2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 7.8s

Alternatives: 4

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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin
   (fma -0.3333333333333333 (fma (PI) 2.0 (acos (/ (- g) h))) (/ (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. lift-/.f64 N/A

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

   3. lift-acos.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. frac-add N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 98.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

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

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

   5. lower-fma.f64 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-PI.f64 98.5

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

6. Applied rewrites 98.5%

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

8. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, \mathsf{fma}\left(-0.6666666666666666, \mathsf{PI}\left(\right), 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin
   (fma
    (acos (/ (- g) h))
    -0.3333333333333333
    (fma -0.6666666666666666 (PI) (* 0.5 (PI)))))))

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

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

   3. lift-acos.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. frac-add N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 98.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

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

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

   5. lower-fma.f64 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-PI.f64 98.5

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

6. Applied rewrites 98.5%

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in g around 0

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

    1. distribute-lft-in N/A

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

    2. mul-1-neg N/A

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

    3. distribute-frac-neg N/A

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

    4. lower-acos.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. lift-neg.f64 N/A

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

    7. *-commutative N/A

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

    8. associate-+l+ N/A

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

    9. *-commutative N/A

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

    10. lower-fma.f64 N/A

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

11. Applied rewrites 100.0%

    \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, \mathsf{fma}\left(-0.6666666666666666, \mathsf{PI}\left(\right), 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
12. Add Preprocessing

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

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. 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. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lower-acos.f64 N/A

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

   8. distribute-frac-neg N/A

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

   9. mul-1-neg N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. mul-1-neg N/A

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

   14. distribute-frac-neg N/A

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

   15. lower-acos.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. lift-neg.f64 N/A

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

   18. lower-*.f64 98.5

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

7. Applied rewrites 98.5%

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

Alternative 4: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

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. 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. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. associate-*r* N/A

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

   7. distribute-lft-in N/A

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

   8. *-commutative N/A

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

   9. lift-acos.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

7. Applied rewrites 98.5%

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

Reproduce

herbie shell --seed 2025045 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))