bug323 (missed optimization)

Percentage Accurate: 6.7% → 10.3%

Time: 7.8s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathsf{fma}\left({\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right)}^{0.75}, 0.5 \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}}, \frac{\frac{-2}{\mathsf{PI}\left(\right)}}{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, t\_0, 1\right)} \cdot \left(\sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_0\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (fma
    (pow (pow (cbrt (PI)) 2.0) 0.75)
    (* 0.5 (cbrt (pow (PI) 1.5)))
    (*
     (/ (/ (- 2.0) (PI)) (fma (/ 2.0 (PI)) t_0 1.0))
     (* (asin (- 1.0 x)) (fma 0.5 (PI) t_0))))))

Derivation

1. Initial program 7.0%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 5.2

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

4. Applied rewrites 5.2%

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

6. Applied rewrites 5.1%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, {\mathsf{PI}\left(\right)}^{\color{blue}{\left(\frac{1}{6} + \frac{1}{3}\right)}} \cdot \frac{1}{2}, -\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \sin^{-1} \left(1 - x\right)\right) \cdot \frac{\frac{2}{\mathsf{PI}\left(\right)}}{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, \cos^{-1} \left(1 - x\right), 1\right)}\right) \]

   4. metadata-eval N/A

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

   5. pow-prod-up N/A

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

   6. pow-pow N/A

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

   7. pow1/2 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/3 N/A

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

   10. pow1/3 N/A

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

   11. cbrt-unprod N/A

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

   12. lower-cbrt.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. pow1/2 N/A

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

   15. pow-plus N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-pow.f64 N/A

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

   19. metadata-eval 10.2

       \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt[3]{{\mathsf{PI}\left(\right)}^{\color{blue}{1.5}}} \cdot 0.5, -\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \sin^{-1} \left(1 - x\right)\right) \cdot \frac{\frac{2}{\mathsf{PI}\left(\right)}}{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, \cos^{-1} \left(1 - x\right), 1\right)}\right) \]

8. Applied rewrites 10.2%

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}}} \cdot 0.5, -\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \sin^{-1} \left(1 - x\right)\right) \cdot \frac{\frac{2}{\mathsf{PI}\left(\right)}}{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, \cos^{-1} \left(1 - x\right), 1\right)}\right) \]
9. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. pow1/3 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\color{blue}{\left(2 \cdot \frac{3}{4}\right)}}, \sqrt[3]{{\mathsf{PI}\left(\right)}^{\frac{3}{2}}} \cdot \frac{1}{2}, -\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \sin^{-1} \left(1 - x\right)\right) \cdot \frac{\frac{2}{\mathsf{PI}\left(\right)}}{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, \cos^{-1} \left(1 - x\right), 1\right)}\right) \]

   8. metadata-eval N/A

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

   9. pow-pow N/A

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

   10. lift-pow.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval 10.2

       \[\leadsto \mathsf{fma}\left({\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right)}^{\color{blue}{0.75}}, \sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}} \cdot 0.5, -\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \sin^{-1} \left(1 - x\right)\right) \cdot \frac{\frac{2}{\mathsf{PI}\left(\right)}}{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, \cos^{-1} \left(1 - x\right), 1\right)}\right) \]

10. Applied rewrites 10.2%

    \[\leadsto \mathsf{fma}\left(\color{blue}{{\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right)}^{0.75}}, \sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}} \cdot 0.5, -\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \sin^{-1} \left(1 - x\right)\right) \cdot \frac{\frac{2}{\mathsf{PI}\left(\right)}}{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, \cos^{-1} \left(1 - x\right), 1\right)}\right) \]

11. Final simplification 10.2%

    \[\leadsto \mathsf{fma}\left({\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right)}^{0.75}, 0.5 \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}}, \frac{\frac{-2}{\mathsf{PI}\left(\right)}}{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, \cos^{-1} \left(1 - x\right), 1\right)} \cdot \left(\sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 2: 10.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, 0.5 \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}}, \frac{\frac{-2}{\mathsf{PI}\left(\right)}}{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, t\_0, 1\right)} \cdot \left(\sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_0\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (fma
    (sqrt (PI))
    (* 0.5 (cbrt (pow (PI) 1.5)))
    (*
     (/ (/ (- 2.0) (PI)) (fma (/ 2.0 (PI)) t_0 1.0))
     (* (asin (- 1.0 x)) (fma 0.5 (PI) t_0))))))

Derivation

1. Initial program 7.0%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 5.2

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

4. Applied rewrites 5.2%

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

6. Applied rewrites 5.1%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, {\mathsf{PI}\left(\right)}^{\color{blue}{\left(\frac{1}{6} + \frac{1}{3}\right)}} \cdot \frac{1}{2}, -\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \sin^{-1} \left(1 - x\right)\right) \cdot \frac{\frac{2}{\mathsf{PI}\left(\right)}}{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, \cos^{-1} \left(1 - x\right), 1\right)}\right) \]

   4. metadata-eval N/A

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

   5. pow-prod-up N/A

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

   6. pow-pow N/A

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

   7. pow1/2 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/3 N/A

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

   10. pow1/3 N/A

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

   11. cbrt-unprod N/A

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

   12. lower-cbrt.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. pow1/2 N/A

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

   15. pow-plus N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-pow.f64 N/A

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

   19. metadata-eval 10.2

       \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt[3]{{\mathsf{PI}\left(\right)}^{\color{blue}{1.5}}} \cdot 0.5, -\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \sin^{-1} \left(1 - x\right)\right) \cdot \frac{\frac{2}{\mathsf{PI}\left(\right)}}{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, \cos^{-1} \left(1 - x\right), 1\right)}\right) \]

8. Applied rewrites 10.2%

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}}} \cdot 0.5, -\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \sin^{-1} \left(1 - x\right)\right) \cdot \frac{\frac{2}{\mathsf{PI}\left(\right)}}{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, \cos^{-1} \left(1 - x\right), 1\right)}\right) \]

9. Final simplification 10.2%

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, 0.5 \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{1.5}}, \frac{\frac{-2}{\mathsf{PI}\left(\right)}}{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, \cos^{-1} \left(1 - x\right), 1\right)} \cdot \left(\sin^{-1} \left(1 - x\right) \cdot \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 3: 10.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}^{0.25}, \sqrt{\mathsf{PI}\left(\right)} \cdot 0.5, -\sin^{-1} \left(1 - x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (pow (* (PI) (PI)) 0.25) (* (sqrt (PI)) 0.5) (- (asin (- 1.0 x)))))

Derivation

1. Initial program 7.0%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 5.2

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

4. Applied rewrites 5.2%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({\mathsf{PI}\left(\right)}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, -\sin^{-1} \left(1 - x\right)\right) \]

   4. pow-prod-up N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{4}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{4}}}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, -\sin^{-1} \left(1 - x\right)\right) \]

   5. pow-prod-down N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}^{\frac{1}{4}}}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, -\sin^{-1} \left(1 - x\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left({\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}^{\frac{1}{4}}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, -\sin^{-1} \left(1 - x\right)\right) \]

   7. lower-pow.f64 10.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}^{0.25}}, \sqrt{\mathsf{PI}\left(\right)} \cdot 0.5, -\sin^{-1} \left(1 - x\right)\right) \]

6. Applied rewrites 10.2%

   \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}^{0.25}}, \sqrt{\mathsf{PI}\left(\right)} \cdot 0.5, -\sin^{-1} \left(1 - x\right)\right) \]
7. Add Preprocessing

Alternative 4: 10.2% accurate, 0.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.0%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-asin.f64 7.0

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

4. Applied rewrites 7.0%

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

6. Applied rewrites 10.2%

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

7. Final simplification 10.2%

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

Alternative 5: 10.2% accurate, 0.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.0%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-asin.f64 7.0

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

4. Applied rewrites 7.0%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. lift-asin.f64 N/A

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

   4. asin-acos N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, 0 - \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(1 - x\right)\right)}\right) \]

   5. lift-acos.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, 0 - \left(\frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(1 - x\right)}\right)\right) \]

6. Applied rewrites 10.1%

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

7. Final simplification 10.1%

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

Alternative 6: 10.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 0.9999999999985981:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -\sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), -\sin^{-1} 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 0.9999999999985981)
   (fma (PI) 0.5 (- (asin (- 1.0 x))))
   (fma (/ 2.0 (PI)) (* 0.25 (* (PI) (PI))) (- (asin 1.0)))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 0.999999999998598121

   1. Initial program 69.6%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-asin.f64 69.7

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

   4. Applied rewrites 69.7%

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

   if 0.999999999998598121 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 3.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.9%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. sub-neg N/A

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

   7. Applied rewrites 7.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25, -\sin^{-1} 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 0.9999999999985981:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -\sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), -\sin^{-1} 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 10.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), -\sin^{-1} \left(1 - x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (/ 2.0 (PI)) (* 0.25 (* (PI) (PI))) (- (asin (- 1.0 x)))))

Derivation

1. Initial program 7.0%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 5.2

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

4. Applied rewrites 5.2%

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

   1. lift-fma.f64 N/A

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

6. Applied rewrites 10.1%

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

7. Final simplification 10.1%

   \[\leadsto \mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), -\sin^{-1} \left(1 - x\right)\right) \]
8. Add Preprocessing

Alternative 8: 9.3% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. 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 6.4

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

   5. Applied rewrites 6.4%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 69.6%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-asin.f64 69.7

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

   4. Applied rewrites 69.7%

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

Alternative 9: 9.3% accurate, 0.9× speedup

Math

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

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(Float64(-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 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   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 6.4

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

   5. Applied rewrites 6.4%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 69.6%

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

Alternative 10: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

Alternative 11: 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 7.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 3.9%

   \[\leadsto \cos^{-1} \color{blue}{1} \]
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 2024283 
(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)))