2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 3.7s

Alternatives: 3

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, 1.1× speedup

Math

\[\begin{array}{l} \\ \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.16666666666666666, \cos^{-1} \left(\frac{-g}{h}\right) \cdot -0.3333333333333333\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (sin
   (fma (PI) -0.16666666666666666 (* (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%

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in g around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. metadata-eval N/A

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

   4. mul-1-neg N/A

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

   5. distribute-frac-neg N/A

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

   6. lower-acos.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-+r+ N/A

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

9. Applied rewrites 100.0%

   \[\leadsto \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.16666666666666666, \cos^{-1} \left(\frac{-g}{h}\right) \cdot -0.3333333333333333\right)\right)} \cdot 2 \]
10. Add Preprocessing

Alternative 2: 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 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. div-add-rev N/A

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

   6. frac-add N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. div-add N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. associate-/l* N/A

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

   14. metadata-eval N/A

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

   15. associate-/l* N/A

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

6. Applied rewrites 98.5%

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

Alternative 3: 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 98.5

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

5. Applied rewrites 98.5%

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

Reproduce

herbie shell --seed 2025032 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))