2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 7.0s

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, 0.4× speedup

Math

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

FPCore

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in g around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. distribute-lft-out N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

7. Applied rewrites 98.5%

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

9. Applied rewrites 100.0%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ (- g) h))))
   (-
    (fma
     (* (sin (* (PI) 0.6666666666666666)) 2.0)
     (sin (* t_0 0.3333333333333333))
     (cos (* -0.3333333333333333 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 100.0%

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

5. Taylor expanded in g around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. distribute-lft-out N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

7. Applied rewrites 98.5%

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

Reproduce

herbie shell --seed 2024303 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))