bug323 (missed optimization)

Percentage Accurate: 7.0% → 10.4%

Time: 37.6s

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: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\pi \cdot 0.5}\\ \pi \cdot 0.5 - \mathsf{fma}\left({\left(\sqrt[3]{t_0}\right)}^{2} \cdot \sqrt[3]{{t_0}^{4}}, t_0, \sin^{-1} \left(1 - x\right) - t_0 \cdot \sqrt[3]{{\left(\pi \cdot 0.5\right)}^{2}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (* PI 0.5))))
   (-
    (* PI 0.5)
    (fma
     (* (pow (cbrt t_0) 2.0) (cbrt (pow t_0 4.0)))
     t_0
     (- (asin (- 1.0 x)) (* t_0 (cbrt (pow (* PI 0.5) 2.0))))))))

C

double code(double x) {
	double t_0 = cbrt((((double) M_PI) * 0.5));
	return (((double) M_PI) * 0.5) - fma((pow(cbrt(t_0), 2.0) * cbrt(pow(t_0, 4.0))), t_0, (asin((1.0 - x)) - (t_0 * cbrt(pow((((double) M_PI) * 0.5), 2.0)))));
}

Julia

function code(x)
	t_0 = cbrt(Float64(pi * 0.5))
	return Float64(Float64(pi * 0.5) - fma(Float64((cbrt(t_0) ^ 2.0) * cbrt((t_0 ^ 4.0))), t_0, Float64(asin(Float64(1.0 - x)) - Float64(t_0 * cbrt((Float64(pi * 0.5) ^ 2.0))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(Pi * 0.5), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(Pi * 0.5), $MachinePrecision] - N[(N[(N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Power[t$95$0, 4.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] - N[(t$95$0 * N[Power[N[Power[N[(Pi * 0.5), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. sub-neg 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

4. Applied egg-rr 6.3%

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

   1. sub-neg 6.3%

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

6. Simplified 6.3%

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

   1. asin-acos 6.3%

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

   2. div-inv 6.3%

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

   3. metadata-eval 6.3%

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

   4. sub-neg 6.3%

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

   5. add-cube-cbrt 10.0%

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

   6. fma-def 10.0%

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

   7. pow2 10.0%

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

8. Applied egg-rr 10.0%

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

   1. add-cube-cbrt 10.0%

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

   2. associate-*l* 10.0%

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

   3. unpow2 10.0%

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

   4. cbrt-prod 10.0%

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

   5. pow2 10.0%

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

   6. cbrt-unprod 10.0%

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

   7. pow-prod-up 10.0%

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

   8. metadata-eval 10.0%

      \[\leadsto \pi \cdot 0.5 - \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot 0.5}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{\color{blue}{4}}}, \sqrt[3]{\pi \cdot 0.5}, -\cos^{-1} \left(1 - x\right)\right) \]

10. Applied egg-rr 10.0%

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

    1. acos-asin 10.0%

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

    2. add-cube-cbrt 10.0%

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

    3. *-un-lft-identity 10.0%

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

    4. prod-diff 10.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot 0.5}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{4}}, \sqrt[3]{\pi \cdot 0.5}, -\color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\frac{\pi}{2}} \cdot \sqrt[3]{\frac{\pi}{2}}, \sqrt[3]{\frac{\pi}{2}}, -\sin^{-1} \left(1 - x\right) \cdot 1\right) + \mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), 1, \sin^{-1} \left(1 - x\right) \cdot 1\right)\right)}\right) \]

    5. cbrt-unprod 10.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot 0.5}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{4}}, \sqrt[3]{\pi \cdot 0.5}, -\left(\mathsf{fma}\left(\color{blue}{\sqrt[3]{\frac{\pi}{2} \cdot \frac{\pi}{2}}}, \sqrt[3]{\frac{\pi}{2}}, -\sin^{-1} \left(1 - x\right) \cdot 1\right) + \mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), 1, \sin^{-1} \left(1 - x\right) \cdot 1\right)\right)\right) \]

    6. pow2 10.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot 0.5}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{4}}, \sqrt[3]{\pi \cdot 0.5}, -\left(\mathsf{fma}\left(\sqrt[3]{\color{blue}{{\left(\frac{\pi}{2}\right)}^{2}}}, \sqrt[3]{\frac{\pi}{2}}, -\sin^{-1} \left(1 - x\right) \cdot 1\right) + \mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), 1, \sin^{-1} \left(1 - x\right) \cdot 1\right)\right)\right) \]

    7. div-inv 10.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot 0.5}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{4}}, \sqrt[3]{\pi \cdot 0.5}, -\left(\mathsf{fma}\left(\sqrt[3]{{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)}}^{2}}, \sqrt[3]{\frac{\pi}{2}}, -\sin^{-1} \left(1 - x\right) \cdot 1\right) + \mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), 1, \sin^{-1} \left(1 - x\right) \cdot 1\right)\right)\right) \]

    8. metadata-eval 10.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot 0.5}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{4}}, \sqrt[3]{\pi \cdot 0.5}, -\left(\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot \color{blue}{0.5}\right)}^{2}}, \sqrt[3]{\frac{\pi}{2}}, -\sin^{-1} \left(1 - x\right) \cdot 1\right) + \mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), 1, \sin^{-1} \left(1 - x\right) \cdot 1\right)\right)\right) \]

    9. div-inv 10.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot 0.5}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{4}}, \sqrt[3]{\pi \cdot 0.5}, -\left(\mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{2}}, \sqrt[3]{\color{blue}{\pi \cdot \frac{1}{2}}}, -\sin^{-1} \left(1 - x\right) \cdot 1\right) + \mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), 1, \sin^{-1} \left(1 - x\right) \cdot 1\right)\right)\right) \]

    10. metadata-eval 10.0%

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

12. Applied egg-rr 10.0%

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

    1. +-commutative 10.0%

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

    2. fma-udef 10.0%

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

    3. *-rgt-identity 10.0%

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

    4. *-rgt-identity 10.0%

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

    5. +-commutative 10.0%

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

    6. sub-neg 10.0%

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

    7. +-inverses 10.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot 0.5}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{4}}, \sqrt[3]{\pi \cdot 0.5}, -\left(\color{blue}{0} + \mathsf{fma}\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{2}}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right) \cdot 1\right)\right)\right) \]

    8. fma-udef 10.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot 0.5}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{4}}, \sqrt[3]{\pi \cdot 0.5}, -\left(0 + \color{blue}{\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{2}} \cdot \sqrt[3]{\pi \cdot 0.5} + \left(-\sin^{-1} \left(1 - x\right) \cdot 1\right)\right)}\right)\right) \]

    9. distribute-lft-neg-in 10.0%

       \[\leadsto \pi \cdot 0.5 - \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot 0.5}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{4}}, \sqrt[3]{\pi \cdot 0.5}, -\left(0 + \left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{2}} \cdot \sqrt[3]{\pi \cdot 0.5} + \color{blue}{\left(-\sin^{-1} \left(1 - x\right)\right) \cdot 1}\right)\right)\right) \]

    10. cancel-sign-sub-inv 10.0%

        \[\leadsto \pi \cdot 0.5 - \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot 0.5}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{4}}, \sqrt[3]{\pi \cdot 0.5}, -\left(0 + \color{blue}{\left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{2}} \cdot \sqrt[3]{\pi \cdot 0.5} - \sin^{-1} \left(1 - x\right) \cdot 1\right)}\right)\right) \]

    11. *-rgt-identity 10.0%

        \[\leadsto \pi \cdot 0.5 - \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot 0.5}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{4}}, \sqrt[3]{\pi \cdot 0.5}, -\left(0 + \left(\sqrt[3]{{\left(\pi \cdot 0.5\right)}^{2}} \cdot \sqrt[3]{\pi \cdot 0.5} - \color{blue}{\sin^{-1} \left(1 - x\right)}\right)\right)\right) \]

14. Simplified 10.0%

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

15. Final simplification 10.0%

    \[\leadsto \pi \cdot 0.5 - \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{\pi \cdot 0.5}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{\pi \cdot 0.5}\right)}^{4}}, \sqrt[3]{\pi \cdot 0.5}, \sin^{-1} \left(1 - x\right) - \sqrt[3]{\pi \cdot 0.5} \cdot \sqrt[3]{{\left(\pi \cdot 0.5\right)}^{2}}\right) \]
16. Add Preprocessing

Alternative 2: 10.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return fma(Float64(sqrt(pi) * sqrt(0.5)), sqrt(Float64(pi * 0.5)), Float64(-asin(Float64(1.0 - x))))
end

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. add-sqr-sqrt 4.5%

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

   3. fma-neg 4.5%

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

   4. div-inv 4.5%

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

   5. metadata-eval 4.5%

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

   6. div-inv 4.5%

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

   7. metadata-eval 4.5%

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

4. Applied egg-rr 4.5%

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

   1. sqrt-prod 10.0%

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

6. Applied egg-rr 10.0%

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

7. Final simplification 10.0%

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

Alternative 3: 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 6.3%

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

   1. acos-asin 6.3%

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

   2. sub-neg 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

4. Applied egg-rr 6.3%

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

   1. sub-neg 6.3%

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

6. Simplified 6.3%

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

   1. add-cube-cbrt 10.0%

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

   2. pow3 10.0%

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

8. Applied egg-rr 10.0%

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

9. Final simplification 10.0%

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

Alternative 4: 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 6.3%

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

   1. acos-asin 6.3%

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

   2. sub-neg 6.3%

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

   3. div-inv 6.3%

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

   4. metadata-eval 6.3%

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

4. Applied egg-rr 6.3%

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

   1. sub-neg 6.3%

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

6. Simplified 6.3%

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

   1. add-sqr-sqrt 10.0%

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

   2. pow2 10.0%

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

8. Applied egg-rr 10.0%

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

9. Final simplification 10.0%

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

Alternative 5: 10.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. acos-asin 6.3%

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

   2. add-sqr-sqrt 4.5%

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

   3. fma-neg 4.5%

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

   4. div-inv 4.5%

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

   5. metadata-eval 4.5%

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

   6. div-inv 4.5%

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

   7. metadata-eval 4.5%

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

4. Applied egg-rr 4.5%

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

5. Taylor expanded in x around 0 10.0%

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

6. Final simplification 10.0%

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

Alternative 6: 9.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \cos^{-1} \left(1 - x\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-17)
   (hypot (* PI 0.5) (asin (- 1.0 x)))
   (+ (+ 1.0 (acos (- 1.0 x))) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = hypot((((double) M_PI) * 0.5), asin((1.0 - x)));
	} else {
		tmp = (1.0 + acos((1.0 - x))) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = Math.hypot((Math.PI * 0.5), Math.asin((1.0 - x)));
	} else {
		tmp = (1.0 + Math.acos((1.0 - x))) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.6e-17:
		tmp = math.hypot((math.pi * 0.5), math.asin((1.0 - x)))
	else:
		tmp = (1.0 + math.acos((1.0 - x))) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = hypot(Float64(pi * 0.5), asin(Float64(1.0 - x)));
	else
		tmp = Float64(Float64(1.0 + acos(Float64(1.0 - x))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = hypot((pi * 0.5), asin((1.0 - x)));
	else
		tmp = (1.0 + acos((1.0 - x))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[Sqrt[N[(Pi * 0.5), $MachinePrecision] ^ 2 + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision], N[(N[(1.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.8%

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

      1. acos-asin 3.8%

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

      2. sub-neg 3.8%

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

      3. div-inv 3.8%

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

      4. metadata-eval 3.8%

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

   4. Applied egg-rr 3.8%

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

      1. sub-neg 3.8%

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

   6. Simplified 3.8%

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

      1. add-sqr-sqrt 3.8%

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

      2. sqrt-unprod 3.8%

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

      3. add-sqr-sqrt 3.8%

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

      4. cancel-sign-sub-inv 3.8%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 3.8%

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

      7. sqr-neg 3.8%

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

      8. add-sqr-sqrt 3.8%

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

      9. add-sqr-sqrt 3.8%

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

   8. Applied egg-rr 6.6%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 52.5%

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

      1. expm1-log1p-u 52.5%

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

      2. expm1-udef 52.5%

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

      3. log1p-udef 52.5%

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

      4. rem-exp-log 52.5%

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

   4. Applied egg-rr 52.5%

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

4. Final simplification 9.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{hypot}\left(\pi \cdot 0.5, \sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \cos^{-1} \left(1 - x\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 7.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.3%

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

      1. expm1-log1p-u 6.3%

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

      2. expm1-udef 6.3%

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

      3. log1p-udef 6.3%

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

      4. rem-exp-log 6.3%

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

   4. Applied egg-rr 6.3%

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

   if 1 < (-.f64 1 x)

   1. Initial program 6.3%

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

      1. acos-asin 6.3%

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

      2. add-sqr-sqrt 4.5%

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

      3. fma-neg 4.5%

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

      4. div-inv 4.5%

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

      5. metadata-eval 4.5%

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

      6. div-inv 4.5%

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

      7. metadata-eval 4.5%

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

   4. Applied egg-rr 4.5%

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

      1. fma-udef 4.5%

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

      2. add-sqr-sqrt 6.3%

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

      3. add-cube-cbrt 10.0%

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

      4. unpow2 10.0%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 6.9%

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

      7. sqr-neg 6.9%

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

      8. sqrt-unprod 6.9%

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

   6. Applied egg-rr 6.9%

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

      1. distribute-lft-out 6.9%

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

      2. metadata-eval 6.9%

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

      3. *-commutative 6.9%

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

      4. *-lft-identity 6.9%

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

   8. Simplified 6.9%

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

4. Final simplification 6.3%

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

Alternative 8: 7.0% 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 6.3%

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

   if 1 < (-.f64 1 x)

   1. Initial program 6.3%

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

      1. acos-asin 6.3%

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

      2. add-sqr-sqrt 4.5%

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

      3. fma-neg 4.5%

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

      4. div-inv 4.5%

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

      5. metadata-eval 4.5%

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

      6. div-inv 4.5%

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

      7. metadata-eval 4.5%

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

   4. Applied egg-rr 4.5%

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

      1. fma-udef 4.5%

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

      2. add-sqr-sqrt 6.3%

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

      3. add-cube-cbrt 10.0%

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

      4. unpow2 10.0%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 6.9%

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

      7. sqr-neg 6.9%

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

      8. sqrt-unprod 6.9%

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

   6. Applied egg-rr 6.9%

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

      1. distribute-lft-out 6.9%

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

      2. metadata-eval 6.9%

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

      3. *-commutative 6.9%

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

      4. *-lft-identity 6.9%

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

   8. Simplified 6.9%

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

4. Final simplification 6.3%

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

Alternative 9: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

3. Final simplification 6.3%

   \[\leadsto \cos^{-1} \left(1 - x\right) \]
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 2024014 
(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)))