bug323 (missed optimization)

Percentage Accurate: 6.7% → 10.2%

Time: 7.8s

Alternatives: 12

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

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (+
    (fma (cbrt (pow (* PI 0.5) 2.0)) (cbrt (* PI 0.5)) (- t_0))
    (fma (- (sqrt t_0)) (sqrt (asin (log (exp (- 1.0 x))))) t_0))))

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[N[Power[N[(Pi * 0.5), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(Pi * 0.5), $MachinePrecision], 1/3], $MachinePrecision] + (-t$95$0)), $MachinePrecision] + N[((-N[Sqrt[t$95$0], $MachinePrecision]) * N[Sqrt[N[ArcSin[N[Log[N[Exp[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + t$95$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. *-un-lft-identity 5.9%

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

   3. add-sqr-sqrt 9.4%

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

   4. prod-diff 9.4%

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

   5. add-sqr-sqrt 9.4%

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

   6. fmm-def 9.4%

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

   7. *-un-lft-identity 9.4%

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

   8. acos-asin 9.4%

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

   9. add-sqr-sqrt 9.4%

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

4. Applied egg-rr 9.4%

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

   1. acos-asin 9.4%

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

   2. add-cube-cbrt 4.1%

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

   3. fmm-def 4.0%

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

   4. cbrt-unprod 9.4%

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

   5. pow2 9.4%

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

   6. div-inv 9.4%

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

   7. metadata-eval 9.4%

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

   8. div-inv 9.4%

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

   9. metadata-eval 9.4%

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

6. Applied egg-rr 9.4%

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

   1. add-log-exp 9.4%

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

8. Applied egg-rr 9.4%

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

Alternative 2: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{t\_0}\\ \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{0.6666666666666666}, \sqrt[3]{\pi \cdot 0.5}, -t\_0\right) + \mathsf{fma}\left(-t\_1, t\_1, t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (sqrt t_0)))
   (+
    (fma (pow (* PI 0.5) 0.6666666666666666) (cbrt (* PI 0.5)) (- t_0))
    (fma (- t_1) t_1 t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = sqrt(t_0);
	return fma(pow((((double) M_PI) * 0.5), 0.6666666666666666), cbrt((((double) M_PI) * 0.5)), -t_0) + fma(-t_1, t_1, t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = sqrt(t_0)
	return Float64(fma((Float64(pi * 0.5) ^ 0.6666666666666666), cbrt(Float64(pi * 0.5)), Float64(-t_0)) + fma(Float64(-t_1), t_1, t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, N[(N[(N[Power[N[(Pi * 0.5), $MachinePrecision], 0.6666666666666666], $MachinePrecision] * N[Power[N[(Pi * 0.5), $MachinePrecision], 1/3], $MachinePrecision] + (-t$95$0)), $MachinePrecision] + N[((-t$95$1) * t$95$1 + t$95$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. *-un-lft-identity 5.9%

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

   3. add-sqr-sqrt 9.4%

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

   4. prod-diff 9.4%

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

   5. add-sqr-sqrt 9.4%

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

   6. fmm-def 9.4%

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

   7. *-un-lft-identity 9.4%

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

   8. acos-asin 9.4%

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

   9. add-sqr-sqrt 9.4%

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

4. Applied egg-rr 9.4%

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

   1. acos-asin 9.4%

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

   2. add-cube-cbrt 4.1%

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

   3. fmm-def 4.0%

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

   4. cbrt-unprod 9.4%

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

   5. pow2 9.4%

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

   6. div-inv 9.4%

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

   7. metadata-eval 9.4%

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

   8. div-inv 9.4%

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

   9. metadata-eval 9.4%

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

6. Applied egg-rr 9.4%

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

   1. pow1/3 9.4%

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

   2. pow-pow 9.4%

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

   3. metadata-eval 9.4%

      \[\leadsto \mathsf{fma}\left({\left(\pi \cdot 0.5\right)}^{\color{blue}{0.6666666666666666}}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right) \]

8. Applied egg-rr 9.4%

   \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\pi \cdot 0.5\right)}^{0.6666666666666666}}, \sqrt[3]{\pi \cdot 0.5}, -\sin^{-1} \left(1 - x\right)\right) + \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \left(1 - x\right)}, \sin^{-1} \left(1 - x\right)\right) \]
9. Add Preprocessing

Alternative 3: 10.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathsf{fma}\left(-\sqrt{t\_0}, \sqrt{\sin^{-1} \log \left(e^{1 - x}\right)}, t\_0\right) + \cos^{-1} \left(1 - x\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[((-N[Sqrt[t$95$0], $MachinePrecision]) * N[Sqrt[N[ArcSin[N[Log[N[Exp[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] + N[ArcCos[N[(1.0 - x), $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. *-un-lft-identity 5.9%

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

   3. add-sqr-sqrt 9.4%

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

   4. prod-diff 9.4%

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

   5. add-sqr-sqrt 9.4%

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

   6. fmm-def 9.4%

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

   7. *-un-lft-identity 9.4%

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

   8. acos-asin 9.4%

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

   9. add-sqr-sqrt 9.4%

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

4. Applied egg-rr 9.4%

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

   1. add-log-exp 9.4%

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

6. Applied egg-rr 9.4%

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

7. Final simplification 9.4%

   \[\leadsto \mathsf{fma}\left(-\sqrt{\sin^{-1} \left(1 - x\right)}, \sqrt{\sin^{-1} \log \left(e^{1 - x}\right)}, \sin^{-1} \left(1 - x\right)\right) + \cos^{-1} \left(1 - x\right) \]
8. Add Preprocessing

Alternative 4: 10.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := 2 + t\_0\\ {\left(\sqrt[3]{\frac{{\left(1 + t\_0\right)}^{2}}{t\_1}}\right)}^{3} + \frac{-1}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))) (t_1 (+ 2.0 t_0)))
   (+ (pow (cbrt (/ (pow (+ 1.0 t_0) 2.0) t_1)) 3.0) (/ -1.0 t_1))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double t_1 = 2.0 + t_0;
	return pow(cbrt((pow((1.0 + t_0), 2.0) / t_1)), 3.0) + (-1.0 / t_1);
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double t_1 = 2.0 + t_0;
	return Math.pow(Math.cbrt((Math.pow((1.0 + t_0), 2.0) / t_1)), 3.0) + (-1.0 / t_1);
}

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	t_1 = Float64(2.0 + t_0)
	return Float64((cbrt(Float64((Float64(1.0 + t_0) ^ 2.0) / t_1)) ^ 3.0) + Float64(-1.0 / t_1))
end

Wolfram

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

Derivation

1. Initial program 5.9%

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

   1. add-cbrt-cube 5.9%

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

   2. pow1/3 5.9%

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

   3. pow3 5.9%

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

4. Applied egg-rr 5.9%

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. unpow1/3 5.9%

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

   2. rem-cbrt-cube 5.9%

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

   3. expm1-log1p-u 5.9%

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

   4. log1p-define 5.9%

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

   5. expm1-define 5.9%

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

   6. add-exp-log 5.9%

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

   7. flip-- 5.9%

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

   8. metadata-eval 5.9%

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

   9. div-sub 5.9%

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

   10. pow2 5.9%

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

   11. +-commutative 5.9%

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

   12. associate-+l+ 5.9%

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

   13. metadata-eval 5.9%

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

   14. +-commutative 5.9%

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

6. Applied egg-rr 5.9%

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

   1. add-cube-cbrt 9.4%

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

   2. pow3 9.4%

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

   3. +-commutative 9.4%

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

8. Applied egg-rr 9.4%

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

9. Final simplification 9.4%

   \[\leadsto {\left(\sqrt[3]{\frac{{\left(1 + \cos^{-1} \left(1 - x\right)\right)}^{2}}{2 + \cos^{-1} \left(1 - x\right)}}\right)}^{3} + \frac{-1}{2 + \cos^{-1} \left(1 - x\right)} \]
10. Add Preprocessing

Alternative 5: 10.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := 2 + t\_0\\ \frac{{\left(1 + t\_0\right)}^{2}}{t\_1} - {\left(\sqrt{t\_1}\right)}^{-2} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))) (t_1 (+ 2.0 t_0)))
   (- (/ (pow (+ 1.0 t_0) 2.0) t_1) (pow (sqrt t_1) -2.0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double t_1 = 2.0 + t_0;
	return (pow((1.0 + t_0), 2.0) / t_1) - pow(sqrt(t_1), -2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = acos((1.0d0 - x))
    t_1 = 2.0d0 + t_0
    code = (((1.0d0 + t_0) ** 2.0d0) / t_1) - (sqrt(t_1) ** (-2.0d0))
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double t_1 = 2.0 + t_0;
	return (Math.pow((1.0 + t_0), 2.0) / t_1) - Math.pow(Math.sqrt(t_1), -2.0);
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	t_1 = 2.0 + t_0
	return (math.pow((1.0 + t_0), 2.0) / t_1) - math.pow(math.sqrt(t_1), -2.0)

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	t_1 = Float64(2.0 + t_0)
	return Float64(Float64((Float64(1.0 + t_0) ^ 2.0) / t_1) - (sqrt(t_1) ^ -2.0))
end

MATLAB

function tmp = code(x)
	t_0 = acos((1.0 - x));
	t_1 = 2.0 + t_0;
	tmp = (((1.0 + t_0) ^ 2.0) / t_1) - (sqrt(t_1) ^ -2.0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, N[(N[(N[Power[N[(1.0 + t$95$0), $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision] - N[Power[N[Sqrt[t$95$1], $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. add-cbrt-cube 5.9%

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

   2. pow1/3 5.9%

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

   3. pow3 5.9%

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

4. Applied egg-rr 5.9%

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(1 - x\right)}^{3}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. unpow1/3 5.9%

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

   2. rem-cbrt-cube 5.9%

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

   3. expm1-log1p-u 5.9%

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

   4. log1p-define 5.9%

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

   5. expm1-define 5.9%

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

   6. add-exp-log 5.9%

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

   7. flip-- 5.9%

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

   8. metadata-eval 5.9%

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

   9. div-sub 5.9%

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

   10. pow2 5.9%

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

   11. +-commutative 5.9%

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

   12. associate-+l+ 5.9%

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

   13. metadata-eval 5.9%

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

   14. +-commutative 5.9%

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

6. Applied egg-rr 5.9%

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

   1. inv-pow 5.9%

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

   2. add-sqr-sqrt 9.4%

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

   3. unpow-prod-down 9.4%

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

   4. +-commutative 9.4%

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

   5. +-commutative 9.4%

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

8. Applied egg-rr 9.4%

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

   1. pow-sqr 9.4%

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

   2. +-commutative 9.4%

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

   3. metadata-eval 9.4%

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

10. Simplified 9.4%

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

11. Final simplification 9.4%

    \[\leadsto \frac{{\left(1 + \cos^{-1} \left(1 - x\right)\right)}^{2}}{2 + \cos^{-1} \left(1 - x\right)} - {\left(\sqrt{2 + \cos^{-1} \left(1 - x\right)}\right)}^{-2} \]
12. Add Preprocessing

Alternative 6: 6.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0)
   (+ (+ 1.0 (log (exp (acos (- 1.0 x))))) -1.0)
   (acos (- x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[(N[(1.0 + N[Log[N[Exp[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[ArcCos[(-x)], $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%

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

      1. add-log-exp 5.9%

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

   6. Applied egg-rr 5.9%

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

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

   1. Initial program 5.9%

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

   3. Taylor expanded in x around inf 6.7%

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

      1. neg-mul-1 6.7%

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

   5. Simplified 6.7%

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

4. Final simplification 5.9%

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

Alternative 7: 10.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{0.5} \cdot \left(\pi \cdot \sqrt[3]{0.25}\right) - \sin^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* (cbrt 0.5) (* PI (cbrt 0.25))) (asin (- 1.0 x))))

C

double code(double x) {
	return (cbrt(0.5) * (((double) M_PI) * cbrt(0.25))) - asin((1.0 - x));
}

Java

public static double code(double x) {
	return (Math.cbrt(0.5) * (Math.PI * Math.cbrt(0.25))) - Math.asin((1.0 - x));
}

Julia

function code(x)
	return Float64(Float64(cbrt(0.5) * Float64(pi * cbrt(0.25))) - asin(Float64(1.0 - x)))
end

Wolfram

code[x_] := N[(N[(N[Power[0.5, 1/3], $MachinePrecision] * N[(Pi * N[Power[0.25, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[ArcSin[N[(1.0 - x), $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. *-un-lft-identity 5.9%

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

   3. add-sqr-sqrt 9.4%

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

   4. prod-diff 9.4%

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

   5. add-sqr-sqrt 9.4%

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

   6. fmm-def 9.4%

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

   7. *-un-lft-identity 9.4%

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

   8. acos-asin 9.4%

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

   9. add-sqr-sqrt 9.4%

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

4. Applied egg-rr 9.4%

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

   1. acos-asin 9.4%

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

   2. add-cube-cbrt 4.1%

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

   3. fmm-def 4.0%

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

   4. cbrt-unprod 9.4%

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

   5. pow2 9.4%

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

   6. div-inv 9.4%

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

   7. metadata-eval 9.4%

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

   8. div-inv 9.4%

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

   9. metadata-eval 9.4%

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

6. Applied egg-rr 9.4%

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

   1. add-log-exp 9.4%

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

8. Applied egg-rr 9.4%

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

9. Taylor expanded in x around 0 5.9%

   \[\leadsto \color{blue}{-1 \cdot \sin^{-1} \left(1 - x\right) + \pi \cdot \left(\sqrt[3]{0.25} \cdot \sqrt[3]{0.5}\right)} \]
10. Step-by-step derivation

    1. neg-mul-1 5.9%

       \[\leadsto \color{blue}{\left(-\sin^{-1} \left(1 - x\right)\right)} + \pi \cdot \left(\sqrt[3]{0.25} \cdot \sqrt[3]{0.5}\right) \]

    2. +-commutative 5.9%

       \[\leadsto \color{blue}{\pi \cdot \left(\sqrt[3]{0.25} \cdot \sqrt[3]{0.5}\right) + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

    3. unsub-neg 5.9%

       \[\leadsto \color{blue}{\pi \cdot \left(\sqrt[3]{0.25} \cdot \sqrt[3]{0.5}\right) - \sin^{-1} \left(1 - x\right)} \]

    4. associate-*r* 9.4%

       \[\leadsto \color{blue}{\left(\pi \cdot \sqrt[3]{0.25}\right) \cdot \sqrt[3]{0.5}} - \sin^{-1} \left(1 - x\right) \]

    5. *-commutative 9.4%

       \[\leadsto \color{blue}{\sqrt[3]{0.5} \cdot \left(\pi \cdot \sqrt[3]{0.25}\right)} - \sin^{-1} \left(1 - x\right) \]

11. Simplified 9.4%

    \[\leadsto \color{blue}{\sqrt[3]{0.5} \cdot \left(\pi \cdot \sqrt[3]{0.25}\right) - \sin^{-1} \left(1 - x\right)} \]
12. Add Preprocessing

Alternative 8: 9.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 + t\_0\right) + -1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0) (acos (- x)) (+ (+ (+ 2.0 t_0) -1.0) -1.0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = acos(-x);
	} else {
		tmp = ((2.0 + t_0) + -1.0) + -1.0;
	}
	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 (t_0 <= 0.0d0) then
        tmp = acos(-x)
    else
        tmp = ((2.0d0 + t_0) + (-1.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.acos(-x);
	} else {
		tmp = ((2.0 + t_0) + -1.0) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.acos(-x)
	else:
		tmp = ((2.0 + t_0) + -1.0) + -1.0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = acos(Float64(-x));
	else
		tmp = Float64(Float64(Float64(2.0 + t_0) + -1.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = acos(-x);
	else
		tmp = ((2.0 + t_0) + -1.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[ArcCos[(-x)], $MachinePrecision], N[(N[(N[(2.0 + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf 6.5%

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

      1. neg-mul-1 6.5%

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

   5. Simplified 6.5%

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

   if 0.0 < (acos.f64 (-.f64 #s(literal 1 binary64) x))

   1. Initial program 55.4%

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

      1. expm1-log1p-u 55.4%

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

      2. expm1-undefine 55.4%

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

      3. log1p-undefine 55.4%

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

      4. rem-exp-log 55.4%

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

   4. Applied egg-rr 55.4%

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

      1. expm1-log1p-u 55.4%

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

      2. expm1-undefine 55.4%

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

      3. log1p-undefine 55.4%

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

      4. +-commutative 55.4%

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

      5. add-exp-log 55.4%

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

      6. +-commutative 55.4%

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

      7. associate-+l+ 55.4%

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

      8. metadata-eval 55.4%

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

   6. Applied egg-rr 55.4%

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

4. Final simplification 8.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 + \cos^{-1} \left(1 - x\right)\right) + -1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 9.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (acos (- 1.0 x)) 0.0) (acos (- x)) (- (* PI 0.5) (asin (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (acos((1.0 - x)) <= 0.0) {
		tmp = acos(-x);
	} else {
		tmp = (((double) M_PI) * 0.5) - asin((1.0 - x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.acos((1.0 - x)) <= 0.0) {
		tmp = Math.acos(-x);
	} else {
		tmp = (Math.PI * 0.5) - Math.asin((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.acos((1.0 - x)) <= 0.0:
		tmp = math.acos(-x)
	else:
		tmp = (math.pi * 0.5) - math.asin((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (acos(Float64(1.0 - x)) <= 0.0)
		tmp = acos(Float64(-x));
	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 (acos((1.0 - x)) <= 0.0)
		tmp = acos(-x);
	else
		tmp = (pi * 0.5) - asin((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 0.0], N[ArcCos[(-x)], $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 (acos.f64 (-.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf 6.5%

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

      1. neg-mul-1 6.5%

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

   5. Simplified 6.5%

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

   if 0.0 < (acos.f64 (-.f64 #s(literal 1 binary64) x))

   1. Initial program 55.4%

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

      1. acos-asin 55.4%

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

      2. sub-neg 55.4%

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

      3. div-inv 55.4%

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

      4. metadata-eval 55.4%

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

   4. Applied egg-rr 55.4%

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

      1. sub-neg 55.4%

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

   6. Simplified 55.4%

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

Alternative 10: 9.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x)))) (if (<= t_0 0.0) (acos (- x)) t_0)))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = acos(-x);
	} else {
		tmp = 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 (t_0 <= 0.0d0) then
        tmp = acos(-x)
    else
        tmp = 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 (t_0 <= 0.0) {
		tmp = Math.acos(-x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.acos(-x)
	else:
		tmp = t_0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[ArcCos[(-x)], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf 6.5%

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

      1. neg-mul-1 6.5%

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

   5. Simplified 6.5%

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

   if 0.0 < (acos.f64 (-.f64 #s(literal 1 binary64) x))

   1. Initial program 55.4%

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

Alternative 11: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.9%

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

3. Taylor expanded in x around inf 6.7%

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

   1. neg-mul-1 6.7%

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

5. Simplified 6.7%

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

Alternative 12: 3.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return acos(1.0)
end

MATLAB

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

Wolfram

code[x_] := N[ArcCos[1.0], $MachinePrecision]

Derivation

1. Initial program 5.9%

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

3. Taylor expanded in x around 0 3.8%

   \[\leadsto \cos^{-1} \color{blue}{1} \]
4. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024172 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

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

  (acos (- 1.0 x)))