bug323 (missed optimization)

Percentage Accurate: 6.8% → 10.3%

Time: 9.0s

Alternatives: 7

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: 6.8% 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: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt[3]{t\_0}\\ t_2 := {t\_1}^{2}\\ \mathsf{fma}\left(\pi, 0.5, \sqrt[3]{\sin^{-1} \left(\frac{1 - {x}^{3}}{1 + x \cdot \left(1 + x\right)}\right)} \cdot \left(-t\_2\right)\right) + \mathsf{fma}\left(-t\_1, t\_2, \sqrt[3]{{t\_0}^{3}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (cbrt t_0)) (t_2 (pow t_1 2.0)))
   (+
    (fma
     PI
     0.5
     (* (cbrt (asin (/ (- 1.0 (pow x 3.0)) (+ 1.0 (* x (+ 1.0 x)))))) (- t_2)))
    (fma (- t_1) t_2 (cbrt (pow t_0 3.0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = cbrt(t_0);
	double t_2 = pow(t_1, 2.0);
	return fma(((double) M_PI), 0.5, (cbrt(asin(((1.0 - pow(x, 3.0)) / (1.0 + (x * (1.0 + x)))))) * -t_2)) + fma(-t_1, t_2, cbrt(pow(t_0, 3.0)));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = cbrt(t_0)
	t_2 = t_1 ^ 2.0
	return Float64(fma(pi, 0.5, Float64(cbrt(asin(Float64(Float64(1.0 - (x ^ 3.0)) / Float64(1.0 + Float64(x * Float64(1.0 + x)))))) * Float64(-t_2))) + fma(Float64(-t_1), t_2, cbrt((t_0 ^ 3.0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, N[(N[(Pi * 0.5 + N[(N[Power[N[ArcSin[N[(N[(1.0 - N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] * (-t$95$2)), $MachinePrecision]), $MachinePrecision] + N[((-t$95$1) * t$95$2 + N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 6.8%

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

   1. acos-asin 6.8%

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

   2. div-inv 6.8%

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

   3. metadata-eval 6.8%

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

   4. add-cube-cbrt 10.4%

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

   5. prod-diff 10.4%

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

   6. pow2 10.4%

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

   7. pow2 10.4%

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

4. Applied egg-rr 10.4%

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

   1. flip3-- 10.5%

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

   2. div-inv 10.5%

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

   3. metadata-eval 10.5%

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

   4. metadata-eval 10.5%

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

   5. +-commutative 10.5%

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

   6. distribute-rgt-out 10.5%

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

   7. +-commutative 10.5%

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

   8. fma-define 10.5%

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

6. Applied egg-rr 10.5%

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

   1. associate-*r/ 10.5%

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

   2. *-rgt-identity 10.5%

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

8. Simplified 10.5%

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

   1. unpow2 10.5%

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

   2. rem-3cbrt-rft 10.5%

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

   3. add-cbrt-cube 10.5%

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

   4. pow3 10.5%

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

10. Applied egg-rr 10.5%

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

11. Taylor expanded in x around 0 10.5%

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

12. Final simplification 10.5%

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

Alternative 2: 10.3% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (-
  (* PI 0.5)
  (pow
   (cbrt
    (cbrt
     (*
      (asin (/ (- 1.0 (pow x 3.0)) (fma x (+ 1.0 x) 1.0)))
      (pow (asin (- 1.0 x)) 2.0))))
   3.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(cbrt(cbrt((asin(((1.0 - pow(x, 3.0)) / fma(x, (1.0 + x), 1.0))) * pow(asin((1.0 - x)), 2.0)))), 3.0);
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (cbrt(cbrt(Float64(asin(Float64(Float64(1.0 - (x ^ 3.0)) / fma(x, Float64(1.0 + x), 1.0))) * (asin(Float64(1.0 - x)) ^ 2.0)))) ^ 3.0))
end

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. acos-asin 6.8%

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

   2. div-inv 6.8%

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

   3. metadata-eval 6.8%

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

   4. add-cube-cbrt 10.4%

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

   5. prod-diff 10.4%

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

   6. pow2 10.4%

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

   7. pow2 10.4%

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

4. Applied egg-rr 10.4%

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

   1. flip3-- 10.5%

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

   2. div-inv 10.5%

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

   3. metadata-eval 10.5%

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

   4. metadata-eval 10.5%

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

   5. +-commutative 10.5%

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

   6. distribute-rgt-out 10.5%

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

   7. +-commutative 10.5%

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

   8. fma-define 10.5%

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

6. Applied egg-rr 10.5%

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

   1. associate-*r/ 10.5%

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

   2. *-rgt-identity 10.5%

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

8. Simplified 10.5%

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

9. Taylor expanded in x around 0 5.1%

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

11. Simplified 5.1%

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

    1. cbrt-prod 10.5%

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

    2. unpow2 10.5%

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

    3. cbrt-prod 10.5%

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

    4. unpow2 10.5%

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

    5. *-commutative 10.5%

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

    6. add-cube-cbrt 10.5%

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

    7. pow3 10.5%

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

13. Applied egg-rr 10.5%

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

14. Final simplification 10.5%

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

Alternative 3: 10.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. acos-asin 6.8%

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

   2. sub-neg 6.8%

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

   3. div-inv 6.8%

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

   4. metadata-eval 6.8%

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

4. Applied egg-rr 6.8%

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

   1. sub-neg 6.8%

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

6. Simplified 6.8%

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

   1. add-cbrt-cube 10.4%

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

   2. pow3 10.4%

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

8. Applied egg-rr 10.4%

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

9. Final simplification 10.4%

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

Alternative 4: 6.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 1

   1. Initial program 6.8%

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

      1. expm1-log1p-u 6.8%

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

      2. expm1-undefine 6.9%

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

      3. log1p-undefine 6.9%

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

      4. rem-exp-log 6.9%

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

   4. Applied egg-rr 6.9%

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

   if 1 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 6.8%

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

      1. acos-asin 6.8%

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

      2. sub-neg 6.8%

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

      3. div-inv 6.8%

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

      4. metadata-eval 6.8%

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

   4. Applied egg-rr 6.8%

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

      1. sub-neg 6.8%

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

   6. Simplified 6.8%

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

      1. sub-neg 6.8%

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

      2. add-sqr-sqrt 10.4%

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

      3. distribute-rgt-neg-out 10.4%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 6.9%

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

      6. distribute-rgt-neg-out 6.9%

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

      7. add-sqr-sqrt 6.9%

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

      8. distribute-rgt-neg-out 6.9%

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

      9. add-sqr-sqrt 6.9%

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

      10. sqr-neg 6.9%

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

      11. sqrt-unprod 6.9%

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

      12. add-sqr-sqrt 6.9%

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

   8. Applied egg-rr 6.9%

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

4. Final simplification 6.9%

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

Alternative 5: 6.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 1

   1. Initial program 6.8%

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

   if 1 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 6.8%

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

      1. acos-asin 6.8%

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

      2. sub-neg 6.8%

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

      3. div-inv 6.8%

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

      4. metadata-eval 6.8%

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

   4. Applied egg-rr 6.8%

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

      1. sub-neg 6.8%

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

   6. Simplified 6.8%

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

      1. sub-neg 6.8%

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

      2. add-sqr-sqrt 10.4%

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

      3. distribute-rgt-neg-out 10.4%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 6.9%

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

      6. distribute-rgt-neg-out 6.9%

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

      7. add-sqr-sqrt 6.9%

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

      8. distribute-rgt-neg-out 6.9%

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

      9. add-sqr-sqrt 6.9%

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

      10. sqr-neg 6.9%

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

      11. sqrt-unprod 6.9%

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

      12. add-sqr-sqrt 6.9%

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

   8. Applied egg-rr 6.9%

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

4. Final simplification 6.8%

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

Alternative 6: 7.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

3. Taylor expanded in x around inf 7.3%

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

4. Final simplification 7.3%

   \[\leadsto \cos^{-1} \left(x \cdot \left(\frac{1}{x} + -1\right)\right) \]
5. Add Preprocessing

Alternative 7: 6.8% 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 6.8%

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

Developer Target 1: 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 2024179 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :alt
  (! :herbie-platform default (* 2 (asin (sqrt (/ x 2)))))

  (acos (- 1.0 x)))