bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.5%

Time: 16.7s

Alternatives: 9

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: 6.9% 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, 0.2× 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 6.6%

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

   1. acos-asin 6.6%

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

   2. add-cube-cbrt 4.8%

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

   3. fma-neg 4.8%

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

   4. pow2 4.8%

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

   5. div-inv 4.8%

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

   6. metadata-eval 4.8%

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

   7. div-inv 4.8%

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

   8. metadata-eval 4.8%

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

4. Applied egg-rr 4.8%

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

   1. pow1/3 10.6%

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

6. Applied egg-rr 10.6%

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

7. Final simplification 10.6%

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

Alternative 2: 6.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;e^{0.3333333333333333 \cdot \left(3 \cdot \log t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= (- 1.0 x) 1.0)
     (exp (* 0.3333333333333333 (* 3.0 (log 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 = exp((0.3333333333333333 * (3.0 * log(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 = Math.exp((0.3333333333333333 * (3.0 * Math.log(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 = math.exp((0.3333333333333333 * (3.0 * math.log(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 = exp(Float64(0.3333333333333333 * Float64(3.0 * log(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 = exp((0.3333333333333333 * (3.0 * log(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[Exp[N[(0.3333333333333333 * N[(3.0 * N[Log[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 6.6%

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

      1. add-cbrt-cube 6.6%

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

      2. pow1/3 6.6%

         \[\leadsto \color{blue}{{\left(\left(\cos^{-1} \left(1 - x\right) \cdot \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \left(1 - x\right)\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 6.6%

         \[\leadsto \color{blue}{e^{\log \left(\left(\cos^{-1} \left(1 - x\right) \cdot \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \left(1 - x\right)\right) \cdot 0.3333333333333333}} \]

      4. pow3 6.6%

         \[\leadsto e^{\log \color{blue}{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 6.6%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \cos^{-1} \left(1 - x\right)\right)} \cdot 0.3333333333333333} \]

   4. Applied egg-rr 6.6%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \log \cos^{-1} \left(1 - x\right)\right) \cdot 0.3333333333333333}} \]

   if 1 < (-.f64 1 x)

   1. Initial program 6.6%

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

      1. acos-asin 6.6%

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

      2. add-cube-cbrt 4.8%

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

      3. fma-neg 4.8%

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

      4. pow2 4.8%

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

      5. div-inv 4.8%

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

      6. metadata-eval 4.8%

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

      7. div-inv 4.8%

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

      8. metadata-eval 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. pow1/3 10.6%

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

   6. Applied egg-rr 10.6%

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

      1. fma-undefine 6.6%

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

      2. pow1/3 4.8%

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

      3. unpow2 4.8%

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

      4. add-cube-cbrt 6.6%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 7.1%

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

      7. sqr-neg 7.1%

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

      8. sqrt-unprod 7.1%

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

      9. add-sqr-sqrt 7.1%

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

      10. +-commutative 7.1%

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

      11. asin-acos 7.1%

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

      12. div-inv 7.1%

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

      13. metadata-eval 7.1%

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

      14. associate-+l- 7.1%

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

   8. Applied egg-rr 7.1%

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

      1. sub-neg 7.1%

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

      2. neg-sub0 7.1%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\left(0 - \left(\cos^{-1} \left(1 - x\right) - \pi \cdot 0.5\right)\right)} \]

      3. associate-+l- 7.1%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\left(\left(0 - \cos^{-1} \left(1 - x\right)\right) + \pi \cdot 0.5\right)} \]

      4. neg-sub0 7.1%

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

      5. +-commutative 7.1%

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

      6. associate-+r+ 7.1%

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

      7. sub-neg 7.1%

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

      8. distribute-lft-out 7.1%

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

      9. metadata-eval 7.1%

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

      10. *-rgt-identity 7.1%

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

   10. Simplified 7.1%

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

4. Final simplification 6.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;e^{0.3333333333333333 \cdot \left(3 \cdot \log \cos^{-1} \left(1 - x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 6.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\pi \cdot 0.5 - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0)
   (- (* PI 0.5) (expm1 (log1p (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) - expm1(log1p(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.expm1(Math.log1p(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.expm1(math.log1p(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) - expm1(log1p(asin(Float64(1.0 - x)))));
	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[(Exp[N[Log[1 + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $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 6.6%

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

      1. acos-asin 6.6%

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

      2. sub-neg 6.6%

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

      3. div-inv 6.6%

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

      4. metadata-eval 6.6%

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

   4. Applied egg-rr 6.6%

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

      1. sub-neg 6.6%

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

   6. Simplified 6.6%

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

      1. expm1-log1p-u 6.6%

         \[\leadsto \pi \cdot 0.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)} \]

   8. Applied egg-rr 6.6%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)} \]

   if 1 < (-.f64 1 x)

   1. Initial program 6.6%

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

      1. acos-asin 6.6%

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

      2. add-cube-cbrt 4.8%

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

      3. fma-neg 4.8%

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

      4. pow2 4.8%

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

      5. div-inv 4.8%

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

      6. metadata-eval 4.8%

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

      7. div-inv 4.8%

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

      8. metadata-eval 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. pow1/3 10.6%

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

   6. Applied egg-rr 10.6%

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

      1. fma-undefine 6.6%

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

      2. pow1/3 4.8%

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

      3. unpow2 4.8%

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

      4. add-cube-cbrt 6.6%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 7.1%

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

      7. sqr-neg 7.1%

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

      8. sqrt-unprod 7.1%

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

      9. add-sqr-sqrt 7.1%

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

      10. +-commutative 7.1%

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

      11. asin-acos 7.1%

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

      12. div-inv 7.1%

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

      13. metadata-eval 7.1%

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

      14. associate-+l- 7.1%

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

   8. Applied egg-rr 7.1%

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

      1. sub-neg 7.1%

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

      2. neg-sub0 7.1%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\left(0 - \left(\cos^{-1} \left(1 - x\right) - \pi \cdot 0.5\right)\right)} \]

      3. associate-+l- 7.1%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\left(\left(0 - \cos^{-1} \left(1 - x\right)\right) + \pi \cdot 0.5\right)} \]

      4. neg-sub0 7.1%

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

      5. +-commutative 7.1%

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

      6. associate-+r+ 7.1%

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

      7. sub-neg 7.1%

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

      8. distribute-lft-out 7.1%

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

      9. metadata-eval 7.1%

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

      10. *-rgt-identity 7.1%

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

   10. Simplified 7.1%

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

4. Final simplification 6.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\pi \cdot 0.5 - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 6.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;{\left(\sqrt{t\_0}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\pi - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= (- 1.0 x) 1.0) (pow (sqrt t_0) 2.0) (- PI t_0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = pow(sqrt(t_0), 2.0);
	} 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 = Math.pow(Math.sqrt(t_0), 2.0);
	} 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 = math.pow(math.sqrt(t_0), 2.0)
	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 = sqrt(t_0) ^ 2.0;
	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 = sqrt(t_0) ^ 2.0;
	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[Power[N[Sqrt[t$95$0], $MachinePrecision], 2.0], $MachinePrecision], N[(Pi - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.6%

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

      1. add-sqr-sqrt 6.6%

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

      2. pow2 6.6%

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

   4. Applied egg-rr 6.6%

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

   if 1 < (-.f64 1 x)

   1. Initial program 6.6%

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

      1. acos-asin 6.6%

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

      2. add-cube-cbrt 4.8%

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

      3. fma-neg 4.8%

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

      4. pow2 4.8%

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

      5. div-inv 4.8%

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

      6. metadata-eval 4.8%

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

      7. div-inv 4.8%

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

      8. metadata-eval 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. pow1/3 10.6%

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

   6. Applied egg-rr 10.6%

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

      1. fma-undefine 6.6%

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

      2. pow1/3 4.8%

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

      3. unpow2 4.8%

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

      4. add-cube-cbrt 6.6%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 7.1%

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

      7. sqr-neg 7.1%

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

      8. sqrt-unprod 7.1%

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

      9. add-sqr-sqrt 7.1%

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

      10. +-commutative 7.1%

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

      11. asin-acos 7.1%

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

      12. div-inv 7.1%

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

      13. metadata-eval 7.1%

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

      14. associate-+l- 7.1%

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

   8. Applied egg-rr 7.1%

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

      1. sub-neg 7.1%

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

      2. neg-sub0 7.1%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\left(0 - \left(\cos^{-1} \left(1 - x\right) - \pi \cdot 0.5\right)\right)} \]

      3. associate-+l- 7.1%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\left(\left(0 - \cos^{-1} \left(1 - x\right)\right) + \pi \cdot 0.5\right)} \]

      4. neg-sub0 7.1%

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

      5. +-commutative 7.1%

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

      6. associate-+r+ 7.1%

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

      7. sub-neg 7.1%

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

      8. distribute-lft-out 7.1%

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

      9. metadata-eval 7.1%

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

      10. *-rgt-identity 7.1%

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

   10. Simplified 7.1%

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

4. Final simplification 6.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;{\left(\sqrt{\cos^{-1} \left(1 - x\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 6.6%

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

   1. acos-asin 6.6%

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

   2. sub-neg 6.6%

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

   3. div-inv 6.6%

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

   4. metadata-eval 6.6%

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

4. Applied egg-rr 6.6%

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

   1. sub-neg 6.6%

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

6. Simplified 6.6%

   \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
7. 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%

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

8. Applied egg-rr 10.5%

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

9. Final simplification 10.5%

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

Alternative 6: 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 6.6%

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

   1. acos-asin 6.6%

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

   2. sub-neg 6.6%

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

   3. div-inv 6.6%

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

   4. metadata-eval 6.6%

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

4. Applied egg-rr 6.6%

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

   1. sub-neg 6.6%

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

6. Simplified 6.6%

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

   1. add-sqr-sqrt 10.5%

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

   2. pow2 10.5%

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

8. Applied egg-rr 10.5%

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

9. Final simplification 10.5%

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

Alternative 7: 6.9% accurate, 0.9× 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 6.6%

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

      1. acos-asin 6.6%

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

      2. sub-neg 6.6%

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

      3. div-inv 6.6%

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

      4. metadata-eval 6.6%

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

   4. Applied egg-rr 6.6%

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

      1. sub-neg 6.6%

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

   6. Simplified 6.6%

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

   if 1 < (-.f64 1 x)

   1. Initial program 6.6%

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

      1. acos-asin 6.6%

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

      2. add-cube-cbrt 4.8%

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

      3. fma-neg 4.8%

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

      4. pow2 4.8%

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

      5. div-inv 4.8%

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

      6. metadata-eval 4.8%

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

      7. div-inv 4.8%

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

      8. metadata-eval 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. pow1/3 10.6%

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

   6. Applied egg-rr 10.6%

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

      1. fma-undefine 6.6%

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

      2. pow1/3 4.8%

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

      3. unpow2 4.8%

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

      4. add-cube-cbrt 6.6%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 7.1%

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

      7. sqr-neg 7.1%

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

      8. sqrt-unprod 7.1%

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

      9. add-sqr-sqrt 7.1%

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

      10. +-commutative 7.1%

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

      11. asin-acos 7.1%

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

      12. div-inv 7.1%

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

      13. metadata-eval 7.1%

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

      14. associate-+l- 7.1%

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

   8. Applied egg-rr 7.1%

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

      1. sub-neg 7.1%

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

      2. neg-sub0 7.1%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\left(0 - \left(\cos^{-1} \left(1 - x\right) - \pi \cdot 0.5\right)\right)} \]

      3. associate-+l- 7.1%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\left(\left(0 - \cos^{-1} \left(1 - x\right)\right) + \pi \cdot 0.5\right)} \]

      4. neg-sub0 7.1%

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

      5. +-commutative 7.1%

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

      6. associate-+r+ 7.1%

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

      7. sub-neg 7.1%

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

      8. distribute-lft-out 7.1%

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

      9. metadata-eval 7.1%

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

      10. *-rgt-identity 7.1%

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

   10. Simplified 7.1%

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

4. Final simplification 6.6%

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

Alternative 8: 6.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\pi - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x)))) (if (<= (- 1.0 x) 1.0) 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 = 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 = 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 = 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 = 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 = 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], t$95$0, N[(Pi - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.6%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing

   if 1 < (-.f64 1 x)

   1. Initial program 6.6%

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

      1. acos-asin 6.6%

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

      2. add-cube-cbrt 4.8%

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

      3. fma-neg 4.8%

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

      4. pow2 4.8%

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

      5. div-inv 4.8%

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

      6. metadata-eval 4.8%

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

      7. div-inv 4.8%

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

      8. metadata-eval 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. pow1/3 10.6%

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

   6. Applied egg-rr 10.6%

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

      1. fma-undefine 6.6%

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

      2. pow1/3 4.8%

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

      3. unpow2 4.8%

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

      4. add-cube-cbrt 6.6%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 7.1%

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

      7. sqr-neg 7.1%

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

      8. sqrt-unprod 7.1%

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

      9. add-sqr-sqrt 7.1%

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

      10. +-commutative 7.1%

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

      11. asin-acos 7.1%

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

      12. div-inv 7.1%

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

      13. metadata-eval 7.1%

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

      14. associate-+l- 7.1%

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

   8. Applied egg-rr 7.1%

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

      1. sub-neg 7.1%

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

      2. neg-sub0 7.1%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\left(0 - \left(\cos^{-1} \left(1 - x\right) - \pi \cdot 0.5\right)\right)} \]

      3. associate-+l- 7.1%

         \[\leadsto \pi \cdot 0.5 + \color{blue}{\left(\left(0 - \cos^{-1} \left(1 - x\right)\right) + \pi \cdot 0.5\right)} \]

      4. neg-sub0 7.1%

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

      5. +-commutative 7.1%

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

      6. associate-+r+ 7.1%

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

      7. sub-neg 7.1%

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

      8. distribute-lft-out 7.1%

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

      9. metadata-eval 7.1%

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

      10. *-rgt-identity 7.1%

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

   10. Simplified 7.1%

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

4. Final simplification 6.6%

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

Alternative 9: 6.9% 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 6.6%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing

3. Final simplification 6.6%

   \[\leadsto \cos^{-1} \left(1 - x\right) \]
4. Add Preprocessing

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 2024034 
(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)))