GTR1 distribution

Percentage Accurate: 98.5% → 98.7%
Time: 10.9s
Alternatives: 9
Speedup: 1.1×

Specification

?
\[\begin{array}{l} t_0 := \alpha \cdot \alpha\\ t_1 := t_0 - 1\\ \frac{t_1}{\left(\pi \cdot \log t_0\right) \cdot \left(1 + \left(t_1 \cdot cosTheta\right) \cdot cosTheta\right)} \end{array} \]

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 9 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: 98.7% accurate, 0.7× speedup?

\[\begin{array}{l} t_0 := \alpha \cdot \alpha + -1\\ \frac{t_0}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + cosTheta \cdot \left(t_0 \cdot cosTheta\right)\right)} \end{array} \]
Derivation

  1. Initial program 98.5%

    \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
  2. Step-by-step derivation

    1. add-log-exp98.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left(e^{\pi \cdot \log \left(\alpha \cdot \alpha\right)}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

    2. *-commutative98.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left(e^{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \pi}}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

    3. exp-to-pow98.6%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \color{blue}{\left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

  3. Applied egg-rr98.6%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

  4. Final simplification98.6%

    \[\leadsto \frac{\alpha \cdot \alpha + -1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + cosTheta \cdot \left(\left(\alpha \cdot \alpha + -1\right) \cdot cosTheta\right)\right)} \]

Alternative 2: 98.5% accurate, 1.0× speedup?

\[\frac{\alpha \cdot \alpha + -1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha\right) \cdot \left(cosTheta \cdot cosTheta\right) - cosTheta \cdot cosTheta\right)\right)} \]
Derivation

  1. Initial program 98.5%

    \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
  2. Step-by-step derivation

    1. fma-neg98.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

    2. metadata-eval98.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\mathsf{fma}\left(\alpha, \alpha, \color{blue}{-1}\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

    3. associate-*r*98.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)}\right)} \]

    4. *-commutative98.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(cosTheta \cdot cosTheta\right) \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)}\right)} \]

    5. fma-udef98.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(cosTheta \cdot cosTheta\right) \cdot \color{blue}{\left(\alpha \cdot \alpha + -1\right)}\right)} \]

    6. distribute-lft-in98.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(\left(cosTheta \cdot cosTheta\right) \cdot \left(\alpha \cdot \alpha\right) + \left(cosTheta \cdot cosTheta\right) \cdot -1\right)}\right)} \]

  3. Applied egg-rr98.5%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(\left(cosTheta \cdot cosTheta\right) \cdot \left(\alpha \cdot \alpha\right) + \left(cosTheta \cdot cosTheta\right) \cdot -1\right)}\right)} \]

  4. Final simplification98.5%

    \[\leadsto \frac{\alpha \cdot \alpha + -1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha\right) \cdot \left(cosTheta \cdot cosTheta\right) - cosTheta \cdot cosTheta\right)\right)} \]

Alternative 3: 98.5% accurate, 1.0× speedup?

\[\begin{array}{l} t_0 := \alpha \cdot \alpha + -1\\ \frac{t_0}{\left(1 + cosTheta \cdot \left(t_0 \cdot cosTheta\right)\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \end{array} \]
Derivation

  1. Initial program 98.5%

    \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

  2. Final simplification98.5%

    \[\leadsto \frac{\alpha \cdot \alpha + -1}{\left(1 + cosTheta \cdot \left(\left(\alpha \cdot \alpha + -1\right) \cdot cosTheta\right)\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]

Alternative 4: 97.5% accurate, 1.0× speedup?

\[\frac{\alpha \cdot \alpha + -1}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\log \alpha \cdot \left(\pi \cdot 2\right)\right)} \]
Derivation

  1. Initial program 98.5%

    \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
  2. Step-by-step derivation

    1. add-log-exp98.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left(e^{\pi \cdot \log \left(\alpha \cdot \alpha\right)}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

    2. *-commutative98.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left(e^{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \pi}}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

    3. exp-to-pow98.6%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \color{blue}{\left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

  3. Applied egg-rr98.6%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

  4. Taylor expanded in alpha around 0 96.8%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + \color{blue}{\left(-1 \cdot cosTheta\right)} \cdot cosTheta\right)} \]

  6. Simplified96.8%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + \color{blue}{\left(-cosTheta\right)} \cdot cosTheta\right)} \]
  7. Step-by-step derivation

    1. distribute-rgt-in96.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{1 \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) + \left(\left(-cosTheta\right) \cdot cosTheta\right) \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)}} \]

    2. *-un-lft-identity96.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} + \left(\left(-cosTheta\right) \cdot cosTheta\right) \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \]

    3. *-commutative96.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) + \color{blue}{\left(cosTheta \cdot \left(-cosTheta\right)\right)} \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \]

  8. Applied egg-rr96.8%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) + \left(cosTheta \cdot \left(-cosTheta\right)\right) \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)}} \]
  9. Step-by-step derivation

    1. distribute-rgt-neg-out96.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) + \color{blue}{\left(-cosTheta \cdot cosTheta\right)} \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \]

    2. unpow296.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) + \left(-\color{blue}{{cosTheta}^{2}}\right) \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \]

    3. mul-1-neg96.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) + \color{blue}{\left(-1 \cdot {cosTheta}^{2}\right)} \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \]

    4. distribute-rgt1-in96.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(-1 \cdot {cosTheta}^{2} + 1\right) \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)}} \]

    5. +-commutative96.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(1 + -1 \cdot {cosTheta}^{2}\right)} \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \]

    6. mul-1-neg96.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(1 + \color{blue}{\left(-{cosTheta}^{2}\right)}\right) \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \]

    7. unsub-neg96.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(1 - {cosTheta}^{2}\right)} \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \]

    8. unpow296.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(1 - \color{blue}{cosTheta \cdot cosTheta}\right) \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \]

    9. unpow296.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \log \left({\color{blue}{\left({\alpha}^{2}\right)}}^{\pi}\right)} \]

    10. log-pow96.7%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \color{blue}{\left(\pi \cdot \log \left({\alpha}^{2}\right)\right)}} \]

    11. *-commutative96.7%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \color{blue}{\left(\log \left({\alpha}^{2}\right) \cdot \pi\right)}} \]

    12. log-pow96.6%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\color{blue}{\left(2 \cdot \log \alpha\right)} \cdot \pi\right)} \]

    13. *-commutative96.6%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\color{blue}{\left(\log \alpha \cdot 2\right)} \cdot \pi\right)} \]

    14. associate-*l*96.6%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \color{blue}{\left(\log \alpha \cdot \left(2 \cdot \pi\right)\right)}} \]

  10. Simplified96.6%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\log \alpha \cdot \left(2 \cdot \pi\right)\right)}} \]

  11. Final simplification96.6%

    \[\leadsto \frac{\alpha \cdot \alpha + -1}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\log \alpha \cdot \left(\pi \cdot 2\right)\right)} \]

Alternative 5: 97.5% accurate, 1.0× speedup?

\[\frac{\alpha \cdot \alpha + -1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 - cosTheta \cdot cosTheta\right)} \]
Derivation

  1. Initial program 98.5%

    \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

  2. Taylor expanded in alpha around 0 96.7%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(-1 \cdot cosTheta\right)} \cdot cosTheta\right)} \]

  4. Simplified96.7%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(-cosTheta\right)} \cdot cosTheta\right)} \]

  5. Final simplification96.7%

    \[\leadsto \frac{\alpha \cdot \alpha + -1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 - cosTheta \cdot cosTheta\right)} \]

Alternative 6: 66.9% accurate, 1.1× speedup?

\[\frac{-0.5}{\log \alpha \cdot \left(\pi \cdot \left(1 - cosTheta \cdot cosTheta\right)\right)} \]
Derivation

  1. Initial program 98.5%

    \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
  2. Step-by-step derivation

    1. add-log-exp98.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left(e^{\pi \cdot \log \left(\alpha \cdot \alpha\right)}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

    2. *-commutative98.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left(e^{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \pi}}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

    3. exp-to-pow98.6%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \color{blue}{\left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

  3. Applied egg-rr98.6%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

  4. Taylor expanded in alpha around 0 67.6%

    \[\leadsto \color{blue}{\frac{-0.5}{\log \alpha \cdot \left(\left(1 + -1 \cdot {cosTheta}^{2}\right) \cdot \pi\right)}} \]
  5. Step-by-step derivation

    1. *-commutative67.6%

      \[\leadsto \frac{-0.5}{\log \alpha \cdot \color{blue}{\left(\pi \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}} \]

    2. neg-mul-167.6%

      \[\leadsto \frac{-0.5}{\log \alpha \cdot \left(\pi \cdot \left(1 + \color{blue}{\left(-{cosTheta}^{2}\right)}\right)\right)} \]

    3. unsub-neg67.6%

      \[\leadsto \frac{-0.5}{\log \alpha \cdot \left(\pi \cdot \color{blue}{\left(1 - {cosTheta}^{2}\right)}\right)} \]

    4. unpow267.6%

      \[\leadsto \frac{-0.5}{\log \alpha \cdot \left(\pi \cdot \left(1 - \color{blue}{cosTheta \cdot cosTheta}\right)\right)} \]

  6. Simplified67.6%

    \[\leadsto \color{blue}{\frac{-0.5}{\log \alpha \cdot \left(\pi \cdot \left(1 - cosTheta \cdot cosTheta\right)\right)}} \]

  7. Final simplification67.6%

    \[\leadsto \frac{-0.5}{\log \alpha \cdot \left(\pi \cdot \left(1 - cosTheta \cdot cosTheta\right)\right)} \]

Alternative 7: 66.9% accurate, 1.1× speedup?

\[\frac{-0.5}{\pi \cdot \left(\left(1 - cosTheta \cdot cosTheta\right) \cdot \log \alpha\right)} \]
Derivation

  1. Initial program 98.5%

    \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
  2. Step-by-step derivation

    1. associate-/r*98.5%

      \[\leadsto \color{blue}{\frac{\frac{\alpha \cdot \alpha - 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta}} \]

    2. difference-of-sqr-198.1%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(\alpha - 1\right)}}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    3. *-commutative98.1%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha - 1\right) \cdot \left(\alpha + 1\right)}}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    4. times-frac98.1%

      \[\leadsto \frac{\color{blue}{\frac{\alpha - 1}{\pi} \cdot \frac{\alpha + 1}{\log \left(\alpha \cdot \alpha\right)}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    5. *-commutative98.1%

      \[\leadsto \frac{\color{blue}{\frac{\alpha + 1}{\log \left(\alpha \cdot \alpha\right)} \cdot \frac{\alpha - 1}{\pi}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    6. times-frac98.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\alpha - 1\right)}{\log \left(\alpha \cdot \alpha\right) \cdot \pi}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    7. difference-of-sqr-198.5%

      \[\leadsto \frac{\frac{\color{blue}{\alpha \cdot \alpha - 1}}{\log \left(\alpha \cdot \alpha\right) \cdot \pi}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    8. associate-/l/98.5%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{\log \left(\alpha \cdot \alpha\right)}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    9. log-prod98.5%

      \[\leadsto \frac{\frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{\color{blue}{\log \alpha + \log \alpha}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    10. count-298.5%

      \[\leadsto \frac{\frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{\color{blue}{2 \cdot \log \alpha}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    11. *-commutative98.5%

      \[\leadsto \frac{\frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{\color{blue}{\log \alpha \cdot 2}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    12. fma-neg98.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right)}}{\pi}}{\log \alpha \cdot 2}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    13. metadata-eval98.5%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, \color{blue}{-1}\right)}{\pi}}{\log \alpha \cdot 2}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    14. +-commutative98.5%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\pi}}{\log \alpha \cdot 2}}{\color{blue}{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta + 1}} \]

  3. Simplified98.5%

    \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\pi}}{\log \alpha \cdot 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)}} \]
  4. Step-by-step derivation

    1. fma-udef98.5%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\pi}}{\log \alpha \cdot 2}}{\color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right) + 1}} \]

  5. Applied egg-rr98.5%

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\pi}}{\log \alpha \cdot 2}}{\color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right) + 1}} \]

  6. Taylor expanded in alpha around 0 67.6%

    \[\leadsto \color{blue}{\frac{-0.5}{\log \alpha \cdot \left(\left(1 + -1 \cdot {cosTheta}^{2}\right) \cdot \pi\right)}} \]
  7. Step-by-step derivation

    1. associate-*r*67.6%

      \[\leadsto \frac{-0.5}{\color{blue}{\left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right) \cdot \pi}} \]

    2. mul-1-neg67.6%

      \[\leadsto \frac{-0.5}{\left(\log \alpha \cdot \left(1 + \color{blue}{\left(-{cosTheta}^{2}\right)}\right)\right) \cdot \pi} \]

    3. unsub-neg67.6%

      \[\leadsto \frac{-0.5}{\left(\log \alpha \cdot \color{blue}{\left(1 - {cosTheta}^{2}\right)}\right) \cdot \pi} \]

    4. unpow267.6%

      \[\leadsto \frac{-0.5}{\left(\log \alpha \cdot \left(1 - \color{blue}{cosTheta \cdot cosTheta}\right)\right) \cdot \pi} \]

  8. Simplified67.6%

    \[\leadsto \color{blue}{\frac{-0.5}{\left(\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)\right) \cdot \pi}} \]

  9. Final simplification67.6%

    \[\leadsto \frac{-0.5}{\pi \cdot \left(\left(1 - cosTheta \cdot cosTheta\right) \cdot \log \alpha\right)} \]

Alternative 8: 95.0% accurate, 1.1× speedup?

\[\frac{\alpha \cdot \alpha + -1}{\log \alpha \cdot \left(\pi \cdot 2\right)} \]
Derivation

  1. Initial program 98.5%

    \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
  2. Step-by-step derivation

    1. add-log-exp98.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left(e^{\pi \cdot \log \left(\alpha \cdot \alpha\right)}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

    2. *-commutative98.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left(e^{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \pi}}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

    3. exp-to-pow98.6%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \color{blue}{\left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

  3. Applied egg-rr98.6%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

  4. Taylor expanded in alpha around 0 96.8%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + \color{blue}{\left(-1 \cdot cosTheta\right)} \cdot cosTheta\right)} \]

  6. Simplified96.8%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + \color{blue}{\left(-cosTheta\right)} \cdot cosTheta\right)} \]

  7. Taylor expanded in cosTheta around 0 93.9%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left({\alpha}^{2}\right) \cdot \pi}} \]
  8. Step-by-step derivation

    1. log-pow93.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(2 \cdot \log \alpha\right)} \cdot \pi} \]

    2. *-commutative93.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\log \alpha \cdot 2\right)} \cdot \pi} \]

    3. associate-*l*93.8%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \alpha \cdot \left(2 \cdot \pi\right)}} \]

  9. Simplified93.8%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \alpha \cdot \left(2 \cdot \pi\right)}} \]

  10. Final simplification93.8%

    \[\leadsto \frac{\alpha \cdot \alpha + -1}{\log \alpha \cdot \left(\pi \cdot 2\right)} \]

Alternative 9: 65.5% accurate, 1.1× speedup?

\[\frac{-0.5}{\pi \cdot \log \alpha} \]
Derivation

  1. Initial program 98.5%

    \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

  2. Taylor expanded in alpha around inf 93.9%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(cosTheta \cdot {\alpha}^{2}\right)} \cdot cosTheta\right)} \]

  4. Simplified93.9%

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(\alpha \cdot \left(\alpha \cdot cosTheta\right)\right)} \cdot cosTheta\right)} \]

  5. Taylor expanded in alpha around 0 66.3%

    \[\leadsto \color{blue}{\frac{-0.5}{\log \alpha \cdot \pi}} \]

  6. Final simplification66.3%

    \[\leadsto \frac{-0.5}{\pi \cdot \log \alpha} \]

Reproduce

?
herbie shell --seed 2023167 
(FPCore (cosTheta alpha)
  :name "GTR1 distribution"
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0)) (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ (- (* alpha alpha) 1.0) (* (* PI (log (* alpha alpha))) (+ 1.0 (* (* (- (* alpha alpha) 1.0) cosTheta) cosTheta)))))