. Simplified 8.0%

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

13. Final simplification 8.0%

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

Alternative 2: 8.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.5 - x \cdot 0.5}\\ \frac{0.25 \cdot {\pi}^{2} - 4 \cdot {\left(0.5 \cdot \pi - \cos^{-1} t\_0\right)}^{2}}{0.5 \cdot \pi + 2 \cdot \sin^{-1} t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (- 0.5 (* x 0.5)))))
   (/
    (- (* 0.25 (pow PI 2.0)) (* 4.0 (pow (- (* 0.5 PI) (acos t_0)) 2.0)))
    (+ (* 0.5 PI) (* 2.0 (asin t_0))))))

C

double code(double x) {
	double t_0 = sqrt((0.5 - (x * 0.5)));
	return ((0.25 * pow(((double) M_PI), 2.0)) - (4.0 * pow(((0.5 * ((double) M_PI)) - acos(t_0)), 2.0))) / ((0.5 * ((double) M_PI)) + (2.0 * asin(t_0)));
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((0.5 - (x * 0.5)));
	return ((0.25 * Math.pow(Math.PI, 2.0)) - (4.0 * Math.pow(((0.5 * Math.PI) - Math.acos(t_0)), 2.0))) / ((0.5 * Math.PI) + (2.0 * Math.asin(t_0)));
}

Python

def code(x):
	t_0 = math.sqrt((0.5 - (x * 0.5)))
	return ((0.25 * math.pow(math.pi, 2.0)) - (4.0 * math.pow(((0.5 * math.pi) - math.acos(t_0)), 2.0))) / ((0.5 * math.pi) + (2.0 * math.asin(t_0)))

Julia

function code(x)
	t_0 = sqrt(Float64(0.5 - Float64(x * 0.5)))
	return Float64(Float64(Float64(0.25 * (pi ^ 2.0)) - Float64(4.0 * (Float64(Float64(0.5 * pi) - acos(t_0)) ^ 2.0))) / Float64(Float64(0.5 * pi) + Float64(2.0 * asin(t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((0.5 - (x * 0.5)));
	tmp = ((0.25 * (pi ^ 2.0)) - (4.0 * (((0.5 * pi) - acos(t_0)) ^ 2.0))) / ((0.5 * pi) + (2.0 * asin(t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] - N[(4.0 * N[Power[N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcCos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * Pi), $MachinePrecision] + N[(2.0 * N[ArcSin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.5%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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. div-inv 6.5%

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

4. Applied egg-rr 6.5%

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

   1. asin-acos 8.0%

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

   2. div-inv 8.0%

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

   3. metadata-eval 8.0%

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

   4. sub-neg 8.0%

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

   5. distribute-rgt-neg-in 8.0%

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

   6. metadata-eval 8.0%

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

6. Applied egg-rr 8.0%

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

7. Taylor expanded in x around inf 8.0%

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

8. Final simplification 8.0%

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

Alternative 3: 5.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-300}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e-300)
   (- (/ PI 2.0) (* 2.0 (asin (sqrt 0.5))))
   (+ (* 0.5 PI) (* 2.0 (asin (sqrt (+ 0.5 (* -0.5 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e-300) {
		tmp = (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(0.5)));
	} else {
		tmp = (0.5 * ((double) M_PI)) + (2.0 * asin(sqrt((0.5 + (-0.5 * x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e-300) {
		tmp = (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(0.5)));
	} else {
		tmp = (0.5 * Math.PI) + (2.0 * Math.asin(Math.sqrt((0.5 + (-0.5 * x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.35e-300:
		tmp = (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(0.5)))
	else:
		tmp = (0.5 * math.pi) + (2.0 * math.asin(math.sqrt((0.5 + (-0.5 * x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e-300)
		tmp = Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(0.5))));
	else
		tmp = Float64(Float64(0.5 * pi) + Float64(2.0 * asin(sqrt(Float64(0.5 + Float64(-0.5 * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.35e-300)
		tmp = (pi / 2.0) - (2.0 * asin(sqrt(0.5)));
	else
		tmp = (0.5 * pi) + (2.0 * asin(sqrt((0.5 + (-0.5 * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e-300], N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $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]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34999999999999998e-300

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0 5.7%

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

   if 1.34999999999999998e-300 < x

   1. Initial program 4.7%

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

      1. asin-acos 7.3%

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

      2. div-inv 7.3%

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

      3. metadata-eval 7.3%

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

      4. div-sub 7.3%

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

      5. metadata-eval 7.3%

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

      6. div-inv 7.3%

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

      7. metadata-eval 7.3%

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

   4. Applied egg-rr 7.3%

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

      1. div-inv 7.3%

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

      2. metadata-eval 7.3%

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

      3. cancel-sign-sub-inv 7.3%

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

      4. metadata-eval 7.3%

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

      5. metadata-eval 7.3%

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

      6. div-inv 7.3%

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

      7. asin-acos 4.7%

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

      8. *-commutative 4.7%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 5.5%

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

      11. *-commutative 5.5%

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

      12. *-commutative 5.5%

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

   6. Applied egg-rr 5.5%

      \[\leadsto \color{blue}{\pi \cdot 0.5 + 2 \cdot \sin^{-1} \left(\sqrt{0.5 + x \cdot -0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-300}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 + -0.5 \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 8.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) + (2.0 * (acos(sqrt((0.5 - (x * 0.5)))) - (0.5 * pi)));
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[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

   1. asin-acos 7.9%

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

   2. div-inv 7.9%

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

   3. metadata-eval 7.9%

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

   4. div-sub 7.9%

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

   5. metadata-eval 7.9%

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

   6. div-inv 7.9%

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

   7. metadata-eval 7.9%

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

4. Applied egg-rr 7.9%

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

5. Final simplification 7.9%

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

Alternative 5: 6.7% 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 6.5%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 6: 5.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{if}\;x \leq 1.35 \cdot 10^{-300}:\\ \;\;\;\;\frac{\pi}{2} - t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \pi + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 2.0 (asin (sqrt 0.5)))))
   (if (<= x 1.35e-300) (- (/ PI 2.0) t_0) (+ (* 0.5 PI) t_0))))

C

double code(double x) {
	double t_0 = 2.0 * asin(sqrt(0.5));
	double tmp;
	if (x <= 1.35e-300) {
		tmp = (((double) M_PI) / 2.0) - t_0;
	} else {
		tmp = (0.5 * ((double) M_PI)) + t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 2.0 * Math.asin(Math.sqrt(0.5));
	double tmp;
	if (x <= 1.35e-300) {
		tmp = (Math.PI / 2.0) - t_0;
	} else {
		tmp = (0.5 * Math.PI) + t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 2.0 * math.asin(math.sqrt(0.5))
	tmp = 0
	if x <= 1.35e-300:
		tmp = (math.pi / 2.0) - t_0
	else:
		tmp = (0.5 * math.pi) + t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(2.0 * asin(sqrt(0.5)))
	tmp = 0.0
	if (x <= 1.35e-300)
		tmp = Float64(Float64(pi / 2.0) - t_0);
	else
		tmp = Float64(Float64(0.5 * pi) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 2.0 * asin(sqrt(0.5));
	tmp = 0.0;
	if (x <= 1.35e-300)
		tmp = (pi / 2.0) - t_0;
	else
		tmp = (0.5 * pi) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(2.0 * N[ArcSin[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.35e-300], N[(N[(Pi / 2.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34999999999999998e-300

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0 5.7%

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

   if 1.34999999999999998e-300 < x

   1. Initial program 4.7%

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

      1. asin-acos 7.3%

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

      2. div-inv 7.3%

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

      3. metadata-eval 7.3%

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

      4. div-sub 7.3%

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

      5. metadata-eval 7.3%

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

      6. div-inv 7.3%

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

      7. metadata-eval 7.3%

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

   4. Applied egg-rr 7.3%

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

      1. div-inv 7.3%

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

      2. metadata-eval 7.3%

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

      3. cancel-sign-sub-inv 7.3%

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

      4. metadata-eval 7.3%

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

      5. metadata-eval 7.3%

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

      6. div-inv 7.3%

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

      7. asin-acos 4.7%

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

      8. *-commutative 4.7%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 5.5%

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

      11. *-commutative 5.5%

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

      12. *-commutative 5.5%

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

   6. Applied egg-rr 5.5%

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

   7. Taylor expanded in x around 0 5.5%

      \[\leadsto \pi \cdot 0.5 + 2 \cdot \sin^{-1} \color{blue}{\left(\sqrt{0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-300}:\\ \;\;\;\;\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 6.7% 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 6.5%

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

3. Final simplification 6.5%

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

Alternative 8: 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 6.5%

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

   1. asin-acos 7.9%

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

   2. div-inv 7.9%

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

   3. metadata-eval 7.9%

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

   4. div-sub 7.9%

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

   5. metadata-eval 7.9%

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

   6. div-inv 7.9%

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

   7. metadata-eval 7.9%

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

4. Applied egg-rr 7.9%

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

   1. div-inv 7.9%

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

   2. metadata-eval 7.9%

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

   3. cancel-sign-sub-inv 7.9%

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

   4. metadata-eval 7.9%

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

   5. metadata-eval 7.9%

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

   6. div-inv 7.9%

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

   7. asin-acos 6.5%

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

   8. *-commutative 6.5%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 3.9%

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

   11. *-commutative 3.9%

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

   12. *-commutative 3.9%

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

6. Applied egg-rr 3.9%

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

7. Taylor expanded in x around 0 3.9%

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

8. Final simplification 3.9%

   \[\leadsto 0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5}\right) \]
9. 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 2024031 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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