bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.5%

Time: 5.8s

Alternatives: 8

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: 6.9% 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: 10.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := {t\_0}^{2} + 0 \cdot t\_0\\ \frac{\mathsf{fma}\left(-{t\_0}^{3}, \frac{2}{\pi}, t\_1\right)}{\frac{2}{\pi} \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (+ (pow t_0 2.0) (* 0.0 t_0))))
   (/ (fma (- (pow t_0 3.0)) (/ 2.0 PI) t_1) (* (/ 2.0 PI) t_1))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = pow(t_0, 2.0) + (0.0 * t_0);
	return fma(-pow(t_0, 3.0), (2.0 / ((double) M_PI)), t_1) / ((2.0 / ((double) M_PI)) * t_1);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = Float64((t_0 ^ 2.0) + Float64(0.0 * t_0))
	return Float64(fma(Float64(-(t_0 ^ 3.0)), Float64(2.0 / pi), t_1) / Float64(Float64(2.0 / pi) * t_1))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(0.0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(N[((-N[Power[t$95$0, 3.0], $MachinePrecision]) * N[(2.0 / Pi), $MachinePrecision] + t$95$1), $MachinePrecision] / N[(N[(2.0 / Pi), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 8.2%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 7.1

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

5. Applied rewrites 7.1%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 7.1

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

7. Applied rewrites 7.1%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\sin^{-1} \color{blue}{\left(1 + -1 \cdot x\right)}\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. sub-neg N/A

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

   3. lower--.f64 6.3

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

10. Applied rewrites 6.3%

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

    1. lift-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-neg.f64 N/A

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

    4. neg-sub0 N/A

       \[\leadsto \color{blue}{\left(0 - \sin^{-1} \left(1 - x\right)\right)} + \sqrt{\mathsf{PI}\left(\right)} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}\right) \]

    5. flip3-- N/A

       \[\leadsto \color{blue}{\frac{{0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)}} + \sqrt{\mathsf{PI}\left(\right)} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}\right) \]

    6. lift-*.f64 N/A

       \[\leadsto \frac{{0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)} + \sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}\right)} \]

    7. associate-*r* N/A

       \[\leadsto \frac{{0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)} + \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{2}} \]

    8. lift-sqrt.f64 N/A

       \[\leadsto \frac{{0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)} + \left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{2} \]

    9. lift-sqrt.f64 N/A

       \[\leadsto \frac{{0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)} + \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{2} \]

    10. rem-square-sqrt N/A

        \[\leadsto \frac{{0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)} + \color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{2} \]

    11. metadata-eval N/A

        \[\leadsto \frac{{0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)} + \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} \]

    12. div-inv N/A

        \[\leadsto \frac{{0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)} + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} \]

    13. clear-num N/A

        \[\leadsto \frac{{0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)} + \color{blue}{\frac{1}{\frac{2}{\mathsf{PI}\left(\right)}}} \]

12. Applied rewrites 11.9%

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

13. Final simplification 11.9%

    \[\leadsto \frac{\mathsf{fma}\left(-{\sin^{-1} \left(1 - x\right)}^{3}, \frac{2}{\pi}, {\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)}{\frac{2}{\pi} \cdot \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)} \]
14. Add Preprocessing

Alternative 2: 10.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt PI))))
   (fma (* t_0 t_0) (* (sqrt PI) 0.5) (- (asin (- 1.0 x))))))

C

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

Julia

function code(x)
	t_0 = sqrt(sqrt(pi))
	return fma(Float64(t_0 * t_0), Float64(sqrt(pi) * 0.5), Float64(-asin(Float64(1.0 - x))))
end

Wolfram

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

Derivation

1. Initial program 8.2%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 7.1

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

5. Applied rewrites 7.1%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 7.1

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

7. Applied rewrites 7.1%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\sin^{-1} \color{blue}{\left(1 + -1 \cdot x\right)}\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. sub-neg N/A

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

   3. lower--.f64 6.3

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

10. Applied rewrites 6.3%

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

    1. lift-sqrt.f64 N/A

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

    2. pow1/2 N/A

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

    3. sqr-pow N/A

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

    4. lower-*.f64 N/A

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

    5. sqrt-pow1 N/A

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

    6. pow1/2 N/A

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

    7. lift-sqrt.f64 N/A

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

    8. lower-sqrt.f64 N/A

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

    9. sqrt-pow1 N/A

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

    10. pow1/2 N/A

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

    11. lift-sqrt.f64 N/A

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

    12. lower-sqrt.f64 11.8

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

12. Applied rewrites 11.8%

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

Alternative 3: 10.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (sqrt PI) 0.5)))
   (fma (sqrt PI) t_0 (- (fma t_0 (sqrt PI) (- (acos (- 1.0 x))))))))

C

double code(double x) {
	double t_0 = sqrt(((double) M_PI)) * 0.5;
	return fma(sqrt(((double) M_PI)), t_0, -fma(t_0, sqrt(((double) M_PI)), -acos((1.0 - x))));
}

Julia

function code(x)
	t_0 = Float64(sqrt(pi) * 0.5)
	return fma(sqrt(pi), t_0, Float64(-fma(t_0, sqrt(pi), Float64(-acos(Float64(1.0 - x))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[Pi], $MachinePrecision] * 0.5), $MachinePrecision]}, N[(N[Sqrt[Pi], $MachinePrecision] * t$95$0 + (-N[(t$95$0 * N[Sqrt[Pi], $MachinePrecision] + (-N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Initial program 8.2%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 7.1

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

5. Applied rewrites 7.1%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 7.1

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

7. Applied rewrites 7.1%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\sin^{-1} \color{blue}{\left(1 + -1 \cdot x\right)}\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. sub-neg N/A

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

   3. lower--.f64 6.3

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

10. Applied rewrites 6.3%

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

    1. lift-asin.f64 N/A

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

    2. asin-acos N/A

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

    3. lift-PI.f64 N/A

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

    4. div-inv N/A

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

    5. metadata-eval N/A

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

    6. rem-square-sqrt N/A

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

    7. lift-sqrt.f64 N/A

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

    8. lift-sqrt.f64 N/A

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

    9. associate-*r* N/A

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

    10. lift-*.f64 N/A

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

    11. sub-neg N/A

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

    12. *-commutative N/A

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

    13. lower-fma.f64 N/A

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

    14. lift-*.f64 N/A

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

    15. *-commutative N/A

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

    16. lower-*.f64 N/A

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

    17. lower-neg.f64 N/A

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

    18. lower-acos.f64 11.8

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

12. Applied rewrites 11.8%

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

13. Final simplification 11.8%

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

Alternative 4: 10.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 7.1

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

5. Applied rewrites 7.1%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 7.1

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

7. Applied rewrites 7.1%

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

8. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-asin.f64 N/A

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

   7. mul-1-neg N/A

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

   8. sub-neg N/A

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

   9. lower--.f64 8.2

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

10. Applied rewrites 8.2%

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

12. Applied rewrites 11.7%

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

13. Final simplification 11.7%

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

Alternative 5: 9.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], 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 x < 5.50000000000000001e-17

   1. Initial program 3.8%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.7

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

   5. Applied rewrites 6.7%

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

   7. Applied rewrites 6.7%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

   if 5.50000000000000001e-17 < x

   1. Initial program 77.4%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 14.3

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

   5. Applied rewrites 14.3%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. add-sqr-sqrt N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-asin.f64 14.3

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

   7. Applied rewrites 14.4%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-asin.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 77.7

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

   10. Applied rewrites 77.7%

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

4. Final simplification 10.8%

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

Alternative 6: 9.5% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (if (<= x 5.5e-17) (acos x) (acos (- 1.0 x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.8%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.7

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

   5. Applied rewrites 6.7%

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

   7. Applied rewrites 6.7%

      \[\leadsto \color{blue}{\cos^{-1} x} \]

   if 5.50000000000000001e-17 < x

   1. Initial program 77.4%

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

Alternative 7: 6.9% accurate, 1.0× speedup

Math

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

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(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 7.1

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

5. Applied rewrites 7.1%

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

7. Applied rewrites 7.1%

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

Alternative 8: 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 8.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \cos^{-1} \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 3.8%

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

Developer Target 1: 100.0% accurate, 0.8× 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 2024220 
(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)))