bug323 (missed optimization)

Percentage Accurate: 6.8% → 10.4%

Time: 7.4s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ t_1 := 0.5 \cdot t\_0\\ \mathsf{fma}\left(\sqrt{t\_1} \cdot \sqrt{t\_0 \cdot 2}, t\_1, -\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -\cos^{-1} \left(1 - x\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))) (t_1 (* 0.5 t_0)))
   (fma
    (* (sqrt t_1) (sqrt (* t_0 2.0)))
    t_1
    (- (fma (PI) 0.5 (- (acos (- 1.0 x))))))))

Derivation

1. Initial program 6.7%

   \[\cos^{-1} \left(1 - x\right) \]
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 4.8

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

4. Applied rewrites 4.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot 0.5, -\sin^{-1} \left(1 - x\right)\right)} \]
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. *-lft-identity N/A

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

   3. metadata-eval N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. sqrt-prod N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 9.9

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 9.9

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

6. Applied rewrites 9.9%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lift-acos.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 9.9

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

8. Applied rewrites 9.9%

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

9. Final simplification 9.9%

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

Alternative 2: 9.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.5 \cdot t\_0, -\sin^{-1} \left(1 - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (if (<= (acos (- 1.0 x)) 0.0)
     (acos (- x))
     (fma t_0 (* 0.5 t_0) (- (asin (- 1.0 x)))))))

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \cos^{-1} \color{blue}{\left(-1 \cdot x\right)} \]
   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 0.0 < (acos.f64 (-.f64 #s(literal 1 binary64) x))

   1. Initial program 63.0%

      \[\cos^{-1} \left(1 - x\right) \]
   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 63.1

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

   4. Applied rewrites 63.1%

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

4. Final simplification 9.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, 0.5 \cdot \sqrt{\mathsf{PI}\left(\right)}, -\sin^{-1} \left(1 - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 9.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, \mathsf{fma}\left(\mathsf{PI}\left(\right), -0.5, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0) (acos (- x)) (fma (PI) 0.5 (fma (PI) -0.5 t_0)))))

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \cos^{-1} \color{blue}{\left(-1 \cdot x\right)} \]
   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 0.0 < (acos.f64 (-.f64 #s(literal 1 binary64) x))

   1. Initial program 63.0%

      \[\cos^{-1} \left(1 - x\right) \]
   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 63.0

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

   4. Applied rewrites 63.0%

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

      1. lift-asin.f64 N/A

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

      2. asin-acos N/A

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

      3. lift-PI.f64 N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lift-acos.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 63.0

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

   6. Applied rewrites 63.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-neg-in N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lift-neg.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 63.0

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

   8. Applied rewrites 63.0%

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

Alternative 4: 10.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ t_1 := 0.5 \cdot t\_0\\ \mathsf{fma}\left(\sqrt{t\_1} \cdot \sqrt{t\_0 \cdot 2}, t\_1, -\sin^{-1} \left(1 - x\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))) (t_1 (* 0.5 t_0)))
   (fma (* (sqrt t_1) (sqrt (* t_0 2.0))) t_1 (- (asin (- 1.0 x))))))

Derivation

1. Initial program 6.7%

   \[\cos^{-1} \left(1 - x\right) \]
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 4.8

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

4. Applied rewrites 4.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot 0.5, -\sin^{-1} \left(1 - x\right)\right)} \]
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. *-lft-identity N/A

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

   3. metadata-eval N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. sqrt-prod N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 9.9

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 9.9

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

6. Applied rewrites 9.9%

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

7. Final simplification 9.9%

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

Alternative 5: 10.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))) (t_1 (* 0.5 t_0)))
   (fma t_0 t_1 (- (fma t_1 t_0 (- (acos (- 1.0 x))))))))

Derivation

1. Initial program 6.7%

   \[\cos^{-1} \left(1 - x\right) \]
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 4.8

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

4. Applied rewrites 4.8%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lift-acos.f64 N/A

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

   8. sub-neg N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower-neg.f64 9.9

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

6. Applied rewrites 9.9%

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

7. Final simplification 9.9%

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

Alternative 6: 9.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;1 - x \leq 0.9999999999999998:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot t\_0, 0.5, -\sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (if (<= (- 1.0 x) 0.9999999999999998)
     (fma (* t_0 t_0) 0.5 (- (asin (- 1.0 x))))
     (acos (- x)))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 0.99999999999999978

   1. Initial program 63.0%

      \[\cos^{-1} \left(1 - x\right) \]
   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 63.0

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

   4. Applied rewrites 63.0%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lower-*.f64 63.3

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

   6. Applied rewrites 63.3%

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

   if 0.99999999999999978 < (-.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 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%

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

Alternative 7: 10.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fma (* (sqrt 0.5) 0.5) (* (sqrt 2.0) (PI)) (- (asin (- 1.0 x)))))

Derivation

1. Initial program 6.7%

   \[\cos^{-1} \left(1 - x\right) \]
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 4.8

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

4. Applied rewrites 4.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot 0.5, -\sin^{-1} \left(1 - x\right)\right)} \]
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. *-lft-identity N/A

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

   3. metadata-eval N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. sqrt-prod N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 9.9

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 9.9

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

6. Applied rewrites 9.9%

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

7. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-asin.f64 N/A

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

   16. mul-1-neg N/A

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

   17. sub-neg N/A

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

   18. lower--.f64 9.9

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

9. Applied rewrites 9.9%

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

Alternative 8: 10.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fma (fma (sqrt 2.0) (sqrt 0.5) -1.0) (* 0.5 (PI)) (acos (- 1.0 x))))

Derivation

1. Initial program 6.7%

   \[\cos^{-1} \left(1 - x\right) \]
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 4.8

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

4. Applied rewrites 4.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot 0.5, -\sin^{-1} \left(1 - x\right)\right)} \]
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. *-lft-identity N/A

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

   3. metadata-eval N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. sqrt-prod N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 9.9

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 9.9

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

6. Applied rewrites 9.9%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lift-acos.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 9.9

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

8. Applied rewrites 9.9%

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

9. Taylor expanded in x around 0

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

    1. associate--l+ N/A

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

    2. sub-neg N/A

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

    3. mul-1-neg N/A

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

    4. +-commutative N/A

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

    5. sub-neg N/A

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

    6. associate-*r* N/A

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

    7. neg-mul-1 N/A

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

    8. *-commutative N/A

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

    9. distribute-lft-in N/A

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

    10. *-commutative N/A

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

    11. lower-fma.f64 N/A

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

11. Applied rewrites 9.8%

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

12. Final simplification 9.8%

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

Alternative 9: 9.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 0.9999999999999998:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -\sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 0.9999999999999998)
   (fma (PI) 0.5 (- (asin (- 1.0 x))))
   (acos (- x))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 0.99999999999999978

   1. Initial program 63.0%

      \[\cos^{-1} \left(1 - x\right) \]
   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 63.0

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

   4. Applied rewrites 63.0%

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

   if 0.99999999999999978 < (-.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 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%

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

Alternative 10: 9.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 0.9999999999999998:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 0.9999999999999998) (acos (- 1.0 x)) (acos (- x))))

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 0.9999999999999998) {
		tmp = acos((1.0 - x));
	} else {
		tmp = acos(-x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((1.0d0 - x) <= 0.9999999999999998d0) then
        tmp = acos((1.0d0 - x))
    else
        tmp = acos(-x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - x) <= 0.9999999999999998) {
		tmp = Math.acos((1.0 - x));
	} else {
		tmp = Math.acos(-x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - x) <= 0.9999999999999998:
		tmp = math.acos((1.0 - x))
	else:
		tmp = math.acos(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - x) <= 0.9999999999999998)
		tmp = acos(Float64(1.0 - x));
	else
		tmp = acos(Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - x) <= 0.9999999999999998)
		tmp = acos((1.0 - x));
	else
		tmp = acos(-x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - x), $MachinePrecision], 0.9999999999999998], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], N[ArcCos[(-x)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 0.99999999999999978

   1. Initial program 63.0%

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

   if 0.99999999999999978 < (-.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 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%

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

Alternative 11: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

   \[\cos^{-1} \left(1 - x\right) \]
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 12: 3.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return acos(1.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

   \[\cos^{-1} \left(1 - x\right) \]
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 2024331 
(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)))