bug323 (missed optimization)

Percentage Accurate: 7.0% → 10.5%

Time: 12.9s

Alternatives: 10

Speedup: 1.0×

Specification

\[0 \leq x \land x \leq 0.5\]

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Initial Program: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Alternative 1: 10.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({\left({\left(\pi \cdot 0.5\right)}^{0.3333333333333333}\right)}^{2}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (pow (pow (* PI 0.5) 0.3333333333333333) 2.0)
  (cbrt (* PI 0.5))
  (- (asin (- 1.0 x)))))

C

double code(double x) {
	return fma(pow(pow((((double) M_PI) * 0.5), 0.3333333333333333), 2.0), cbrt((((double) M_PI) * 0.5)), -asin((1.0 - x)));
}

Julia

function code(x)
	return fma(((Float64(pi * 0.5) ^ 0.3333333333333333) ^ 2.0), cbrt(Float64(pi * 0.5)), Float64(-asin(Float64(1.0 - x))))
end

Wolfram

code[x_] := N[(N[Power[N[Power[N[(Pi * 0.5), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(Pi * 0.5), $MachinePrecision], 1/3], $MachinePrecision] + (-N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 7.2%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin 7.2%

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

   2. flip-- 7.2%

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

   3. div-inv 7.2%

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

   4. metadata-eval 7.2%

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

   5. div-inv 7.2%

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

   6. metadata-eval 7.2%

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

   7. div-inv 7.2%

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

   8. metadata-eval 7.2%

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

3. Applied egg-rr 7.2%

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

   1. flip-- 7.2%

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

   2. add-cube-cbrt 5.4%

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

   3. fma-neg 5.4%

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

   4. pow2 5.4%

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

5. Applied egg-rr 5.4%

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

   1. pow1/3 10.6%

      \[\leadsto \mathsf{fma}\left({\color{blue}{\left({\left(\pi \cdot 0.5\right)}^{0.3333333333333333}\right)}}^{2}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) \]

7. Applied egg-rr 10.6%

   \[\leadsto \mathsf{fma}\left({\color{blue}{\left({\left(\pi \cdot 0.5\right)}^{0.3333333333333333}\right)}}^{2}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) \]

8. Final simplification 10.6%

   \[\leadsto \mathsf{fma}\left({\left({\left(\pi \cdot 0.5\right)}^{0.3333333333333333}\right)}^{2}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) \]

Alternative 2: 10.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - {\left(\sqrt[3]{\sqrt[3]{\sin^{-1} \left(1 - x\right)}}\right)}^{9} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (cbrt (cbrt (asin (- 1.0 x)))) 9.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(cbrt(cbrt(asin((1.0 - x)))), 9.0);
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.cbrt(Math.cbrt(Math.asin((1.0 - x)))), 9.0);
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (cbrt(cbrt(asin(Float64(1.0 - x)))) ^ 9.0))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 9.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin 7.2%

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

   2. sub-neg 7.2%

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

   3. div-inv 7.2%

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

   4. metadata-eval 7.2%

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

3. Applied egg-rr 7.2%

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

   1. sub-neg 7.2%

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

5. Simplified 7.2%

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

   1. add-cube-cbrt 10.5%

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

   2. pow3 10.5%

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

7. Applied egg-rr 10.5%

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

   1. add-cube-cbrt 10.5%

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

   2. pow3 10.5%

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

9. Applied egg-rr 10.5%

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

    1. pow-pow 10.6%

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

    2. pow-to-exp 10.6%

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

    3. metadata-eval 10.6%

       \[\leadsto \pi \cdot 0.5 - e^{\log \left(\sqrt[3]{\sqrt[3]{\sin^{-1} \left(1 - x\right)}}\right) \cdot \color{blue}{9}} \]

11. Applied egg-rr 10.6%

    \[\leadsto \pi \cdot 0.5 - \color{blue}{e^{\log \left(\sqrt[3]{\sqrt[3]{\sin^{-1} \left(1 - x\right)}}\right) \cdot 9}} \]
12. Step-by-step derivation

    1. exp-to-pow 10.6%

       \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sqrt[3]{\sin^{-1} \left(1 - x\right)}}\right)}^{9}} \]

13. Simplified 10.6%

    \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sqrt[3]{\sin^{-1} \left(1 - x\right)}}\right)}^{9}} \]

14. Final simplification 10.6%

    \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{\sqrt[3]{\sin^{-1} \left(1 - x\right)}}\right)}^{9} \]

Alternative 3: 5.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\pi \cdot 0.5 - \sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0)
   (- (* PI 0.5) (cbrt (pow (asin (- 1.0 x)) 3.0)))
   (- PI (acos (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = (((double) M_PI) * 0.5) - cbrt(pow(asin((1.0 - x)), 3.0));
	} else {
		tmp = ((double) M_PI) - acos((1.0 - x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = (Math.PI * 0.5) - Math.cbrt(Math.pow(Math.asin((1.0 - x)), 3.0));
	} else {
		tmp = Math.PI - Math.acos((1.0 - x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = Float64(Float64(pi * 0.5) - cbrt((asin(Float64(1.0 - x)) ^ 3.0)));
	else
		tmp = Float64(pi - acos(Float64(1.0 - x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(Pi - N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 x) < 1

   1. Initial program 7.2%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 7.2%

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

      2. sub-neg 7.2%

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

      3. div-inv 7.2%

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

      4. metadata-eval 7.2%

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

   3. Applied egg-rr 7.2%

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

      1. sub-neg 7.2%

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

   5. Simplified 7.2%

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

      1. add-cbrt-cube 5.4%

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

      2. pow3 5.4%

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

   7. Applied egg-rr 5.4%

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

   if 1 < (-.f64 1 x)

   1. Initial program 7.2%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 7.2%

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

      2. flip-- 7.2%

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

      3. div-inv 7.2%

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

      4. metadata-eval 7.2%

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

      5. div-inv 7.2%

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

      6. metadata-eval 7.2%

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

      7. div-inv 7.2%

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

      8. metadata-eval 7.2%

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

   3. Applied egg-rr 7.2%

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

      1. flip-- 7.2%

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

      2. metadata-eval 7.2%

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

      3. div-inv 7.2%

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

      4. acos-asin 7.2%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      5. expm1-log1p-u 7.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      6. expm1-udef 7.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      7. log1p-udef 7.2%

         \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

      8. add-exp-log 7.2%

         \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

      9. associate--l+ 7.2%

         \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      10. +-commutative 7.2%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) - 1\right) + 1} \]

      11. sub-neg 7.2%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1 \]

      12. metadata-eval 7.2%

          \[\leadsto \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1 \]

   5. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]
   6. Step-by-step derivation

      1. associate-+l+ 7.2%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right) + \left(-1 + 1\right)} \]

      2. metadata-eval 7.2%

         \[\leadsto \cos^{-1} \left(1 - x\right) + \color{blue}{0} \]

      3. +-rgt-identity 7.2%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      4. acos-asin 7.2%

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

      5. div-inv 7.2%

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

      6. metadata-eval 7.2%

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

      7. add-cube-cbrt 10.5%

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

      8. unpow3 10.5%

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

      9. sub-neg 10.5%

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

      10. unpow3 10.5%

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

      11. add-cube-cbrt 7.2%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.8%

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

      14. add-cube-cbrt 6.8%

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

      15. unpow3 6.8%

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

      16. add-cube-cbrt 6.8%

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

      17. unpow3 6.8%

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

   7. Applied egg-rr 6.8%

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

      1. distribute-lft-out 6.8%

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

      2. metadata-eval 6.8%

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

      3. *-rgt-identity 6.8%

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

   9. Simplified 6.8%

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

4. Final simplification 5.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\pi \cdot 0.5 - \sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 4: 10.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (cbrt (asin (- 1.0 x))) 3.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(cbrt(asin((1.0 - x))), 3.0);
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.cbrt(Math.asin((1.0 - x))), 3.0);
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (cbrt(asin(Float64(1.0 - x))) ^ 3.0))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin 7.2%

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

   2. sub-neg 7.2%

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

   3. div-inv 7.2%

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

   4. metadata-eval 7.2%

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

3. Applied egg-rr 7.2%

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

   1. sub-neg 7.2%

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

5. Simplified 7.2%

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

   1. add-cube-cbrt 10.5%

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

   2. pow3 10.5%

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

7. Applied egg-rr 10.5%

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

8. Final simplification 10.5%

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

Alternative 5: 10.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	return (math.pi * 0.5) - math.pow(math.sqrt(math.asin((1.0 - x))), 2.0)

Julia

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

MATLAB

function tmp = code(x)
	tmp = (pi * 0.5) - (sqrt(asin((1.0 - x))) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 7.2%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin 7.2%

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

   2. sub-neg 7.2%

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

   3. div-inv 7.2%

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

   4. metadata-eval 7.2%

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

3. Applied egg-rr 7.2%

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

   1. sub-neg 7.2%

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

5. Simplified 7.2%

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

   1. add-sqr-sqrt 10.6%

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

   2. pow2 10.6%

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

7. Applied egg-rr 10.6%

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

8. Final simplification 10.6%

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

Alternative 6: 7.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;2 \cdot \log \left(e^{0.5 \cdot t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= (- 1.0 x) 1.0) (* 2.0 (log (exp (* 0.5 t_0)))) (- PI t_0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 2.0 * log(exp((0.5 * t_0)));
	} else {
		tmp = ((double) M_PI) - t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 2.0 * Math.log(Math.exp((0.5 * t_0)));
	} else {
		tmp = Math.PI - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = 2.0 * math.log(math.exp((0.5 * t_0)))
	else:
		tmp = math.pi - t_0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = Float64(2.0 * log(exp(Float64(0.5 * t_0))));
	else
		tmp = Float64(pi - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = 2.0 * log(exp((0.5 * t_0)));
	else
		tmp = pi - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[(2.0 * N[Log[N[Exp[N[(0.5 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(Pi - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 x) < 1

   1. Initial program 7.2%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 7.2%

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

      2. flip-- 7.2%

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

      3. div-inv 7.2%

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

      4. metadata-eval 7.2%

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

      5. div-inv 7.2%

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

      6. metadata-eval 7.2%

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

      7. div-inv 7.2%

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

      8. metadata-eval 7.2%

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

   3. Applied egg-rr 7.2%

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

      1. flip-- 7.2%

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

      2. metadata-eval 7.2%

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

      3. div-inv 7.2%

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

      4. acos-asin 7.2%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      5. add-log-exp 7.2%

         \[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)} \]

      6. add-sqr-sqrt 7.2%

         \[\leadsto \log \color{blue}{\left(\sqrt{e^{\cos^{-1} \left(1 - x\right)}} \cdot \sqrt{e^{\cos^{-1} \left(1 - x\right)}}\right)} \]

      7. log-prod 7.2%

         \[\leadsto \color{blue}{\log \left(\sqrt{e^{\cos^{-1} \left(1 - x\right)}}\right) + \log \left(\sqrt{e^{\cos^{-1} \left(1 - x\right)}}\right)} \]

   5. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{\log \left(\sqrt{e^{\cos^{-1} \left(1 - x\right)}}\right) + \log \left(\sqrt{e^{\cos^{-1} \left(1 - x\right)}}\right)} \]
   6. Step-by-step derivation

      1. count-2 7.2%

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

   7. Simplified 7.2%

      \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt{e^{\cos^{-1} \left(1 - x\right)}}\right)} \]
   8. Step-by-step derivation

      1. pow1/2 7.2%

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

      2. pow-exp 7.2%

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

   9. Applied egg-rr 7.2%

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

   if 1 < (-.f64 1 x)

   1. Initial program 7.2%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 7.2%

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

      2. flip-- 7.2%

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

      3. div-inv 7.2%

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

      4. metadata-eval 7.2%

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

      5. div-inv 7.2%

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

      6. metadata-eval 7.2%

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

      7. div-inv 7.2%

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

      8. metadata-eval 7.2%

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

   3. Applied egg-rr 7.2%

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

      1. flip-- 7.2%

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

      2. metadata-eval 7.2%

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

      3. div-inv 7.2%

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

      4. acos-asin 7.2%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      5. expm1-log1p-u 7.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      6. expm1-udef 7.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      7. log1p-udef 7.2%

         \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

      8. add-exp-log 7.2%

         \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

      9. associate--l+ 7.2%

         \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      10. +-commutative 7.2%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) - 1\right) + 1} \]

      11. sub-neg 7.2%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1 \]

      12. metadata-eval 7.2%

          \[\leadsto \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1 \]

   5. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]
   6. Step-by-step derivation

      1. associate-+l+ 7.2%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right) + \left(-1 + 1\right)} \]

      2. metadata-eval 7.2%

         \[\leadsto \cos^{-1} \left(1 - x\right) + \color{blue}{0} \]

      3. +-rgt-identity 7.2%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      4. acos-asin 7.2%

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

      5. div-inv 7.2%

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

      6. metadata-eval 7.2%

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

      7. add-cube-cbrt 10.5%

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

      8. unpow3 10.5%

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

      9. sub-neg 10.5%

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

      10. unpow3 10.5%

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

      11. add-cube-cbrt 7.2%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.8%

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

      14. add-cube-cbrt 6.8%

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

      15. unpow3 6.8%

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

      16. add-cube-cbrt 6.8%

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

      17. unpow3 6.8%

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

   7. Applied egg-rr 6.8%

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

      1. distribute-lft-out 6.8%

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

      2. metadata-eval 6.8%

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

      3. *-rgt-identity 6.8%

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

   9. Simplified 6.8%

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

4. Final simplification 7.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;2 \cdot \log \left(e^{0.5 \cdot \cos^{-1} \left(1 - x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 7: 7.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0)
   (- (* PI 0.5) (asin (- 1.0 x)))
   (- PI (acos (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = (((double) M_PI) * 0.5) - asin((1.0 - x));
	} else {
		tmp = ((double) M_PI) - acos((1.0 - x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = (Math.PI * 0.5) - Math.asin((1.0 - x));
	} else {
		tmp = Math.PI - Math.acos((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = (math.pi * 0.5) - math.asin((1.0 - x))
	else:
		tmp = math.pi - math.acos((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = Float64(Float64(pi * 0.5) - asin(Float64(1.0 - x)));
	else
		tmp = Float64(pi - acos(Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = (pi * 0.5) - asin((1.0 - x));
	else
		tmp = pi - acos((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(Pi - N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 x) < 1

   1. Initial program 7.2%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 7.2%

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

      2. sub-neg 7.2%

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

      3. div-inv 7.2%

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

      4. metadata-eval 7.2%

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

   3. Applied egg-rr 7.2%

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

      1. sub-neg 7.2%

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

   5. Simplified 7.2%

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

   if 1 < (-.f64 1 x)

   1. Initial program 7.2%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 7.2%

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

      2. flip-- 7.2%

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

      3. div-inv 7.2%

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

      4. metadata-eval 7.2%

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

      5. div-inv 7.2%

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

      6. metadata-eval 7.2%

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

      7. div-inv 7.2%

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

      8. metadata-eval 7.2%

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

   3. Applied egg-rr 7.2%

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

      1. flip-- 7.2%

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

      2. metadata-eval 7.2%

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

      3. div-inv 7.2%

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

      4. acos-asin 7.2%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      5. expm1-log1p-u 7.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      6. expm1-udef 7.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      7. log1p-udef 7.2%

         \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

      8. add-exp-log 7.2%

         \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

      9. associate--l+ 7.2%

         \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      10. +-commutative 7.2%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) - 1\right) + 1} \]

      11. sub-neg 7.2%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1 \]

      12. metadata-eval 7.2%

          \[\leadsto \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1 \]

   5. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]
   6. Step-by-step derivation

      1. associate-+l+ 7.2%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right) + \left(-1 + 1\right)} \]

      2. metadata-eval 7.2%

         \[\leadsto \cos^{-1} \left(1 - x\right) + \color{blue}{0} \]

      3. +-rgt-identity 7.2%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      4. acos-asin 7.2%

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

      5. div-inv 7.2%

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

      6. metadata-eval 7.2%

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

      7. add-cube-cbrt 10.5%

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

      8. unpow3 10.5%

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

      9. sub-neg 10.5%

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

      10. unpow3 10.5%

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

      11. add-cube-cbrt 7.2%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.8%

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

      14. add-cube-cbrt 6.8%

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

      15. unpow3 6.8%

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

      16. add-cube-cbrt 6.8%

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

      17. unpow3 6.8%

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

   7. Applied egg-rr 6.8%

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

      1. distribute-lft-out 6.8%

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

      2. metadata-eval 6.8%

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

      3. *-rgt-identity 6.8%

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

   9. Simplified 6.8%

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

4. Final simplification 7.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 8: 7.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;1 + \left(t_0 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= (- 1.0 x) 1.0) (+ 1.0 (+ t_0 -1.0)) (- PI t_0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 1.0 + (t_0 + -1.0);
	} else {
		tmp = ((double) M_PI) - t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = 1.0 + (t_0 + -1.0);
	} else {
		tmp = Math.PI - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = 1.0 + (t_0 + -1.0)
	else:
		tmp = math.pi - t_0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = Float64(1.0 + Float64(t_0 + -1.0));
	else
		tmp = Float64(pi - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = 1.0 + (t_0 + -1.0);
	else
		tmp = pi - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[(1.0 + N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(Pi - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 x) < 1

   1. Initial program 7.2%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 7.2%

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

      2. flip-- 7.2%

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

      3. div-inv 7.2%

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

      4. metadata-eval 7.2%

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

      5. div-inv 7.2%

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

      6. metadata-eval 7.2%

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

      7. div-inv 7.2%

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

      8. metadata-eval 7.2%

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

   3. Applied egg-rr 7.2%

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

      1. flip-- 7.2%

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

      2. metadata-eval 7.2%

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

      3. div-inv 7.2%

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

      4. acos-asin 7.2%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      5. expm1-log1p-u 7.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      6. expm1-udef 7.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      7. log1p-udef 7.2%

         \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

      8. add-exp-log 7.2%

         \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

      9. associate--l+ 7.2%

         \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      10. +-commutative 7.2%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) - 1\right) + 1} \]

      11. sub-neg 7.2%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1 \]

      12. metadata-eval 7.2%

          \[\leadsto \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1 \]

   5. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]

   if 1 < (-.f64 1 x)

   1. Initial program 7.2%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 7.2%

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

      2. flip-- 7.2%

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

      3. div-inv 7.2%

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

      4. metadata-eval 7.2%

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

      5. div-inv 7.2%

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

      6. metadata-eval 7.2%

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

      7. div-inv 7.2%

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

      8. metadata-eval 7.2%

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

   3. Applied egg-rr 7.2%

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

      1. flip-- 7.2%

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

      2. metadata-eval 7.2%

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

      3. div-inv 7.2%

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

      4. acos-asin 7.2%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      5. expm1-log1p-u 7.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      6. expm1-udef 7.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      7. log1p-udef 7.2%

         \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

      8. add-exp-log 7.2%

         \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

      9. associate--l+ 7.2%

         \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      10. +-commutative 7.2%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) - 1\right) + 1} \]

      11. sub-neg 7.2%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1 \]

      12. metadata-eval 7.2%

          \[\leadsto \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1 \]

   5. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]
   6. Step-by-step derivation

      1. associate-+l+ 7.2%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right) + \left(-1 + 1\right)} \]

      2. metadata-eval 7.2%

         \[\leadsto \cos^{-1} \left(1 - x\right) + \color{blue}{0} \]

      3. +-rgt-identity 7.2%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      4. acos-asin 7.2%

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

      5. div-inv 7.2%

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

      6. metadata-eval 7.2%

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

      7. add-cube-cbrt 10.5%

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

      8. unpow3 10.5%

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

      9. sub-neg 10.5%

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

      10. unpow3 10.5%

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

      11. add-cube-cbrt 7.2%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 6.8%

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

      14. add-cube-cbrt 6.8%

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

      15. unpow3 6.8%

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

      16. add-cube-cbrt 6.8%

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

      17. unpow3 6.8%

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

   7. Applied egg-rr 6.8%

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

      1. distribute-lft-out 6.8%

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

      2. metadata-eval 6.8%

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

      3. *-rgt-identity 6.8%

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

   9. Simplified 6.8%

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

4. Final simplification 7.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;1 + \left(\cos^{-1} \left(1 - x\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 9: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \left(\cos^{-1} \left(1 - x\right) + -1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (+ (acos (- 1.0 x)) -1.0)))

C

double code(double x) {
	return 1.0 + (acos((1.0 - x)) + -1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + (acos((1.0d0 - x)) + (-1.0d0))
end function

Java

public static double code(double x) {
	return 1.0 + (Math.acos((1.0 - x)) + -1.0);
}

Python

def code(x):
	return 1.0 + (math.acos((1.0 - x)) + -1.0)

Julia

function code(x)
	return Float64(1.0 + Float64(acos(Float64(1.0 - x)) + -1.0))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (acos((1.0 - x)) + -1.0);
end

Wolfram

code[x_] := N[(1.0 + N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin 7.2%

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

   2. flip-- 7.2%

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

   3. div-inv 7.2%

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

   4. metadata-eval 7.2%

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

   5. div-inv 7.2%

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

   6. metadata-eval 7.2%

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

   7. div-inv 7.2%

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

   8. metadata-eval 7.2%

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

3. Applied egg-rr 7.2%

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

   1. flip-- 7.2%

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

   2. metadata-eval 7.2%

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

   3. div-inv 7.2%

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

   4. acos-asin 7.2%

      \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

   5. expm1-log1p-u 7.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

   6. expm1-udef 7.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

   7. log1p-udef 7.2%

      \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

   8. add-exp-log 7.2%

      \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

   9. associate--l+ 7.2%

      \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

   10. +-commutative 7.2%

       \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) - 1\right) + 1} \]

   11. sub-neg 7.2%

       \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1 \]

   12. metadata-eval 7.2%

       \[\leadsto \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1 \]

5. Applied egg-rr 7.2%

   \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]

6. Final simplification 7.2%

   \[\leadsto 1 + \left(\cos^{-1} \left(1 - x\right) + -1\right) \]

Alternative 10: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.2%

   \[\cos^{-1} \left(1 - x\right) \]

2. Final simplification 7.2%

   \[\leadsto \cos^{-1} \left(1 - x\right) \]

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023274 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :herbie-target
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))