bug323 (missed optimization)

Percentage Accurate: 6.8% → 10.4%

Time: 12.6s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := {t_0}^{2}\\ \frac{\left(0.25 \cdot {\pi}^{2} - t_1\right) + \mathsf{fma}\left(-t_0, t_0, t_1\right)}{t_0 + \pi \cdot 0.5} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (pow t_0 2.0)))
   (/
    (+ (- (* 0.25 (pow PI 2.0)) t_1) (fma (- t_0) t_0 t_1))
    (+ t_0 (* PI 0.5)))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = pow(t_0, 2.0);
	return (((0.25 * pow(((double) M_PI), 2.0)) - t_1) + fma(-t_0, t_0, t_1)) / (t_0 + (((double) M_PI) * 0.5));
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = t_0 ^ 2.0
	return Float64(Float64(Float64(Float64(0.25 * (pi ^ 2.0)) - t_1) + fma(Float64(-t_0), t_0, t_1)) / Float64(t_0 + Float64(pi * 0.5)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, N[(N[(N[(N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] + N[((-t$95$0) * t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 8.6%

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

   1. acos-asin 8.6%

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

   2. flip-- 8.6%

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

   3. div-inv 8.6%

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

   4. metadata-eval 8.6%

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

   5. div-inv 8.6%

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

   6. metadata-eval 8.6%

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

   7. div-inv 8.6%

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

   8. metadata-eval 8.6%

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

3. Applied egg-rr 8.6%

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

   1. expm1-log1p-u 8.6%

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

   2. expm1-udef 12.2%

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

   3. pow2 12.2%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \left(e^{\mathsf{log1p}\left(\color{blue}{{\sin^{-1} \left(1 - x\right)}^{2}}\right)} - 1\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

5. Applied egg-rr 12.2%

   \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{\left(e^{\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{2}\right)} - 1\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
6. Step-by-step derivation

   1. expm1-def 8.6%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{2}\right)\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   2. expm1-log1p-u 8.6%

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

   3. unpow2 8.6%

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

   4. prod-diff 8.6%

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

   5. unpow2 8.6%

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

   6. fma-neg 12.3%

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

   7. pow2 12.3%

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

   8. *-commutative 12.3%

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

   9. unpow-prod-down 12.3%

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

   10. metadata-eval 12.3%

       \[\leadsto \frac{\left(\color{blue}{0.25} \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right) + \mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   11. unpow2 12.3%

       \[\leadsto \frac{\left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right) + \mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), \color{blue}{{\sin^{-1} \left(1 - x\right)}^{2}}\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

7. Applied egg-rr 12.3%

   \[\leadsto \frac{\color{blue}{\left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right) + \mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

8. Final simplification 12.3%

   \[\leadsto \frac{\left(0.25 \cdot {\pi}^{2} - {\sin^{-1} \left(1 - x\right)}^{2}\right) + \mathsf{fma}\left(-\sin^{-1} \left(1 - x\right), \sin^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right)}{\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5} \]

Alternative 2: 10.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

   1. add-sqr-sqrt 8.6%

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

   2. pow2 8.6%

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

3. Applied egg-rr 8.6%

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

   1. unpow2 8.6%

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

   2. add-sqr-sqrt 8.6%

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

   3. acos-asin 8.6%

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

   4. div-inv 8.6%

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

   5. metadata-eval 8.6%

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

   6. add-sqr-sqrt 12.2%

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

   7. prod-diff 12.2%

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

   8. add-sqr-sqrt 12.3%

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

   9. fma-neg 12.3%

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

   10. metadata-eval 12.3%

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

   11. div-inv 12.3%

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

   12. acos-asin 12.3%

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

   13. add-sqr-sqrt 12.2%

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

5. Applied egg-rr 12.2%

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

   1. add-sqr-sqrt 12.2%

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

   2. pow2 12.2%

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

7. Applied egg-rr 12.3%

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

8. Final simplification 12.3%

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

Alternative 3: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \left(e^{\mathsf{log1p}\left({t_0}^{2}\right)} + -1\right)}{t_0 + \pi \cdot 0.5} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (- (* (* PI 0.5) (* PI 0.5)) (+ (exp (log1p (pow t_0 2.0))) -1.0))
    (+ t_0 (* PI 0.5)))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return (((((double) M_PI) * 0.5) * (((double) M_PI) * 0.5)) - (exp(log1p(pow(t_0, 2.0))) + -1.0)) / (t_0 + (((double) M_PI) * 0.5));
}

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	return (((Math.PI * 0.5) * (Math.PI * 0.5)) - (Math.exp(Math.log1p(Math.pow(t_0, 2.0))) + -1.0)) / (t_0 + (Math.PI * 0.5));
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	return (((math.pi * 0.5) * (math.pi * 0.5)) - (math.exp(math.log1p(math.pow(t_0, 2.0))) + -1.0)) / (t_0 + (math.pi * 0.5))

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(Float64(Float64(Float64(pi * 0.5) * Float64(pi * 0.5)) - Float64(exp(log1p((t_0 ^ 2.0))) + -1.0)) / Float64(t_0 + Float64(pi * 0.5)))
end

Wolfram

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

Derivation

1. Initial program 8.6%

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

   1. acos-asin 8.6%

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

   2. flip-- 8.6%

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

   3. div-inv 8.6%

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

   4. metadata-eval 8.6%

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

   5. div-inv 8.6%

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

   6. metadata-eval 8.6%

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

   7. div-inv 8.6%

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

   8. metadata-eval 8.6%

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

3. Applied egg-rr 8.6%

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

   1. expm1-log1p-u 8.6%

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

   2. expm1-udef 12.2%

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

   3. pow2 12.2%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \left(e^{\mathsf{log1p}\left(\color{blue}{{\sin^{-1} \left(1 - x\right)}^{2}}\right)} - 1\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

5. Applied egg-rr 12.2%

   \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{\left(e^{\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{2}\right)} - 1\right)}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

6. Final simplification 12.2%

   \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \left(e^{\mathsf{log1p}\left({\sin^{-1} \left(1 - x\right)}^{2}\right)} + -1\right)}{\sin^{-1} \left(1 - x\right) + \pi \cdot 0.5} \]

Alternative 4: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{t_0}\\ \left(\left(1 + \cos^{-1} \left(1 - x\right)\right) + -1\right) + \mathsf{fma}\left(-t_1, t_1, t_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (sqrt t_0)))
   (+ (+ (+ 1.0 (acos (- 1.0 x))) -1.0) (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 ((1.0 + acos((1.0 - x))) + -1.0) + 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(Float64(Float64(1.0 + acos(Float64(1.0 - x))) + -1.0) + 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[(1.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[((-t$95$1) * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 8.6%

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

   1. add-sqr-sqrt 8.6%

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

   2. pow2 8.6%

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

3. Applied egg-rr 8.6%

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

   1. unpow2 8.6%

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

   2. add-sqr-sqrt 8.6%

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

   3. acos-asin 8.6%

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

   4. div-inv 8.6%

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

   5. metadata-eval 8.6%

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

   6. add-sqr-sqrt 12.2%

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

   7. prod-diff 12.2%

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

   8. add-sqr-sqrt 12.3%

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

   9. fma-neg 12.3%

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

   10. metadata-eval 12.3%

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

   11. div-inv 12.3%

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

   12. acos-asin 12.3%

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

   13. add-sqr-sqrt 12.2%

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

5. Applied egg-rr 12.2%

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

   1. expm1-log1p-u 12.2%

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

   2. expm1-udef 12.2%

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

   3. log1p-udef 12.2%

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

   4. add-exp-log 12.2%

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

7. Applied egg-rr 12.2%

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

8. Final simplification 12.2%

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

Alternative 5: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{t_0}\\ \cos^{-1} \left(1 - x\right) + \mathsf{fma}\left(-t_1, t_1, t_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (sqrt t_0)))
   (+ (acos (- 1.0 x)) (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 acos((1.0 - x)) + 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(acos(Float64(1.0 - x)) + 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[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[((-t$95$1) * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 8.6%

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

   1. add-sqr-sqrt 8.6%

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

   2. pow2 8.6%

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

3. Applied egg-rr 8.6%

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

   1. unpow2 8.6%

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

   2. add-sqr-sqrt 8.6%

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

   3. acos-asin 8.6%

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

   4. div-inv 8.6%

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

   5. metadata-eval 8.6%

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

   6. add-sqr-sqrt 12.2%

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

   7. prod-diff 12.2%

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

   8. add-sqr-sqrt 12.3%

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

   9. fma-neg 12.3%

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

   10. metadata-eval 12.3%

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

   11. div-inv 12.3%

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

   12. acos-asin 12.3%

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

   13. add-sqr-sqrt 12.2%

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

5. Applied egg-rr 12.2%

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

6. Final simplification 12.2%

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

Alternative 6: 10.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sqrt{t_0}\\ \cos^{-1} \left(1 - x\right) + \left(t_0 - t_1 \cdot t_1\right) \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = sqrt(t_0);
	return acos((1.0 - x)) + (t_0 - (t_1 * t_1));
}

Fortran

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

Java

public static double code(double x) {
	double t_0 = Math.asin((1.0 - x));
	double t_1 = Math.sqrt(t_0);
	return Math.acos((1.0 - x)) + (t_0 - (t_1 * t_1));
}

Python

def code(x):
	t_0 = math.asin((1.0 - x))
	t_1 = math.sqrt(t_0)
	return math.acos((1.0 - x)) + (t_0 - (t_1 * t_1))

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = sqrt(t_0)
	return Float64(acos(Float64(1.0 - x)) + Float64(t_0 - Float64(t_1 * t_1)))
end

MATLAB

function tmp = code(x)
	t_0 = asin((1.0 - x));
	t_1 = sqrt(t_0);
	tmp = acos((1.0 - x)) + (t_0 - (t_1 * t_1));
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[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[(t$95$0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 8.6%

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

   1. add-sqr-sqrt 8.6%

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

   2. pow2 8.6%

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

3. Applied egg-rr 8.6%

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

   1. unpow2 8.6%

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

   2. add-sqr-sqrt 8.6%

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

   3. acos-asin 8.6%

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

   4. div-inv 8.6%

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

   5. metadata-eval 8.6%

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

   6. add-sqr-sqrt 12.2%

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

   7. prod-diff 12.2%

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

   8. add-sqr-sqrt 12.3%

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

   9. fma-neg 12.3%

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

   10. metadata-eval 12.3%

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

   11. div-inv 12.3%

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

   12. acos-asin 12.3%

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

   13. add-sqr-sqrt 12.2%

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

5. Applied egg-rr 12.2%

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

   1. fma-udef 12.2%

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

7. Applied egg-rr 12.2%

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

8. Final simplification 12.2%

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

Alternative 7: 10.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

   1. add-exp-log 8.6%

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

3. Applied egg-rr 8.6%

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

   1. acos-asin 8.6%

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

   2. sub-neg 8.6%

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

   3. div-inv 8.6%

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

   4. metadata-eval 8.6%

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

5. Applied egg-rr 8.6%

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

   1. sub-neg 8.6%

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

7. Simplified 8.6%

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

   1. add-cube-cbrt 12.2%

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

   2. pow3 12.2%

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

9. Applied egg-rr 12.2%

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

10. Final simplification 12.2%

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

Alternative 8: 10.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

   1. add-exp-log 8.6%

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

3. Applied egg-rr 8.6%

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

   1. acos-asin 8.6%

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

   2. sub-neg 8.6%

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

   3. div-inv 8.6%

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

   4. metadata-eval 8.6%

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

5. Applied egg-rr 8.6%

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

   1. sub-neg 8.6%

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

7. Simplified 8.6%

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

   1. add-sqr-sqrt 12.2%

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

   2. pow2 12.2%

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

9. Applied egg-rr 12.2%

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

10. Final simplification 12.2%

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

Alternative 9: 9.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := 1 - t_0\\ t_2 := t_0 + -1\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;1 + \sqrt{{t_2}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_2 \cdot t_1}{1 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))) (t_1 (- 1.0 t_0)) (t_2 (+ t_0 -1.0)))
   (if (<= t_0 0.0)
     (+ 1.0 (sqrt (pow t_2 2.0)))
     (/ (+ 1.0 (* t_2 t_1)) (+ 1.0 t_1)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double t_1 = 1.0 - t_0;
	double t_2 = t_0 + -1.0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + sqrt(pow(t_2, 2.0));
	} else {
		tmp = (1.0 + (t_2 * t_1)) / (1.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = acos((1.0d0 - x))
    t_1 = 1.0d0 - t_0
    t_2 = t_0 + (-1.0d0)
    if (t_0 <= 0.0d0) then
        tmp = 1.0d0 + sqrt((t_2 ** 2.0d0))
    else
        tmp = (1.0d0 + (t_2 * t_1)) / (1.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double t_1 = 1.0 - t_0;
	double t_2 = t_0 + -1.0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + Math.sqrt(Math.pow(t_2, 2.0));
	} else {
		tmp = (1.0 + (t_2 * t_1)) / (1.0 + t_1);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	t_1 = 1.0 - t_0
	t_2 = t_0 + -1.0
	tmp = 0
	if t_0 <= 0.0:
		tmp = 1.0 + math.sqrt(math.pow(t_2, 2.0))
	else:
		tmp = (1.0 + (t_2 * t_1)) / (1.0 + t_1)
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	t_1 = Float64(1.0 - t_0)
	t_2 = Float64(t_0 + -1.0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 + sqrt((t_2 ^ 2.0)));
	else
		tmp = Float64(Float64(1.0 + Float64(t_2 * t_1)) / Float64(1.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	t_1 = 1.0 - t_0;
	t_2 = t_0 + -1.0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = 1.0 + sqrt((t_2 ^ 2.0));
	else
		tmp = (1.0 + (t_2 * t_1)) / (1.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + -1.0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(1.0 + N[Sqrt[N[Power[t$95$2, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.8%

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

      1. add-sqr-sqrt 3.8%

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

      2. pow2 3.8%

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

   3. Applied egg-rr 3.8%

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

      1. unpow2 3.8%

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

      2. add-sqr-sqrt 3.8%

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

      3. expm1-log1p-u 3.8%

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

      4. expm1-udef 3.8%

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

      5. log1p-udef 3.8%

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

      6. add-exp-log 3.8%

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

      7. associate--l+ 3.8%

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

      8. +-commutative 3.8%

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

      9. sub-neg 3.8%

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

      10. metadata-eval 3.8%

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

   5. Applied egg-rr 3.8%

      \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]
   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 6.8%

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

      3. pow2 6.8%

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

   7. Applied egg-rr 6.8%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 65.8%

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

      1. add-sqr-sqrt 65.9%

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

      2. pow2 65.9%

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

   3. Applied egg-rr 65.9%

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

      1. unpow2 65.9%

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

      2. add-sqr-sqrt 65.8%

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

      3. expm1-log1p-u 65.8%

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

      4. expm1-udef 65.8%

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

      5. log1p-udef 65.8%

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

      6. add-exp-log 65.8%

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

      7. associate--l+ 65.9%

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

      8. flip-+ 65.9%

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

      9. metadata-eval 65.9%

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

      10. sub-neg 65.9%

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

      11. metadata-eval 65.9%

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

      12. sub-neg 65.9%

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

      13. metadata-eval 65.9%

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

      14. sub-neg 65.9%

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

      15. metadata-eval 65.9%

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

   5. Applied egg-rr 65.9%

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

4. Final simplification 11.4%

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

Alternative 10: 9.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + sqrt(pow((t_0 + -1.0), 2.0));
	} else {
		tmp = pow(sqrt(t_0), 2.0);
	}
	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 = 1.0d0 + sqrt(((t_0 + (-1.0d0)) ** 2.0d0))
    else
        tmp = sqrt(t_0) ** 2.0d0
    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 = 1.0 + Math.sqrt(Math.pow((t_0 + -1.0), 2.0));
	} else {
		tmp = Math.pow(Math.sqrt(t_0), 2.0);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 + sqrt((Float64(t_0 + -1.0) ^ 2.0)));
	else
		tmp = sqrt(t_0) ^ 2.0;
	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 = 1.0 + sqrt(((t_0 + -1.0) ^ 2.0));
	else
		tmp = sqrt(t_0) ^ 2.0;
	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[(1.0 + N[Sqrt[N[Power[N[(t$95$0 + -1.0), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Sqrt[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.8%

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

      1. add-sqr-sqrt 3.8%

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

      2. pow2 3.8%

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

   3. Applied egg-rr 3.8%

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

      1. unpow2 3.8%

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

      2. add-sqr-sqrt 3.8%

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

      3. expm1-log1p-u 3.8%

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

      4. expm1-udef 3.8%

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

      5. log1p-udef 3.8%

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

      6. add-exp-log 3.8%

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

      7. associate--l+ 3.8%

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

      8. +-commutative 3.8%

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

      9. sub-neg 3.8%

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

      10. metadata-eval 3.8%

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

   5. Applied egg-rr 3.8%

      \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]
   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 6.8%

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

      3. pow2 6.8%

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

   7. Applied egg-rr 6.8%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 65.8%

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

      1. add-sqr-sqrt 65.9%

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

      2. pow2 65.9%

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

   3. Applied egg-rr 65.9%

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

4. Final simplification 11.4%

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

Alternative 11: 6.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

   1. add-sqr-sqrt 8.6%

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

   2. pow2 8.6%

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

3. Applied egg-rr 8.6%

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

4. Final simplification 8.6%

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

Alternative 12: 6.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

   1. add-log-exp 8.6%

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

3. Applied egg-rr 8.6%

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

4. Final simplification 8.6%

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

Alternative 13: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

   1. add-sqr-sqrt 8.6%

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

   2. pow2 8.6%

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

3. Applied egg-rr 8.6%

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

   1. unpow2 8.6%

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

   2. add-sqr-sqrt 8.6%

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

   3. expm1-log1p-u 8.6%

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

   4. expm1-udef 8.6%

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

   5. log1p-udef 8.6%

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

   6. add-exp-log 8.6%

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

   7. associate--l+ 8.6%

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

   8. +-commutative 8.6%

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

   9. sub-neg 8.6%

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

   10. metadata-eval 8.6%

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

5. Applied egg-rr 8.6%

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

6. Final simplification 8.6%

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

Alternative 14: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

2. Final simplification 8.6%

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

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 2023240 
(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)))