bug323 (missed optimization)

Percentage Accurate: 6.7% → 10.2%

Time: 17.7s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.7% 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: 10.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt[3]{\sqrt[3]{t\_0}}\\ \pi \cdot 0.5 - t\_1 \cdot \left(\sqrt[3]{{t\_0}^{2}} \cdot {\left(\sqrt[3]{t\_1 \cdot \left({\left(\sqrt[3]{t\_1}\right)}^{2} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{{t\_0}^{4}}}}\right)}\right)}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (cbrt (cbrt t_0))))
   (-
    (* PI 0.5)
    (*
     t_1
     (*
      (cbrt (pow t_0 2.0))
      (pow
       (cbrt
        (* t_1 (* (pow (cbrt t_1) 2.0) (cbrt (cbrt (cbrt (pow t_0 4.0)))))))
       2.0))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = cbrt(cbrt(t_0));
	return (((double) M_PI) * 0.5) - (t_1 * (cbrt(pow(t_0, 2.0)) * pow(cbrt((t_1 * (pow(cbrt(t_1), 2.0) * cbrt(cbrt(cbrt(pow(t_0, 4.0))))))), 2.0)));
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	double t_1 = Math.cbrt(Math.cbrt(t_0));
	return (Math.PI * 0.5) - (t_1 * (Math.cbrt(Math.pow(t_0, 2.0)) * Math.pow(Math.cbrt((t_1 * (Math.pow(Math.cbrt(t_1), 2.0) * Math.cbrt(Math.cbrt(Math.cbrt(Math.pow(t_0, 4.0))))))), 2.0)));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = cbrt(cbrt(t_0))
	return Float64(Float64(pi * 0.5) - Float64(t_1 * Float64(cbrt((t_0 ^ 2.0)) * (cbrt(Float64(t_1 * Float64((cbrt(t_1) ^ 2.0) * cbrt(cbrt(cbrt((t_0 ^ 4.0))))))) ^ 2.0))))
end

Wolfram

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

Derivation

1. Initial program 5.9%

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

   1. acos-asin 5.9%

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

   2. sub-neg 5.9%

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

   3. div-inv 5.9%

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

   4. metadata-eval 5.9%

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

4. Applied egg-rr 5.9%

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

   1. sub-neg 5.9%

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

6. Simplified 5.9%

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

   1. add-sqr-sqrt 9.7%

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

   2. pow2 9.7%

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

8. Applied egg-rr 9.7%

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

   1. unpow2 9.7%

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

   2. add-sqr-sqrt 5.9%

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

   3. add-cube-cbrt 9.6%

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

   4. unpow2 9.6%

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

   5. add-cube-cbrt 9.6%

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

   6. unpow2 9.6%

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

   7. cbrt-prod 9.6%

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

   8. associate-*r* 9.6%

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

10. Applied egg-rr 9.6%

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

    1. add-cube-cbrt 9.6%

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

    2. unpow2 9.6%

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

    3. add-cube-cbrt 9.6%

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

    4. associate-*l* 9.6%

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

12. Applied egg-rr 9.7%

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

    1. *-commutative 9.7%

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

    2. *-commutative 9.7%

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

    3. associate-*l* 9.7%

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

14. Simplified 9.7%

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

15. Final simplification 9.7%

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

Alternative 2: 10.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}^{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (cbrt (pow (asin (- 1.0 x)) 1.5)) 2.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(cbrt(pow(asin((1.0 - x)), 1.5)), 2.0);
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.cbrt(Math.pow(Math.asin((1.0 - x)), 1.5)), 2.0);
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (cbrt((asin(Float64(1.0 - x)) ^ 1.5)) ^ 2.0))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.9%

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

   1. acos-asin 5.9%

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

   2. sub-neg 5.9%

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

   3. div-inv 5.9%

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

   4. metadata-eval 5.9%

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

4. Applied egg-rr 5.9%

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

   1. sub-neg 5.9%

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

6. Simplified 5.9%

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

   1. add-sqr-sqrt 9.7%

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

   2. pow2 9.7%

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

8. Applied egg-rr 9.7%

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

   1. add-cbrt-cube 9.7%

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

   2. pow1/3 9.7%

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

   3. add-sqr-sqrt 9.7%

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

   4. pow1 9.7%

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

   5. pow1/2 9.7%

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

   6. pow-prod-up 9.7%

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

   7. metadata-eval 9.7%

      \[\leadsto \pi \cdot 0.5 - {\left({\left({\sin^{-1} \left(1 - x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2} \]

10. Applied egg-rr 9.7%

    \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left({\left({\sin^{-1} \left(1 - x\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \]

12. Simplified 9.7%

    \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}}^{2} \]

13. Final simplification 9.7%

    \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}^{2} \]
14. Add Preprocessing

Alternative 3: 6.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0)
   (- (* PI 0.5) (expm1 (log1p (asin (- 1.0 x)))))
   (- PI (acos (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = (((double) M_PI) * 0.5) - expm1(log1p(asin((1.0 - x))));
	} else {
		tmp = ((double) M_PI) - acos((1.0 - x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = (Math.PI * 0.5) - Math.expm1(Math.log1p(Math.asin((1.0 - x))));
	} else {
		tmp = Math.PI - Math.acos((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = (math.pi * 0.5) - math.expm1(math.log1p(math.asin((1.0 - x))))
	else:
		tmp = math.pi - math.acos((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = Float64(Float64(pi * 0.5) - expm1(log1p(asin(Float64(1.0 - x)))));
	else
		tmp = Float64(pi - acos(Float64(1.0 - x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[(N[(Pi * 0.5), $MachinePrecision] - N[(Exp[N[Log[1 + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(Pi - N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.9%

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

      1. acos-asin 5.9%

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

      2. sub-neg 5.9%

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

      3. div-inv 5.9%

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

      4. metadata-eval 5.9%

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

   4. Applied egg-rr 5.9%

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

      1. sub-neg 5.9%

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

   6. Simplified 5.9%

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

      1. expm1-log1p-u 5.9%

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

   8. Applied egg-rr 5.9%

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

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

   1. Initial program 5.9%

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

      1. acos-asin 5.9%

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

      2. sub-neg 5.9%

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

      3. div-inv 5.9%

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

      4. metadata-eval 5.9%

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

   4. Applied egg-rr 5.9%

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

      1. sub-neg 5.9%

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

   6. Simplified 5.9%

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

      1. add-sqr-sqrt 9.7%

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

      2. pow2 9.7%

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

   8. Applied egg-rr 9.7%

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

      1. unpow2 9.7%

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

      2. add-sqr-sqrt 5.9%

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

      3. sub-neg 5.9%

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

      4. +-commutative 5.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 6.9%

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

      7. sqr-neg 6.9%

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

      8. sqrt-unprod 6.9%

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

      9. add-sqr-sqrt 6.9%

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

      10. asin-acos 6.9%

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

      11. div-inv 6.9%

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

      12. metadata-eval 6.9%

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

      13. associate-+l- 6.9%

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

   10. Applied egg-rr 6.9%

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

       1. fma-undefine 6.9%

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

       2. neg-sub0 6.9%

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

       3. associate-+l- 6.9%

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

       4. neg-sub0 6.9%

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

       5. +-commutative 6.9%

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

       6. associate-+r+ 6.9%

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

       7. sub-neg 6.9%

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

       8. distribute-lft-out 6.9%

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

       9. metadata-eval 6.9%

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

       10. *-rgt-identity 6.9%

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

   12. Simplified 6.9%

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

4. Final simplification 5.9%

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

Alternative 4: 10.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.9%

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

   1. acos-asin 5.9%

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

   2. sub-neg 5.9%

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

   3. div-inv 5.9%

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

   4. metadata-eval 5.9%

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

4. Applied egg-rr 5.9%

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

   1. sub-neg 5.9%

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

6. Simplified 5.9%

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

   1. add-sqr-sqrt 9.7%

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

   2. pow2 9.7%

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

8. Applied egg-rr 9.7%

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

   1. add-cbrt-cube 9.7%

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

   2. pow1/3 9.7%

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

   3. add-sqr-sqrt 9.7%

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

   4. pow1 9.7%

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

   5. pow1/2 9.7%

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

   6. pow-prod-up 9.7%

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

   7. metadata-eval 9.7%

      \[\leadsto \pi \cdot 0.5 - {\left({\left({\sin^{-1} \left(1 - x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2} \]

10. Applied egg-rr 9.7%

    \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left({\left({\sin^{-1} \left(1 - x\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \]

12. Simplified 9.7%

    \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}}^{2} \]
13. Step-by-step derivation

    1. pow1/3 9.7%

       \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left({\left({\sin^{-1} \left(1 - x\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \]

    2. pow-pow 9.7%

       \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}\right)}}^{2} \]

    3. metadata-eval 9.7%

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

    4. metadata-eval 9.7%

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

    5. pow-div 4.1%

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

    6. pow1 4.1%

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

    7. pow1/2 4.1%

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

    8. clear-num 4.1%

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

    9. pow1/2 4.1%

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

    10. pow1 4.1%

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

    11. pow-div 9.7%

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

    12. metadata-eval 9.7%

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

14. Applied egg-rr 9.7%

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

15. Final simplification 9.7%

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

Alternative 5: 10.2% 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 5.9%

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

   1. acos-asin 5.9%

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

   2. sub-neg 5.9%

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

   3. div-inv 5.9%

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

   4. metadata-eval 5.9%

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

4. Applied egg-rr 5.9%

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

   1. sub-neg 5.9%

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

6. Simplified 5.9%

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

   1. add-cube-cbrt 9.6%

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

   2. pow3 9.6%

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

8. Applied egg-rr 9.6%

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

9. Final simplification 9.6%

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

Alternative 6: 10.2% 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 5.9%

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

   1. acos-asin 5.9%

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

   2. sub-neg 5.9%

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

   3. div-inv 5.9%

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

   4. metadata-eval 5.9%

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

4. Applied egg-rr 5.9%

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

   1. sub-neg 5.9%

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

6. Simplified 5.9%

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

   1. add-sqr-sqrt 9.7%

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

   2. pow2 9.7%

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

8. Applied egg-rr 9.7%

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

9. Final simplification 9.7%

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

Alternative 7: 6.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.9%

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

      1. acos-asin 5.9%

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

      2. sub-neg 5.9%

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

      3. div-inv 5.9%

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

      4. metadata-eval 5.9%

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

   4. Applied egg-rr 5.9%

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

      1. sub-neg 5.9%

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

   6. Simplified 5.9%

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

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

   1. Initial program 5.9%

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

      1. acos-asin 5.9%

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

      2. sub-neg 5.9%

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

      3. div-inv 5.9%

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

      4. metadata-eval 5.9%

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

   4. Applied egg-rr 5.9%

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

      1. sub-neg 5.9%

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

   6. Simplified 5.9%

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

      1. add-sqr-sqrt 9.7%

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

      2. pow2 9.7%

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

   8. Applied egg-rr 9.7%

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

      1. unpow2 9.7%

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

      2. add-sqr-sqrt 5.9%

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

      3. sub-neg 5.9%

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

      4. +-commutative 5.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 6.9%

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

      7. sqr-neg 6.9%

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

      8. sqrt-unprod 6.9%

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

      9. add-sqr-sqrt 6.9%

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

      10. asin-acos 6.9%

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

      11. div-inv 6.9%

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

      12. metadata-eval 6.9%

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

      13. associate-+l- 6.9%

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

   10. Applied egg-rr 6.9%

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

       1. fma-undefine 6.9%

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

       2. neg-sub0 6.9%

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

       3. associate-+l- 6.9%

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

       4. neg-sub0 6.9%

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

       5. +-commutative 6.9%

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

       6. associate-+r+ 6.9%

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

       7. sub-neg 6.9%

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

       8. distribute-lft-out 6.9%

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

       9. metadata-eval 6.9%

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

       10. *-rgt-identity 6.9%

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

   12. Simplified 6.9%

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

4. Final simplification 5.9%

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

Alternative 8: 6.7% 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 #s(literal 1 binary64) x) < 1

   1. Initial program 5.9%

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

      1. expm1-log1p-u 5.9%

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

      2. expm1-undefine 5.9%

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

      3. log1p-undefine 5.9%

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

      4. rem-exp-log 5.9%

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

   4. Applied egg-rr 5.9%

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

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

   1. Initial program 5.9%

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

      1. acos-asin 5.9%

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

      2. sub-neg 5.9%

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

      3. div-inv 5.9%

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

      4. metadata-eval 5.9%

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

   4. Applied egg-rr 5.9%

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

      1. sub-neg 5.9%

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

   6. Simplified 5.9%

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

      1. add-sqr-sqrt 9.7%

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

      2. pow2 9.7%

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

   8. Applied egg-rr 9.7%

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

      1. unpow2 9.7%

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

      2. add-sqr-sqrt 5.9%

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

      3. sub-neg 5.9%

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

      4. +-commutative 5.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 6.9%

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

      7. sqr-neg 6.9%

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

      8. sqrt-unprod 6.9%

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

      9. add-sqr-sqrt 6.9%

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

      10. asin-acos 6.9%

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

      11. div-inv 6.9%

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

      12. metadata-eval 6.9%

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

      13. associate-+l- 6.9%

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

   10. Applied egg-rr 6.9%

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

       1. fma-undefine 6.9%

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

       2. neg-sub0 6.9%

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

       3. associate-+l- 6.9%

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

       4. neg-sub0 6.9%

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

       5. +-commutative 6.9%

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

       6. associate-+r+ 6.9%

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

       7. sub-neg 6.9%

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

       8. distribute-lft-out 6.9%

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

       9. metadata-eval 6.9%

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

       10. *-rgt-identity 6.9%

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

   12. Simplified 6.9%

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

4. Final simplification 5.9%

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

Alternative 9: 6.7% 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 #s(literal 1 binary64) x) < 1

   1. Initial program 5.9%

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

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

   1. Initial program 5.9%

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

      1. acos-asin 5.9%

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

      2. sub-neg 5.9%

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

      3. div-inv 5.9%

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

      4. metadata-eval 5.9%

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

   4. Applied egg-rr 5.9%

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

      1. sub-neg 5.9%

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

   6. Simplified 5.9%

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

      1. add-sqr-sqrt 9.7%

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

      2. pow2 9.7%

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

   8. Applied egg-rr 9.7%

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

      1. unpow2 9.7%

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

      2. add-sqr-sqrt 5.9%

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

      3. sub-neg 5.9%

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

      4. +-commutative 5.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 6.9%

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

      7. sqr-neg 6.9%

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

      8. sqrt-unprod 6.9%

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

      9. add-sqr-sqrt 6.9%

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

      10. asin-acos 6.9%

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

      11. div-inv 6.9%

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

      12. metadata-eval 6.9%

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

      13. associate-+l- 6.9%

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

   10. Applied egg-rr 6.9%

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

       1. fma-undefine 6.9%

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

       2. neg-sub0 6.9%

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

       3. associate-+l- 6.9%

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

       4. neg-sub0 6.9%

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

       5. +-commutative 6.9%

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

       6. associate-+r+ 6.9%

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

       7. sub-neg 6.9%

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

       8. distribute-lft-out 6.9%

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

       9. metadata-eval 6.9%

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

       10. *-rgt-identity 6.9%

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

   12. Simplified 6.9%

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

4. Final simplification 5.9%

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

Alternative 10: 6.7% 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 5.9%

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

3. Final simplification 5.9%

   \[\leadsto \cos^{-1} \left(1 - x\right) \]
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 2024110 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :alt
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))