2-ancestry mixing, negative discriminant

Percentage Accurate: 98.4% → 100.0%

Time: 5.2s

Alternatives: 2

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: 98.4% 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

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

FPCore

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

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. mul-1-neg N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. cos-neg-rev N/A

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

   9. cos-+PI-rev N/A

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

   10. remove-double-neg N/A

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

   11. distribute-neg-out N/A

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

   12. cos-neg N/A

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

   13. lower-cos.f64 N/A

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

5. Applied rewrites 98.5%

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in g around 0

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

10. Applied rewrites 100.0%

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

Alternative 2: 98.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (* (cos (* (- (acos (/ (- g) h)) (PI)) -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. Taylor expanded in g around 0

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. mul-1-neg N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. cos-neg-rev N/A

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

   9. cos-+PI-rev N/A

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

   10. remove-double-neg N/A

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

   11. distribute-neg-out N/A

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

   12. cos-neg N/A

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

   13. lower-cos.f64 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in g around 0

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

8. Applied rewrites 98.5%

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

Reproduce

herbie shell --seed 2024360 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))