2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%
Time: 8.4s
Alternatives: 2
Speedup: 1.1×

Specification

?
\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \end{array} \]
(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))
\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}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 2 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 98.5% accurate, 1.0× speedup?

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \end{array} \]
(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))
\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}

Alternative 1: 100.0% accurate, 0.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ t_1 := \cos^{-1} \left(\frac{-g}{h}\right)\\ \left(\cos \left(-0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(t\_1 \cdot -0.3333333333333333\right) - \sin \left(\left(t\_0 \cdot 0.6666666666666666\right) \cdot t\_0\right) \cdot \sin \left(0.3333333333333333 \cdot t\_1\right)\right) \cdot 2 \end{array} \end{array} \]
(FPCore (g h)
 :precision binary64
 (let* ((t_0 (sqrt (PI))) (t_1 (acos (/ (- g) h))))
   (*
    (-
     (* (cos (* -0.6666666666666666 (PI))) (cos (* t_1 -0.3333333333333333)))
     (*
      (sin (* (* t_0 0.6666666666666666) t_0))
      (sin (* 0.3333333333333333 t_1))))
    2.0)))
\begin{array}{l}

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

  4. Applied rewrites98.4%

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

    1. lift-*.f64N/A

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

    2. lift-PI.f64N/A

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

    3. add-sqr-sqrtN/A

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

    4. associate-*r*N/A

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

    5. lower-*.f64N/A

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

    6. lower-*.f64N/A

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

    7. lift-PI.f64N/A

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

    8. lower-sqrt.f64N/A

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

    9. lift-PI.f64N/A

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

    10. lower-sqrt.f64100.0

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

  6. Applied rewrites100.0%

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

  7. Final simplification100.0%

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

Alternative 2: 98.5% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.6666666666666666, 0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)\right) \cdot 2 \end{array} \]
(FPCore (g h)
 :precision binary64
 (*
  (cos (fma (PI) 0.6666666666666666 (* 0.3333333333333333 (acos (/ (- g) h)))))
  2.0))
\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}
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-+.f64N/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-/.f64N/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-invN/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-*.f64N/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. *-commutativeN/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.f64N/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-evalN/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-eval98.5

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

    10. lift-/.f64N/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-numN/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-*.f64N/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-eval98.5

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

  4. Applied rewrites98.5%

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

  5. Final simplification98.5%

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

Reproduce

?
herbie shell --seed 2024267 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))