Average Error: 2.7 → 0.2
Time: 28.9s
Precision: 64
Internal Precision: 576
${\left(\left(1 - \alpha\right) \cdot \mathsf{fmax}\left(0, utility\right)\right)}^{\left(\frac{1}{1 - \alpha}\right)}$
$\begin{array}{l} \mathbf{if}\;{\left(\left(1 - \alpha\right) \cdot \mathsf{fmax}\left(0, utility\right)\right)}^{\left(\frac{1}{1 - \alpha}\right)} \le 4.972007243076791 \cdot 10^{+296}:\\ \;\;\;\;{\left({\left(\left(1 - \alpha\right) \cdot \mathsf{fmax}\left(0, utility\right)\right)}^{\left(\sqrt[3]{\frac{1}{1 - \alpha}} \cdot \sqrt[3]{\frac{1}{1 - \alpha}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{1 - \alpha}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{\frac{\log \left(1 - \alpha\right) + \log \left(\mathsf{fmax}\left(0, utility\right)\right)}{1 - \alpha \cdot \alpha}}\right)}^{\left(1 + \alpha\right)}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if (pow (* (- 1 alpha) (fmax 0 utility)) (/ 1 (- 1 alpha))) < 4.972007243076791e+296

1. Initial program 0.1

${\left(\left(1 - \alpha\right) \cdot \mathsf{fmax}\left(0, utility\right)\right)}^{\left(\frac{1}{1 - \alpha}\right)}$
2. Using strategy rm

$\leadsto {\left(\left(1 - \alpha\right) \cdot \mathsf{fmax}\left(0, utility\right)\right)}^{\color{blue}{\left(\left(\sqrt[3]{\frac{1}{1 - \alpha}} \cdot \sqrt[3]{\frac{1}{1 - \alpha}}\right) \cdot \sqrt[3]{\frac{1}{1 - \alpha}}\right)}}$
4. Applied pow-unpow0.2

$\leadsto \color{blue}{{\left({\left(\left(1 - \alpha\right) \cdot \mathsf{fmax}\left(0, utility\right)\right)}^{\left(\sqrt[3]{\frac{1}{1 - \alpha}} \cdot \sqrt[3]{\frac{1}{1 - \alpha}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{1 - \alpha}}\right)}}$

## if 4.972007243076791e+296 < (pow (* (- 1 alpha) (fmax 0 utility)) (/ 1 (- 1 alpha)))

1. Initial program 52.7

${\left(\left(1 - \alpha\right) \cdot \mathsf{fmax}\left(0, utility\right)\right)}^{\left(\frac{1}{1 - \alpha}\right)}$
2. Using strategy rm

$\leadsto {\left(\left(1 - \alpha\right) \cdot \color{blue}{e^{\log \left(\mathsf{fmax}\left(0, utility\right)\right)}}\right)}^{\left(\frac{1}{1 - \alpha}\right)}$

$\leadsto {\left(\color{blue}{e^{\log \left(1 - \alpha\right)}} \cdot e^{\log \left(\mathsf{fmax}\left(0, utility\right)\right)}\right)}^{\left(\frac{1}{1 - \alpha}\right)}$
5. Applied prod-exp53.8

$\leadsto {\color{blue}{\left(e^{\log \left(1 - \alpha\right) + \log \left(\mathsf{fmax}\left(0, utility\right)\right)}\right)}}^{\left(\frac{1}{1 - \alpha}\right)}$
6. Applied pow-exp1.1

$\leadsto \color{blue}{e^{\left(\log \left(1 - \alpha\right) + \log \left(\mathsf{fmax}\left(0, utility\right)\right)\right) \cdot \frac{1}{1 - \alpha}}}$
7. Applied simplify1.1

$\leadsto e^{\color{blue}{\frac{\log \left(1 - \alpha\right) + \log \left(\mathsf{fmax}\left(0, utility\right)\right)}{1 - \alpha}}}$
8. Using strategy rm
9. Applied flip--1.1

$\leadsto e^{\frac{\log \left(1 - \alpha\right) + \log \left(\mathsf{fmax}\left(0, utility\right)\right)}{\color{blue}{\frac{1 \cdot 1 - \alpha \cdot \alpha}{1 + \alpha}}}}$
10. Applied associate-/r/1.1

$\leadsto e^{\color{blue}{\frac{\log \left(1 - \alpha\right) + \log \left(\mathsf{fmax}\left(0, utility\right)\right)}{1 \cdot 1 - \alpha \cdot \alpha} \cdot \left(1 + \alpha\right)}}$
11. Applied exp-prod1.1

$\leadsto \color{blue}{{\left(e^{\frac{\log \left(1 - \alpha\right) + \log \left(\mathsf{fmax}\left(0, utility\right)\right)}{1 \cdot 1 - \alpha \cdot \alpha}}\right)}^{\left(1 + \alpha\right)}}$
12. Applied simplify1.1

$\leadsto {\color{blue}{\left(e^{\frac{\log \left(1 - \alpha\right) + \log \left(\mathsf{fmax}\left(0, utility\right)\right)}{1 - \alpha \cdot \alpha}}\right)}}^{\left(1 + \alpha\right)}$
3. Recombined 2 regimes into one program.

# Runtime

Time bar (total: 28.9s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (alpha utility)
:name "pow((1 - alpha) * fmax(0, utility), 1 / (1 - alpha))"
(pow (* (- 1 alpha) (fmax 0 utility)) (/ 1 (- 1 alpha))))