Ian Simplification

Percentage Accurate: 7.0% → 8.5%

Time: 17.7s

Alternatives: 7

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: 7.0% 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: 8.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (log1p
  (expm1 (fma (- (* PI 0.5) (acos (sqrt (fma x -0.5 0.5)))) -2.0 (* PI 0.5)))))

C

double code(double x) {
	return log1p(expm1(fma(((((double) M_PI) * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))), -2.0, (((double) M_PI) * 0.5))));
}

Julia

function code(x)
	return log1p(expm1(fma(Float64(Float64(pi * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))), -2.0, Float64(pi * 0.5))))
end

Wolfram

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

Derivation

1. Initial program 7.3%

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

   1. add-cbrt-cube 7.3%

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

   2. pow3 7.3%

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

3. Applied egg-rr 7.3%

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

   1. asin-acos 8.7%

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

   2. div-inv 8.7%

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

   3. metadata-eval 8.7%

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

   4. sub-neg 8.7%

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

   5. distribute-rgt-neg-in 8.7%

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

   6. metadata-eval 8.7%

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

5. Applied egg-rr 8.7%

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

   1. log1p-expm1-u 8.7%

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

   2. rem-cbrt-cube 8.7%

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

   3. +-commutative 8.7%

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

   4. fma-def 8.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(x, -0.5, 0.5\right)}}\right), -2, \pi \cdot 0.5\right)\right)\right) \]

7. Applied egg-rr 8.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), -2, \pi \cdot 0.5\right)\right)\right)} \]

8. Final simplification 8.7%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), -2, \pi \cdot 0.5\right)\right)\right) \]

Alternative 2: 8.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (cbrt (fma (- (* PI 0.5) (acos (sqrt (+ 0.5 (* x -0.5))))) -2.0 (* PI 0.5)))
  3.0))

C

double code(double x) {
	return pow(cbrt(fma(((((double) M_PI) * 0.5) - acos(sqrt((0.5 + (x * -0.5))))), -2.0, (((double) M_PI) * 0.5))), 3.0);
}

Julia

function code(x)
	return cbrt(fma(Float64(Float64(pi * 0.5) - acos(sqrt(Float64(0.5 + Float64(x * -0.5))))), -2.0, Float64(pi * 0.5))) ^ 3.0
end

Wolfram

code[x_] := N[Power[N[Power[N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 7.3%

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

   1. add-cube-cbrt 7.3%

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

   2. pow3 7.3%

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

3. Applied egg-rr 7.3%

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

   1. asin-acos 8.7%

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

   2. div-inv 8.7%

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

   3. metadata-eval 8.7%

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

   4. sub-neg 8.7%

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

   5. distribute-rgt-neg-in 8.7%

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

   6. metadata-eval 8.7%

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

5. Applied egg-rr 8.7%

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

6. Final simplification 8.7%

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

Alternative 3: 8.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (cbrt
  (pow
   (+ (* PI 0.5) (* -2.0 (- (* PI 0.5) (acos (sqrt (+ 0.5 (* x -0.5)))))))
   3.0)))

C

double code(double x) {
	return cbrt(pow(((((double) M_PI) * 0.5) + (-2.0 * ((((double) M_PI) * 0.5) - acos(sqrt((0.5 + (x * -0.5))))))), 3.0));
}

Java

public static double code(double x) {
	return Math.cbrt(Math.pow(((Math.PI * 0.5) + (-2.0 * ((Math.PI * 0.5) - Math.acos(Math.sqrt((0.5 + (x * -0.5))))))), 3.0));
}

Julia

function code(x)
	return cbrt((Float64(Float64(pi * 0.5) + Float64(-2.0 * Float64(Float64(pi * 0.5) - acos(sqrt(Float64(0.5 + Float64(x * -0.5))))))) ^ 3.0))
end

Wolfram

code[x_] := N[Power[N[Power[N[(N[(Pi * 0.5), $MachinePrecision] + N[(-2.0 * N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 7.3%

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

   1. add-cbrt-cube 7.3%

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

   2. pow3 7.3%

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

3. Applied egg-rr 7.3%

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

   1. asin-acos 8.7%

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

   2. div-inv 8.7%

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

   3. metadata-eval 8.7%

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

   4. sub-neg 8.7%

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

   5. distribute-rgt-neg-in 8.7%

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

   6. metadata-eval 8.7%

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

5. Applied egg-rr 8.7%

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

   1. fma-udef 8.7%

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

7. Applied egg-rr 8.7%

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

8. Final simplification 8.7%

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

Alternative 4: 8.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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 * 0.5)));
}

Python

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

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 * 0.5))))
end

MATLAB

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

Derivation

1. Initial program 7.3%

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

   1. asin-acos 8.7%

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

   2. div-inv 8.7%

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

   3. metadata-eval 8.7%

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

   4. div-sub 8.7%

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

   5. metadata-eval 8.7%

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

   6. div-inv 8.7%

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

   7. metadata-eval 8.7%

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

3. Applied egg-rr 8.7%

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

4. Final simplification 8.7%

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

Alternative 5: 7.0% 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 7.3%

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

2. Final simplification 7.3%

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

Alternative 6: 3.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

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

   1. asin-acos 8.7%

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

   2. div-inv 8.7%

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

   3. metadata-eval 8.7%

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

   4. div-sub 8.7%

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

   5. metadata-eval 8.7%

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

   6. div-inv 8.7%

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

   7. metadata-eval 8.7%

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

3. Applied egg-rr 8.7%

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

   1. div-inv 8.7%

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

   2. metadata-eval 8.7%

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

   3. cancel-sign-sub-inv 8.7%

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

   4. metadata-eval 8.7%

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

   5. metadata-eval 8.7%

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

   6. div-inv 8.7%

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

   7. asin-acos 7.3%

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

   8. *-commutative 7.3%

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

   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 3.8%

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

   11. *-commutative 3.8%

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

   12. *-commutative 3.8%

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

   13. swap-sqr 3.8%

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

5. Applied egg-rr 3.8%

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

6. Taylor expanded in x around 0 3.8%

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

7. Final simplification 3.8%

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

Alternative 7: 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 7.3%

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

2. Taylor expanded in x around 0 4.3%

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

3. Final simplification 4.3%

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

Developer target: 100.0% accurate, 3.1× 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 2023230 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

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