bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.6%

Time: 8.1s

Alternatives: 5

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.6% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 5.6%

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

4. Applied rewrites 3.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot 0.5, -\sin^{-1} \left(1 - x\right)\right)} \]
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 9.3

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

   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 9.3

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

6. Applied rewrites 9.3%

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

Alternative 2: 10.4% accurate, 0.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 5.6%

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

4. Applied rewrites 3.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot 0.5, -\sin^{-1} \left(1 - x\right)\right)} \]
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 9.3

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

   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 9.3

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

6. Applied rewrites 9.3%

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

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

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-PI.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-asin.f64 N/A

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

   18. mul-1-neg N/A

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

9. Applied rewrites 9.2%

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

Alternative 3: 10.4% accurate, 0.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 5.6%

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

4. Applied rewrites 5.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, -\sin^{-1} \left(1 - x\right)\right)} \]
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-PI.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. rem-cube-cbrt N/A

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

   9. lift-cbrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. pow-plus N/A

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

   12. lift-pow.f64 N/A

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

   13. associate-*r* N/A

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

   14. lift-*.f64 N/A

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

   15. lift-acos.f64 N/A

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

   16. associate--r- N/A

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

   17. neg-sub0 N/A

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

6. Applied rewrites 5.6%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. rem-square-sqrt N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 9.2

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

8. Applied rewrites 9.2%

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

Alternative 4: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.6%

   \[\cos^{-1} \left(1 - x\right) \]
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 6.8

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

5. Applied rewrites 6.8%

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

Alternative 5: 3.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return acos(1.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 3.8%

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

Developer Target 1: 100.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024322 
(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)))