2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 9.3s

Alternatives: 7

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ t_1 := -0.1111111111111111 \cdot \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right)\\ \left(\cos \left(0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right) \cdot \cos t\_1 - \sin \left(\left(t\_0 \cdot 0.6666666666666666\right) \cdot t\_0\right) \cdot \sin t\_1\right) \cdot 2 \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (sqrt (PI)))
        (t_1 (* -0.1111111111111111 (* (acos (/ (- g) h)) -3.0))))
   (*
    (-
     (* (cos (* 0.6666666666666666 (PI))) (cos t_1))
     (* (sin (* (* t_0 0.6666666666666666) t_0)) (sin t_1)))
    2.0)))

Derivation

1. Initial program 98.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. frac-add N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{\color{blue}{-9}}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + 3 \cdot \left(\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \]

   12. distribute-rgt-neg-out N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + \color{blue}{\left(\mathsf{neg}\left(3 \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}}\right) \]

   13. *-commutative N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot 3}\right)\right)}}\right) \]

   14. distribute-lft-neg-out N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3}}}\right) \]

   15. lower-fma.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \mathsf{neg}\left(3\right), \left(\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3\right)}}}\right) \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{\frac{-9}{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -3, -6 \cdot \mathsf{PI}\left(\right)\right)}}\right)} \]
5. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\frac{1}{\frac{-9}{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -3, -6 \cdot \mathsf{PI}\left(\right)\right)}}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{\frac{-9}{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -3, -6 \cdot \mathsf{PI}\left(\right)\right)}}\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{\color{blue}{\frac{-9}{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -3, -6 \cdot \mathsf{PI}\left(\right)\right)}}}\right) \]

   4. associate-/r/ N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{-9} \cdot \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -3, -6 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   5. lift-fma.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{-9} \cdot \color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3 + -6 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   6. distribute-rgt-in N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right) \cdot \frac{1}{-9} + \left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{-9}\right)} \]

   7. cos-sum N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right) \cdot \frac{1}{-9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{-9}\right) - \sin \left(\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right) \cdot \frac{1}{-9}\right) \cdot \sin \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{-9}\right)\right)} \]

   8. lower--.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right) \cdot \frac{1}{-9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{-9}\right) - \sin \left(\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right) \cdot \frac{1}{-9}\right) \cdot \sin \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{-9}\right)\right)} \]

6. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot -0.1111111111111111\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot -0.1111111111111111\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot -0.1111111111111111\right) \cdot \sin \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot -0.1111111111111111\right)\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \color{blue}{\left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \color{blue}{\left(\frac{-1}{9} \cdot \left(-6 \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \left(\frac{-1}{9} \cdot \color{blue}{\left(-6 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \color{blue}{\left(\left(\frac{-1}{9} \cdot -6\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   5. metadata-eval N/A

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \left(\color{blue}{\frac{2}{3}} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \left(\frac{2}{3} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

   7. add-sqr-sqrt N/A

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \left(\frac{2}{3} \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \color{blue}{\left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \color{blue}{\left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \left(\color{blue}{\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   11. lift-PI.f64 N/A

       \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   13. lift-PI.f64 N/A

       \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right) \]

   14. lower-sqrt.f64 100.0

       \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot -0.1111111111111111\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot -0.1111111111111111\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot -0.1111111111111111\right) \cdot \sin \left(\left(0.6666666666666666 \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \]

8. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot -0.1111111111111111\right) \cdot \cos \left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot -0.1111111111111111\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot -0.1111111111111111\right) \cdot \sin \color{blue}{\left(\left(0.6666666666666666 \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \color{blue}{\left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{9}\right)} - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\color{blue}{\left(-6 \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot -6\right)} \cdot \frac{-1}{9}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(-6 \cdot \frac{-1}{9}\right)\right)} - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{2}{3}}\right) - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \cdot \sin \left(\left(\frac{2}{3} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

   6. lift-*.f64 100.0

      \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot -0.1111111111111111\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 0.6666666666666666\right)} - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot -0.1111111111111111\right) \cdot \sin \left(\left(0.6666666666666666 \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

10. Applied rewrites 100.0%

    \[\leadsto 2 \cdot \left(\cos \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot -0.1111111111111111\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 0.6666666666666666\right)} - \sin \left(\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot -0.1111111111111111\right) \cdot \sin \left(\left(0.6666666666666666 \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

11. Final simplification 100.0%

    \[\leadsto \left(\cos \left(0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(-0.1111111111111111 \cdot \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right)\right) - \sin \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot 0.6666666666666666\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sin \left(-0.1111111111111111 \cdot \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right)\right)\right) \cdot 2 \]
12. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{-g}{h}\right)\\ t_1 := \cos \left(\mathsf{fma}\left(0.3333333333333333, t\_0, -0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)\\ \frac{\cos \left(\mathsf{fma}\left(0.3333333333333333, t\_0, 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2}{t\_1} \cdot t\_1 \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ (- g) h)))
        (t_1 (cos (fma 0.3333333333333333 t_0 (* -0.6666666666666666 (PI))))))
   (*
    (/
     (* (cos (fma 0.3333333333333333 t_0 (* 0.6666666666666666 (PI)))) 2.0)
     t_1)
    t_1)))

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 98.4%

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

6. Applied rewrites 99.9%

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

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.6666666666666666 \cdot \mathsf{PI}\left(\right)\\ t_1 := \cos^{-1} \left(\frac{-g}{h}\right)\\ \left(\cos \left(\mathsf{fma}\left(t\_1, 0.3333333333333333, 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{2}{\cos \left(\mathsf{fma}\left(t\_1, 0.3333333333333333, t\_0\right)\right)}\right) \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, t\_1, t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (* -0.6666666666666666 (PI))) (t_1 (acos (/ (- g) h))))
   (*
    (*
     (cos (fma t_1 0.3333333333333333 (* 0.6666666666666666 (PI))))
     (/ 2.0 (cos (fma t_1 0.3333333333333333 t_0))))
    (cos (fma 0.3333333333333333 t_1 t_0)))))

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 98.4%

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

6. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 99.9

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

8. Applied rewrites 99.9%

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

9. Final simplification 99.9%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{\mathsf{fma}\left(\frac{1.5}{\mathsf{PI}\left(\right)}, \cos^{-1} \left(\frac{-g}{h}\right), 3\right)}{\frac{-4.5}{\mathsf{PI}\left(\right)}}\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* (cos (/ (fma (/ 1.5 (PI)) (acos (/ (- g) h)) 3.0) (/ -4.5 (PI)))) 2.0))

Derivation

1. Initial program 98.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. frac-add N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/r* N/A

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

   17. lower-/.f64 N/A

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

   18. metadata-eval 98.5

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

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1.5}{\mathsf{PI}\left(\right)}, 3\right)}{3 \cdot \frac{1.5}{\mathsf{PI}\left(\right)}}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.5

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

6. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\cos \left(\frac{\mathsf{fma}\left(\frac{1.5}{\mathsf{PI}\left(\right)}, \cos^{-1} \left(\frac{-g}{h}\right), 3\right)}{\frac{-4.5}{\mathsf{PI}\left(\right)}}\right) \cdot 2} \]
7. Add Preprocessing

Alternative 5: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{1}{\frac{-9}{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -3, -6 \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* (cos (/ 1.0 (/ -9.0 (fma (acos (/ (- g) h)) -3.0 (* -6.0 (PI)))))) 2.0))

Derivation

1. Initial program 98.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. frac-add N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{\color{blue}{-9}}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + 3 \cdot \left(\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \]

   12. distribute-rgt-neg-out N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + \color{blue}{\left(\mathsf{neg}\left(3 \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}}\right) \]

   13. *-commutative N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot 3}\right)\right)}}\right) \]

   14. distribute-lft-neg-out N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3}}}\right) \]

   15. lower-fma.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \mathsf{neg}\left(3\right), \left(\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3\right)}}}\right) \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{\frac{-9}{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -3, -6 \cdot \mathsf{PI}\left(\right)\right)}}\right)} \]

5. Final simplification 98.5%

   \[\leadsto \cos \left(\frac{1}{\frac{-9}{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -3, -6 \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot 2 \]
6. Add Preprocessing

Alternative 6: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, 6 \cdot \mathsf{PI}\left(\right)\right)}{9}\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* (cos (/ (fma (acos (/ (- g) h)) 3.0 (* 6.0 (PI))) 9.0)) 2.0))

Derivation

1. Initial program 98.4%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-add N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval 98.5

       \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, 6 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{9}}\right) \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, 6 \cdot \mathsf{PI}\left(\right)\right)}{9}\right)} \]

5. Final simplification 98.5%

   \[\leadsto \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, 6 \cdot \mathsf{PI}\left(\right)\right)}{9}\right) \cdot 2 \]
6. Add Preprocessing

Alternative 7: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.4%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval 98.5

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

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 98.5

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Reproduce

herbie shell --seed 2024253 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))