2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 8.2s

Alternatives: 5

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ 2 \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(-0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right) - \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \sin \left(\left(0.6666666666666666 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (*
    2.0
    (-
     (*
      (cos (/ (- (PI) (acos (/ g h))) 3.0))
      (cos (* -0.6666666666666666 (PI))))
     (*
      (sin (* 0.3333333333333333 (acos (/ (- g) h))))
      (sin (* (* 0.6666666666666666 t_0) t_0)))))))

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing

4. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(-0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right) - \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \sin \left(0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right)\right) - \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \sin \left(\frac{2}{3} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right)\right) - \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \sin \color{blue}{\left(\frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   3. add-sqr-sqrt N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lift-PI.f64 100.0

       \[\leadsto 2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(-0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right) - \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \sin \left(\left(0.6666666666666666 \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right) \]

6. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \left(\cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(-0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right) - \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \sin \color{blue}{\left(\left(0.6666666666666666 \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]
7. Step-by-step derivation

   1. lift-acos.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. distribute-frac-neg N/A

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

   5. acos-neg N/A

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

   6. frac-2neg N/A

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

   7. distribute-frac-neg2 N/A

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

   8. acos-neg-rev N/A

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

   9. lift-/.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-acos.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-acos.f64 N/A

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

   15. lift-neg.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. acos-neg-rev N/A

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

   18. distribute-frac-neg2 N/A

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

   19. frac-2neg N/A

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

   20. lower-acos.f64 N/A

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

   21. lower-/.f64 100.0

       \[\leadsto 2 \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) - \cos^{-1} \color{blue}{\left(\frac{g}{h}\right)}}{3}\right) \cdot \cos \left(-0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right) - \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \sin \left(\left(0.6666666666666666 \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

8. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \left(\cos \left(\frac{\color{blue}{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}}{3}\right) \cdot \cos \left(-0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right) - \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \sin \left(\left(0.6666666666666666 \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

9. Taylor expanded in g around 0

   \[\leadsto 2 \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right)} \cdot \sin \left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]
10. Step-by-step derivation

    1. lower-sin.f64 N/A

       \[\leadsto 2 \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right)\right) - \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

    2. mul-1-neg N/A

       \[\leadsto 2 \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right)\right) - \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

    3. distribute-frac-neg N/A

       \[\leadsto 2 \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right)\right) - \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

    4. lift-/.f64 N/A

       \[\leadsto 2 \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right)\right) - \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

    5. lift-neg.f64 N/A

       \[\leadsto 2 \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right)\right) - \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

    6. lift-acos.f64 N/A

       \[\leadsto 2 \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right)\right) - \sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

    7. lower-*.f64 100.0

       \[\leadsto 2 \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(-0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right) - \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \sin \left(\left(0.6666666666666666 \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

11. Applied rewrites 100.0%

    \[\leadsto 2 \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}{3}\right) \cdot \cos \left(-0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)} \cdot \sin \left(\left(0.6666666666666666 \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]
12. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), -0.6666666666666666, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin
   (fma
    -0.3333333333333333
    (acos (/ (- g) h))
    (fma (PI) -0.6666666666666666 (/ (PI) 2.0))))))

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   2. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-PI.f64 97.6

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

   8. lift-+.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-acos.f64 N/A

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

   12. lift-neg.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{h}\right)}{3}\right)\right) \]

   13. lift-/.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \color{blue}{\left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}}{3}\right)\right) \]

   14. div-add-rev N/A

       \[\leadsto 2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2} + \color{blue}{\frac{2 \cdot \mathsf{PI}\left(\right) + \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}}\right) \]

   15. lower-/.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2} + \color{blue}{\frac{2 \cdot \mathsf{PI}\left(\right) + \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}}\right) \]

4. Applied rewrites 97.6%

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

6. Applied rewrites 97.6%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), 0.3333333333333333, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

8. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), -0.6666666666666666, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)} \]
9. Add Preprocessing

Alternative 3: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), -0.6666666666666666, \mathsf{PI}\left(\right)\right) \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin
   (* (fma (fma (PI) 2.0 (acos (/ (- g) h))) -0.6666666666666666 (PI)) 0.5))))

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\cos \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{3}\right) + \cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(-0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right) - \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \sin \left(0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)} \]

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), \frac{\mathsf{PI}\left(\right)}{2}\right) + \mathsf{fma}\left(-0.3333333333333333, \mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), \frac{\mathsf{PI}\left(\right)}{2}\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), \frac{\mathsf{PI}\left(\right)}{2}\right) - \mathsf{fma}\left(-0.3333333333333333, \mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), \frac{\mathsf{PI}\left(\right)}{2}\right)}{2}\right)\right)} \]

7. Taylor expanded in g around 0

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) + \frac{-2}{3} \cdot \left(\cos^{-1} \left(-1 \cdot \frac{g}{h}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. lower-sin.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) + \frac{-2}{3} \cdot \left(\cos^{-1} \left(-1 \cdot \frac{g}{h}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto 2 \cdot \sin \left(\left(\mathsf{PI}\left(\right) + \frac{-2}{3} \cdot \left(\cos^{-1} \left(-1 \cdot \frac{g}{h}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\left(\mathsf{PI}\left(\right) + \frac{-2}{3} \cdot \left(\cos^{-1} \left(-1 \cdot \frac{g}{h}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}\right) \]

9. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), -0.6666666666666666, \mathsf{PI}\left(\right)\right) \cdot 0.5\right)} \]
10. Add Preprocessing

Alternative 4: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 0.3333333333333333\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* (cos (* (fma (PI) 2.0 (acos (/ (- g) h))) 0.3333333333333333)) 2.0))

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

   3. lower-*.f64 98.5

      \[\leadsto \color{blue}{\cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

4. Applied rewrites 98.5%

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

5. Taylor expanded in g around 0

   \[\leadsto \cos \color{blue}{\left(\frac{1}{3} \cdot \left(\cos^{-1} \left(-1 \cdot \frac{g}{h}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot 2 \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(\cos^{-1} \left(-1 \cdot \frac{g}{h}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{1}{3}}\right) \cdot 2 \]

   2. +-commutative N/A

      \[\leadsto \cos \left(\left(2 \cdot \mathsf{PI}\left(\right) + \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \frac{1}{3}\right) \cdot 2 \]

   3. *-commutative N/A

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2 + \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \frac{1}{3}\right) \cdot 2 \]

   4. mul-1-neg N/A

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2 + \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \frac{1}{3}\right) \cdot 2 \]

   5. distribute-frac-neg N/A

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2 + \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right) \cdot \frac{1}{3}\right) \cdot 2 \]

   6. lift-/.f64 N/A

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2 + \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right) \cdot \frac{1}{3}\right) \cdot 2 \]

   7. lift-neg.f64 N/A

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2 + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right) \cdot 2 \]

   8. lift-acos.f64 N/A

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2 + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right) \cdot 2 \]

   9. lift-fma.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right) \cdot 2 \]

   10. lift-PI.f64 N/A

       \[\leadsto \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right) \cdot 2 \]

   11. lower-*.f64 98.5

       \[\leadsto \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \color{blue}{0.3333333333333333}\right) \cdot 2 \]

7. Applied rewrites 98.5%

   \[\leadsto \cos \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 0.3333333333333333\right)} \cdot 2 \]
8. Add Preprocessing

Alternative 5: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (cos
   (fma 0.3333333333333333 (acos (/ (- g) h)) (* 0.6666666666666666 (PI))))))

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing

3. Taylor expanded in g around 0

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right) + \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) + \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right) \]

   2. distribute-frac-neg N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) + \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right) \]

   3. lower-fma.f64 N/A

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

   4. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right), \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. lift-neg.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. lift-acos.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. lift-PI.f64 98.4

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right) \]

5. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

6. Final simplification 98.4%

   \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right) \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025046 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))