2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 98.5%
Time: 5.0s
Alternatives: 3
Speedup: 1.0×

Specification

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

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 3 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.

Alternative 1: 98.5% accurate, 0.8× speedup?

\[2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right) \]
Derivation

  1. Initial program 98.4%

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

    1. associate-/l*98.4%

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

    2. associate-/r/98.4%

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

    3. *-commutative98.4%

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

    4. fma-def98.5%

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

    5. metadata-eval98.5%

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

  3. Simplified98.5%

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

  4. Final simplification98.5%

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

Alternative 2: 98.5% accurate, 1.0× speedup?

\[2 \cdot \cos \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3} + \pi \cdot 0.6666666666666666\right) \]
Derivation

  1. Initial program 98.4%

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

    1. associate-/l*98.4%

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

    2. associate-/r/98.4%

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

    3. metadata-eval98.4%

      \[\leadsto 2 \cdot \cos \left(\color{blue}{0.6666666666666666} \cdot \pi + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]

  3. Simplified98.4%

    \[\leadsto \color{blue}{2 \cdot \cos \left(0.6666666666666666 \cdot \pi + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

  4. Final simplification98.4%

    \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3} + \pi \cdot 0.6666666666666666\right) \]

Alternative 3: 96.6% accurate, 1.0× speedup?

\[2 \cdot \cos \left(\pi \cdot 0.6666666666666666 + \cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333\right) \]
Derivation

  1. Initial program 98.4%

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

    1. associate-/l*98.4%

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

    2. associate-/r/98.4%

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

    3. *-commutative98.4%

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

    4. fma-def98.5%

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

    5. metadata-eval98.5%

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

  3. Simplified98.5%

    \[\leadsto \color{blue}{2 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)} \]
  4. Step-by-step derivation

    1. fma-udef98.4%

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\pi \cdot 0.6666666666666666 + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

    2. *-commutative98.4%

      \[\leadsto 2 \cdot \cos \left(\color{blue}{0.6666666666666666 \cdot \pi} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]

    3. +-commutative98.4%

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3} + 0.6666666666666666 \cdot \pi\right)} \]

    4. div-inv98.4%

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{1}{3}} + 0.6666666666666666 \cdot \pi\right) \]

    5. add-sqr-sqrt46.5%

      \[\leadsto 2 \cdot \cos \left(\cos^{-1} \left(\frac{\color{blue}{\sqrt{-g} \cdot \sqrt{-g}}}{h}\right) \cdot \frac{1}{3} + 0.6666666666666666 \cdot \pi\right) \]

    6. sqrt-unprod91.2%

      \[\leadsto 2 \cdot \cos \left(\cos^{-1} \left(\frac{\color{blue}{\sqrt{\left(-g\right) \cdot \left(-g\right)}}}{h}\right) \cdot \frac{1}{3} + 0.6666666666666666 \cdot \pi\right) \]

    7. sqr-neg91.2%

      \[\leadsto 2 \cdot \cos \left(\cos^{-1} \left(\frac{\sqrt{\color{blue}{g \cdot g}}}{h}\right) \cdot \frac{1}{3} + 0.6666666666666666 \cdot \pi\right) \]

    8. sqrt-unprod50.7%

      \[\leadsto 2 \cdot \cos \left(\cos^{-1} \left(\frac{\color{blue}{\sqrt{g} \cdot \sqrt{g}}}{h}\right) \cdot \frac{1}{3} + 0.6666666666666666 \cdot \pi\right) \]

    9. add-sqr-sqrt96.0%

      \[\leadsto 2 \cdot \cos \left(\cos^{-1} \left(\frac{\color{blue}{g}}{h}\right) \cdot \frac{1}{3} + 0.6666666666666666 \cdot \pi\right) \]

    10. metadata-eval96.0%

      \[\leadsto 2 \cdot \cos \left(\cos^{-1} \left(\frac{g}{h}\right) \cdot \color{blue}{0.3333333333333333} + 0.6666666666666666 \cdot \pi\right) \]

  5. Applied egg-rr96.0%

    \[\leadsto 2 \cdot \cos \color{blue}{\left(\cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333 + 0.6666666666666666 \cdot \pi\right)} \]

  6. Final simplification96.0%

    \[\leadsto 2 \cdot \cos \left(\pi \cdot 0.6666666666666666 + \cos^{-1} \left(\frac{g}{h}\right) \cdot 0.3333333333333333\right) \]

Reproduce

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herbie shell --seed 2023167 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))