Ian Simplification

Percentage Accurate: 6.6% → 8.1%

Time: 37.5s

Alternatives: 4

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: 6.6% 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: 8.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(0.5 \cdot \pi + -2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}} \cdot \sqrt[3]{0.5 \cdot \pi + -2 \cdot \sin^{-1} \left({\left(0.5 \cdot \left(1 - x\right)\right)}^{0.5}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (cbrt
   (pow
    (+ (* 0.5 PI) (* -2.0 (- (* 0.5 PI) (acos (sqrt (- 0.5 (* 0.5 x)))))))
    2.0))
  (cbrt (+ (* 0.5 PI) (* -2.0 (asin (pow (* 0.5 (- 1.0 x)) 0.5)))))))

C

double code(double x) {
	return cbrt(pow(((0.5 * ((double) M_PI)) + (-2.0 * ((0.5 * ((double) M_PI)) - acos(sqrt((0.5 - (0.5 * x))))))), 2.0)) * cbrt(((0.5 * ((double) M_PI)) + (-2.0 * asin(pow((0.5 * (1.0 - x)), 0.5)))));
}

Java

public static double code(double x) {
	return Math.cbrt(Math.pow(((0.5 * Math.PI) + (-2.0 * ((0.5 * Math.PI) - Math.acos(Math.sqrt((0.5 - (0.5 * x))))))), 2.0)) * Math.cbrt(((0.5 * Math.PI) + (-2.0 * Math.asin(Math.pow((0.5 * (1.0 - x)), 0.5)))));
}

Julia

function code(x)
	return Float64(cbrt((Float64(Float64(0.5 * pi) + Float64(-2.0 * Float64(Float64(0.5 * pi) - acos(sqrt(Float64(0.5 - Float64(0.5 * x))))))) ^ 2.0)) * cbrt(Float64(Float64(0.5 * pi) + Float64(-2.0 * asin((Float64(0.5 * Float64(1.0 - x)) ^ 0.5))))))
end

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. add-cube-cbrt 6.3%

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

4. Applied egg-rr 6.3%

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

   1. sub-neg 6.3%

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

   2. +-commutative 6.3%

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

   3. distribute-rgt1-in 6.3%

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

   4. cancel-sign-sub-inv 6.3%

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

   5. asin-acos 8.2%

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

   6. div-inv 8.2%

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

   7. metadata-eval 8.2%

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

   8. *-commutative 8.2%

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

6. Applied egg-rr 8.2%

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

7. Taylor expanded in x around 0 8.2%

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

8. Final simplification 8.2%

   \[\leadsto \sqrt[3]{{\left(0.5 \cdot \pi + -2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)}^{2}} \cdot \sqrt[3]{0.5 \cdot \pi + -2 \cdot \sin^{-1} \left({\left(0.5 \cdot \left(1 - x\right)\right)}^{0.5}\right)} \]
9. Add Preprocessing

Alternative 2: 8.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 6.3%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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-sqr-sqrt 6.2%

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

   3. associate-/l* 6.2%

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

   4. fma-neg 6.2%

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

   5. pow1/2 6.2%

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

   6. pow1/2 6.2%

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

   7. pow1/2 6.2%

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

   8. div-inv 6.2%

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

   9. metadata-eval 6.2%

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

4. Applied egg-rr 6.2%

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

   1. fma-neg 6.2%

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

   2. associate-*r/ 6.2%

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

   3. unpow1/2 6.2%

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

   4. unpow1/2 6.2%

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

   5. rem-square-sqrt 8.2%

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

   6. sub-neg 8.2%

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

   7. mul-1-neg 8.2%

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

   8. +-commutative 8.2%

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

   9. distribute-rgt1-in 8.2%

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

   10. mul-1-neg 8.2%

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

   11. distribute-lft-neg-out 8.2%

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

   12. unsub-neg 8.2%

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

6. Simplified 8.2%

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

7. Taylor expanded in x around 0 8.2%

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

8. Final simplification 8.2%

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

Alternative 3: 6.6% 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 6.3%

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

Alternative 4: 3.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := 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. Initial program 6.3%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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-sqr-sqrt 6.2%

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

   3. associate-/l* 6.2%

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

   4. fma-neg 6.2%

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

   5. pow1/2 6.2%

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

   6. pow1/2 6.2%

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

   7. pow1/2 6.2%

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

   8. div-inv 6.2%

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

   9. metadata-eval 6.2%

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

4. Applied egg-rr 6.2%

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

   1. fma-neg 6.2%

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

   2. associate-*r/ 6.2%

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

   3. unpow1/2 6.2%

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

   4. unpow1/2 6.2%

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

   5. rem-square-sqrt 8.2%

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

   6. sub-neg 8.2%

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

   7. mul-1-neg 8.2%

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

   8. +-commutative 8.2%

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

   9. distribute-rgt1-in 8.2%

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

   10. mul-1-neg 8.2%

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

   11. distribute-lft-neg-out 8.2%

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

   12. unsub-neg 8.2%

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

6. Simplified 8.2%

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

8. Applied egg-rr 3.8%

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

9. Taylor expanded in x around 0 3.8%

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

10. Final simplification 3.8%

    \[\leadsto 0.5 \cdot \pi + 2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \]
11. 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 2024107 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :alt
  (asin x)

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