bug323 (missed optimization)

Percentage Accurate: 6.7% → 10.2%

Time: 9.7s

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. add-log-exp 8.0%

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

3. Applied egg-rr 8.0%

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

   1. add-log-exp 8.0%

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

   2. acos-asin 8.0%

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

   3. div-inv 8.0%

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

   4. metadata-eval 8.0%

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

   5. add-cube-cbrt 6.2%

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

   6. fma-neg 6.2%

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

   7. pow2 6.2%

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

5. Applied egg-rr 6.2%

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

   1. pow1/3 11.4%

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

7. Applied egg-rr 11.4%

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

8. Final simplification 11.4%

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

Alternative 2: 10.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2 + \cos^{-1} \left(1 - x\right)}\\ \mathsf{fma}\left(t_0, t_0, -2\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 2.0 (acos (- 1.0 x)))))) (fma t_0 t_0 -2.0)))

C

double code(double x) {
	double t_0 = sqrt((2.0 + acos((1.0 - x))));
	return fma(t_0, t_0, -2.0);
}

Julia

function code(x)
	t_0 = sqrt(Float64(2.0 + acos(Float64(1.0 - x))))
	return fma(t_0, t_0, -2.0)
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(2.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 * t$95$0 + -2.0), $MachinePrecision]]

Derivation

1. Initial program 8.0%

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

   1. acos-asin 8.0%

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

   2. sub-neg 8.0%

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

   3. div-inv 8.0%

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

   4. metadata-eval 8.0%

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

3. Applied egg-rr 8.0%

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

   1. sub-neg 8.0%

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

5. Simplified 8.0%

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

   1. add-sqr-sqrt 11.3%

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

   2. pow2 11.3%

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

7. Applied egg-rr 11.3%

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

   1. add-cbrt-cube 11.3%

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

   2. pow1/3 11.3%

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

   3. add-sqr-sqrt 11.3%

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

   4. pow1 11.3%

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

   5. pow1/2 11.3%

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

   6. pow-prod-up 11.3%

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

   7. metadata-eval 11.3%

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

9. Applied egg-rr 11.3%

   \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left({\left({\sin^{-1} \left(1 - x\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \]
10. Step-by-step derivation

    1. unpow1/3 11.3%

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

11. Simplified 11.3%

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

    1. unpow2 11.3%

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

    2. metadata-eval 11.3%

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

    3. div-inv 11.3%

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

    4. cbrt-unprod 6.2%

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

    5. pow-prod-up 6.2%

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

    6. metadata-eval 6.2%

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

    7. add-cube-cbrt 11.3%

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

    8. unpow-prod-down 11.3%

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

    9. pow-prod-down 11.3%

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

    10. add-cbrt-cube 11.3%

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

    11. rem-cube-cbrt 8.0%

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

    12. acos-asin 8.0%

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

    13. expm1-log1p-u 8.0%

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

    14. expm1-udef 7.9%

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

13. Applied egg-rr 11.3%

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

14. Final simplification 11.3%

    \[\leadsto \mathsf{fma}\left(\sqrt{2 + \cos^{-1} \left(1 - x\right)}, \sqrt{2 + \cos^{-1} \left(1 - x\right)}, -2\right) \]

Alternative 3: 10.1% 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 8.0%

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

   1. acos-asin 8.0%

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

   2. sub-neg 8.0%

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

   3. div-inv 8.0%

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

   4. metadata-eval 8.0%

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

3. Applied egg-rr 8.0%

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

   1. sub-neg 8.0%

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

5. Simplified 8.0%

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

   1. add-sqr-sqrt 11.3%

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

   2. pow2 11.3%

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

7. Applied egg-rr 11.3%

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

8. Final simplification 11.3%

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

Alternative 4: 10.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \left({\left(\sqrt[3]{2 + \cos^{-1} \left(1 - x\right)}\right)}^{3} + -1\right) + -1 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (+ (pow (cbrt (+ 2.0 (acos (- 1.0 x)))) 3.0) -1.0) -1.0))

C

double code(double x) {
	return (pow(cbrt((2.0 + acos((1.0 - x)))), 3.0) + -1.0) + -1.0;
}

Java

public static double code(double x) {
	return (Math.pow(Math.cbrt((2.0 + Math.acos((1.0 - x)))), 3.0) + -1.0) + -1.0;
}

Julia

function code(x)
	return Float64(Float64((cbrt(Float64(2.0 + acos(Float64(1.0 - x)))) ^ 3.0) + -1.0) + -1.0)
end

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. expm1-log1p-u 8.0%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)\right)}}^{3}} \]

   2. expm1-udef 8.0%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1\right)}}^{3}} \]

   3. log1p-udef 8.0%

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

   4. add-exp-log 8.0%

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

3. Applied egg-rr 7.9%

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

   1. expm1-log1p-u 7.9%

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

   2. expm1-udef 7.9%

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

   3. log1p-udef 7.9%

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

   4. +-commutative 7.9%

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

   5. add-exp-log 7.9%

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

   6. +-commutative 7.9%

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

   7. associate-+l+ 7.9%

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

   8. metadata-eval 7.9%

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

5. Applied egg-rr 7.9%

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

   1. add-cube-cbrt 11.3%

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

   2. pow3 11.3%

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

   3. +-commutative 11.3%

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

7. Applied egg-rr 11.3%

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

8. Final simplification 11.3%

   \[\leadsto \left({\left(\sqrt[3]{2 + \cos^{-1} \left(1 - x\right)}\right)}^{3} + -1\right) + -1 \]

Alternative 5: 10.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \left({\left(\sqrt{2 + \cos^{-1} \left(1 - x\right)}\right)}^{2} + -1\right) + -1 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (+ (pow (sqrt (+ 2.0 (acos (- 1.0 x)))) 2.0) -1.0) -1.0))

C

double code(double x) {
	return (pow(sqrt((2.0 + acos((1.0 - x)))), 2.0) + -1.0) + -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((sqrt((2.0d0 + acos((1.0d0 - x)))) ** 2.0d0) + (-1.0d0)) + (-1.0d0)
end function

Java

public static double code(double x) {
	return (Math.pow(Math.sqrt((2.0 + Math.acos((1.0 - x)))), 2.0) + -1.0) + -1.0;
}

Python

def code(x):
	return (math.pow(math.sqrt((2.0 + math.acos((1.0 - x)))), 2.0) + -1.0) + -1.0

Julia

function code(x)
	return Float64(Float64((sqrt(Float64(2.0 + acos(Float64(1.0 - x)))) ^ 2.0) + -1.0) + -1.0)
end

MATLAB

function tmp = code(x)
	tmp = ((sqrt((2.0 + acos((1.0 - x)))) ^ 2.0) + -1.0) + -1.0;
end

Wolfram

code[x_] := N[(N[(N[Power[N[Sqrt[N[(2.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 8.0%

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

   1. expm1-log1p-u 8.0%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)\right)}}^{3}} \]

   2. expm1-udef 8.0%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1\right)}}^{3}} \]

   3. log1p-udef 8.0%

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

   4. add-exp-log 8.0%

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

3. Applied egg-rr 7.9%

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

   1. expm1-log1p-u 7.9%

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

   2. expm1-udef 7.9%

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

   3. log1p-udef 7.9%

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

   4. +-commutative 7.9%

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

   5. add-exp-log 7.9%

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

   6. +-commutative 7.9%

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

   7. associate-+l+ 7.9%

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

   8. metadata-eval 7.9%

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

5. Applied egg-rr 7.9%

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

   1. add-sqr-sqrt 11.3%

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

   2. pow2 11.3%

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

   3. +-commutative 11.3%

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

7. Applied egg-rr 11.3%

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

8. Final simplification 11.3%

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

Alternative 6: 9.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|t_0 + -1\right|\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{t_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= x 5.5e-17) (+ 1.0 (fabs (+ t_0 -1.0))) (log (exp t_0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = 1.0 + fabs((t_0 + -1.0));
	} else {
		tmp = log(exp(t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x))
    if (x <= 5.5d-17) then
        tmp = 1.0d0 + abs((t_0 + (-1.0d0)))
    else
        tmp = log(exp(t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = 1.0 + Math.abs((t_0 + -1.0));
	} else {
		tmp = Math.log(Math.exp(t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = 1.0 + math.fabs((t_0 + -1.0))
	else:
		tmp = math.log(math.exp(t_0))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(1.0 + abs(Float64(t_0 + -1.0)));
	else
		tmp = log(exp(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = 1.0 + abs((t_0 + -1.0));
	else
		tmp = log(exp(t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(1.0 + N[Abs[N[(t$95$0 + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. add-log-exp 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. add-log-exp 3.9%

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

      2. expm1-log1p-u 3.9%

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

      3. expm1-udef 3.9%

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

      4. log1p-udef 3.9%

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

      5. add-exp-log 3.9%

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

      6. associate--l+ 3.9%

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

      7. +-commutative 3.9%

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

      8. sub-neg 3.9%

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

      9. metadata-eval 3.9%

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

   5. Applied egg-rr 3.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.6%

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

      3. pow2 6.6%

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

   7. Applied egg-rr 6.6%

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

      1. unpow2 6.6%

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

      2. rem-sqrt-square 6.6%

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

   9. Simplified 6.6%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 62.1%

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

      1. add-log-exp 62.2%

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

   3. Applied egg-rr 62.2%

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

4. Final simplification 10.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\\ \end{array} \]

Alternative 7: 9.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17)
   (+ 1.0 (fabs (+ (acos (- 1.0 x)) -1.0)))
   (- (* PI 0.5) (asin (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = 1.0 + fabs((acos((1.0 - x)) + -1.0));
	} else {
		tmp = (((double) M_PI) * 0.5) - asin((1.0 - x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = 1.0 + Math.abs((Math.acos((1.0 - x)) + -1.0));
	} else {
		tmp = (Math.PI * 0.5) - Math.asin((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = 1.0 + math.fabs((math.acos((1.0 - x)) + -1.0))
	else:
		tmp = (math.pi * 0.5) - math.asin((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(1.0 + abs(Float64(acos(Float64(1.0 - x)) + -1.0)));
	else
		tmp = Float64(Float64(pi * 0.5) - asin(Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = 1.0 + abs((acos((1.0 - x)) + -1.0));
	else
		tmp = (pi * 0.5) - asin((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[(1.0 + N[Abs[N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. add-log-exp 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. add-log-exp 3.9%

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

      2. expm1-log1p-u 3.9%

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

      3. expm1-udef 3.9%

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

      4. log1p-udef 3.9%

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

      5. add-exp-log 3.9%

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

      6. associate--l+ 3.9%

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

      7. +-commutative 3.9%

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

      8. sub-neg 3.9%

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

      9. metadata-eval 3.9%

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

   5. Applied egg-rr 3.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.6%

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

      3. pow2 6.6%

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

   7. Applied egg-rr 6.6%

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

      1. unpow2 6.6%

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

      2. rem-sqrt-square 6.6%

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

   9. Simplified 6.6%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 62.1%

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

      1. acos-asin 62.2%

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

      2. sub-neg 62.2%

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

      3. div-inv 62.2%

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

      4. metadata-eval 62.2%

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

   3. Applied egg-rr 62.2%

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

      1. sub-neg 62.2%

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

   5. Simplified 62.2%

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

4. Final simplification 10.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 8: 9.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17) (+ (* PI 0.5) (asin (- 1.0 x))) (acos (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (((double) M_PI) * 0.5) + asin((1.0 - x));
	} else {
		tmp = acos((1.0 - x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (Math.PI * 0.5) + Math.asin((1.0 - x));
	} else {
		tmp = Math.acos((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = (math.pi * 0.5) + math.asin((1.0 - x))
	else:
		tmp = math.acos((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(Float64(pi * 0.5) + asin(Float64(1.0 - x)));
	else
		tmp = acos(Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = (pi * 0.5) + asin((1.0 - x));
	else
		tmp = acos((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[(N[(Pi * 0.5), $MachinePrecision] + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. add-cbrt-cube 3.9%

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

      2. pow3 3.9%

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

   3. Applied egg-rr 3.9%

      \[\leadsto \color{blue}{\sqrt[3]{{\cos^{-1} \left(1 - x\right)}^{3}}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 3.9%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)\right)}}^{3}} \]

      2. expm1-udef 3.9%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1\right)}}^{3}} \]

      3. log1p-udef 3.9%

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

      4. add-exp-log 3.9%

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

   5. Applied egg-rr 3.9%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(1 + \cos^{-1} \left(1 - x\right)\right) - 1\right)}}^{3}} \]
   6. Step-by-step derivation

      1. rem-cbrt-cube 3.9%

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

      2. add-exp-log 3.9%

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

      3. log1p-udef 3.9%

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

      4. expm1-udef 3.9%

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

      5. expm1-log1p-u 3.9%

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

      6. acos-asin 3.9%

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

      7. div-inv 3.9%

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

      8. metadata-eval 3.9%

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

      9. sub-neg 3.9%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 6.6%

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

      12. sqr-neg 6.6%

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

      13. sqrt-unprod 6.6%

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

      14. add-sqr-sqrt 6.6%

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

   7. Applied egg-rr 6.6%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 62.1%

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

4. Final simplification 10.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 9: 9.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot 0.5 + t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (if (<= x 5.5e-17) (+ (* PI 0.5) t_0) (- (* PI 0.5) t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (((double) M_PI) * 0.5) + t_0;
	} else {
		tmp = (((double) M_PI) * 0.5) - t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = (Math.PI * 0.5) + t_0;
	} else {
		tmp = (Math.PI * 0.5) - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = (math.pi * 0.5) + t_0
	else:
		tmp = (math.pi * 0.5) - t_0
	return tmp

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(Float64(pi * 0.5) + t_0);
	else
		tmp = Float64(Float64(pi * 0.5) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = asin((1.0 - x));
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = (pi * 0.5) + t_0;
	else
		tmp = (pi * 0.5) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(N[(Pi * 0.5), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. add-cbrt-cube 3.9%

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

      2. pow3 3.9%

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

   3. Applied egg-rr 3.9%

      \[\leadsto \color{blue}{\sqrt[3]{{\cos^{-1} \left(1 - x\right)}^{3}}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 3.9%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)\right)}}^{3}} \]

      2. expm1-udef 3.9%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1\right)}}^{3}} \]

      3. log1p-udef 3.9%

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

      4. add-exp-log 3.9%

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

   5. Applied egg-rr 3.9%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(1 + \cos^{-1} \left(1 - x\right)\right) - 1\right)}}^{3}} \]
   6. Step-by-step derivation

      1. rem-cbrt-cube 3.9%

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

      2. add-exp-log 3.9%

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

      3. log1p-udef 3.9%

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

      4. expm1-udef 3.9%

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

      5. expm1-log1p-u 3.9%

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

      6. acos-asin 3.9%

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

      7. div-inv 3.9%

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

      8. metadata-eval 3.9%

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

      9. sub-neg 3.9%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 6.6%

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

      12. sqr-neg 6.6%

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

      13. sqrt-unprod 6.6%

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

      14. add-sqr-sqrt 6.6%

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

   7. Applied egg-rr 6.6%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 62.1%

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

      1. acos-asin 62.2%

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

      2. sub-neg 62.2%

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

      3. div-inv 62.2%

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

      4. metadata-eval 62.2%

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

   3. Applied egg-rr 62.2%

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

      1. sub-neg 62.2%

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

   5. Simplified 62.2%

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

4. Final simplification 10.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\\ \end{array} \]

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 8.0%

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

2. Final simplification 8.0%

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

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023297 
(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)))