Rust f64::asinh Percentage Accurate: 29.6% → 99.8%
Time: 5.5s
Alternatives: 12
Speedup: N/A×
Specification ? \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
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? 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: 99.8% accurate, 0.3× speedup? \[\begin{array}{l}
t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\
\mathbf{if}\;t_0 \leq -1:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\
\mathbf{elif}\;t_0 \leq 2 \cdot 10^{-8}:\\
\;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < -1 Initial program 52.4%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation +-commutative52.4%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right)
\]
hypot-1-def100.0%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right)
\]
Simplified100.0%
\[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)}
\]
if -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 2e-8 Initial program 8.9%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity8.9%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative8.9%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod8.9%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt3.5%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr3.5%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt8.9%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative8.9%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def8.9%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval8.9%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 8.9%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity8.9%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified8.9%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around 0 100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)}, x\right)
\]
if 2e-8 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) Initial program 47.7%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity47.7%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative47.7%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod47.7%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def100.0%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval100.0%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Recombined 3 regimes into one program. Final simplification100.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\
\mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 2 \cdot 10^{-8}:\\
\;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\
\end{array}
\]
Alternative 2: 99.8% accurate, 0.4× speedup? \[\begin{array}{l}
t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\
\mathbf{if}\;t_0 \leq -0.01:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\
\mathbf{elif}\;t_0 \leq 2 \cdot 10^{-8}:\\
\;\;\;\;\mathsf{copysign}\left(2 \cdot \left(x \cdot 0.5 + \left({x}^{3} \cdot -0.08333333333333333 + {x}^{5} \cdot 0.0375\right)\right), x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < -0.0100000000000000002 Initial program 53.0%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation +-commutative53.0%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right)
\]
hypot-1-def99.9%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right)
\]
Simplified99.9%
\[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)}
\]
if -0.0100000000000000002 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 2e-8 Initial program 8.3%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation add-sqr-sqrt8.1%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right)
\]
pow28.1%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{2}\right)}, x\right)
\]
log-pow8.2%
\[\leadsto \mathsf{copysign}\left(\color{blue}{2 \cdot \log \left(\sqrt{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right)
\]
add-sqr-sqrt3.5%
\[\leadsto \mathsf{copysign}\left(2 \cdot \log \left(\sqrt{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right), x\right)
\]
fabs-sqr3.5%
\[\leadsto \mathsf{copysign}\left(2 \cdot \log \left(\sqrt{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right), x\right)
\]
add-sqr-sqrt8.2%
\[\leadsto \mathsf{copysign}\left(2 \cdot \log \left(\sqrt{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right), x\right)
\]
+-commutative8.2%
\[\leadsto \mathsf{copysign}\left(2 \cdot \log \left(\sqrt{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right), x\right)
\]
hypot-1-def8.2%
\[\leadsto \mathsf{copysign}\left(2 \cdot \log \left(\sqrt{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right), x\right)
\]
Applied egg-rr 8.2%
\[\leadsto \mathsf{copysign}\left(\color{blue}{2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)}, x\right)
\]
Taylor expanded in x around 0 100.0%
\[\leadsto \mathsf{copysign}\left(2 \cdot \color{blue}{\left(0.5 \cdot x + \left(-0.08333333333333333 \cdot {x}^{3} + 0.0375 \cdot {x}^{5}\right)\right)}, x\right)
\]
if 2e-8 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) Initial program 47.7%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity47.7%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative47.7%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod47.7%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def100.0%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval100.0%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Recombined 3 regimes into one program. Final simplification100.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.01:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\
\mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 2 \cdot 10^{-8}:\\
\;\;\;\;\mathsf{copysign}\left(2 \cdot \left(x \cdot 0.5 + \left({x}^{3} \cdot -0.08333333333333333 + {x}^{5} \cdot 0.0375\right)\right), x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\
\end{array}
\]
Alternative 3: 99.8% accurate, 1.3× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq -1.3:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\
\mathbf{elif}\;x \leq 0.0072:\\
\;\;\;\;\mathsf{copysign}\left(2 \cdot \left(x \cdot 0.5 + \left({x}^{3} \cdot -0.08333333333333333 + {x}^{5} \cdot 0.0375\right)\right), x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if x < -1.30000000000000004 Initial program 51.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity51.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative51.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod51.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt0.0%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr0.0%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around -inf 100.0%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)}, x\right)
\]
if -1.30000000000000004 < x < 0.0071999999999999998 Initial program 9.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation add-sqr-sqrt9.4%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right)
\]
pow29.4%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{2}\right)}, x\right)
\]
log-pow9.4%
\[\leadsto \mathsf{copysign}\left(\color{blue}{2 \cdot \log \left(\sqrt{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right)
\]
add-sqr-sqrt3.5%
\[\leadsto \mathsf{copysign}\left(2 \cdot \log \left(\sqrt{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right), x\right)
\]
fabs-sqr3.5%
\[\leadsto \mathsf{copysign}\left(2 \cdot \log \left(\sqrt{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right), x\right)
\]
add-sqr-sqrt9.6%
\[\leadsto \mathsf{copysign}\left(2 \cdot \log \left(\sqrt{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right), x\right)
\]
+-commutative9.6%
\[\leadsto \mathsf{copysign}\left(2 \cdot \log \left(\sqrt{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right), x\right)
\]
hypot-1-def9.6%
\[\leadsto \mathsf{copysign}\left(2 \cdot \log \left(\sqrt{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right), x\right)
\]
Applied egg-rr 9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)}, x\right)
\]
Taylor expanded in x around 0 99.3%
\[\leadsto \mathsf{copysign}\left(2 \cdot \color{blue}{\left(0.5 \cdot x + \left(-0.08333333333333333 \cdot {x}^{3} + 0.0375 \cdot {x}^{5}\right)\right)}, x\right)
\]
if 0.0071999999999999998 < x Initial program 47.7%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity47.7%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative47.7%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod47.7%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def100.0%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval100.0%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Recombined 3 regimes into one program. Final simplification99.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -1.3:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\
\mathbf{elif}\;x \leq 0.0072:\\
\;\;\;\;\mathsf{copysign}\left(2 \cdot \left(x \cdot 0.5 + \left({x}^{3} \cdot -0.08333333333333333 + {x}^{5} \cdot 0.0375\right)\right), x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\
\end{array}
\]
Alternative 4: 99.8% accurate, 1.3× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq -1.3:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\
\mathbf{elif}\;x \leq 0.0072:\\
\;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if x < -1.30000000000000004 Initial program 51.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity51.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative51.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod51.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt0.0%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr0.0%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around -inf 100.0%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)}, x\right)
\]
if -1.30000000000000004 < x < 0.0071999999999999998 Initial program 9.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity9.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative9.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt3.5%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr3.5%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around 0 99.3%
\[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + x\right)}, x\right)
\]
if 0.0071999999999999998 < x Initial program 47.7%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity47.7%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative47.7%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod47.7%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def100.0%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval100.0%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Recombined 3 regimes into one program. Final simplification99.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -1.3:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\
\mathbf{elif}\;x \leq 0.0072:\\
\;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\
\end{array}
\]
Alternative 5: 99.7% accurate, 1.3× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq -1.25:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\
\mathbf{elif}\;x \leq 0.0011:\\
\;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if x < -1.25 Initial program 51.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity51.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative51.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod51.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt0.0%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr0.0%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around -inf 100.0%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)}, x\right)
\]
if -1.25 < x < 0.00110000000000000007 Initial program 9.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity9.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative9.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt3.5%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr3.5%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around 0 98.9%
\[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + x}, x\right)
\]
if 0.00110000000000000007 < x Initial program 47.7%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity47.7%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative47.7%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod47.7%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative47.7%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def100.0%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval100.0%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified100.0%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Recombined 3 regimes into one program. Final simplification99.4%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -1.25:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\
\mathbf{elif}\;x \leq 0.0011:\\
\;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\
\end{array}
\]
Alternative 6: 99.5% accurate, 1.9× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq -1.25:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\
\mathbf{elif}\;x \leq 0.95:\\
\;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if x < -1.25 Initial program 51.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity51.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative51.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod51.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt0.0%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr0.0%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around -inf 100.0%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)}, x\right)
\]
if -1.25 < x < 0.94999999999999996 Initial program 9.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity9.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative9.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt3.5%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr3.5%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around 0 98.9%
\[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + x}, x\right)
\]
if 0.94999999999999996 < x Initial program 47.7%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Taylor expanded in x around inf 99.7%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0.5 \cdot \frac{1}{x} + \left(\left|x\right| + x\right)\right)}, x\right)
\]
Step-by-step derivation +-commutative99.7%
\[\leadsto \mathsf{copysign}\left(\log \left(0.5 \cdot \frac{1}{x} + \color{blue}{\left(x + \left|x\right|\right)}\right), x\right)
\]
rem-square-sqrt99.7%
\[\leadsto \mathsf{copysign}\left(\log \left(0.5 \cdot \frac{1}{x} + \left(x + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right), x\right)
\]
fabs-sqr99.7%
\[\leadsto \mathsf{copysign}\left(\log \left(0.5 \cdot \frac{1}{x} + \left(x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right)
\]
rem-square-sqrt99.7%
\[\leadsto \mathsf{copysign}\left(\log \left(0.5 \cdot \frac{1}{x} + \left(x + \color{blue}{x}\right)\right), x\right)
\]
associate-*r/99.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + \left(x + x\right)\right), x\right)
\]
metadata-eval99.7%
\[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{0.5}}{x} + \left(x + x\right)\right), x\right)
\]
Simplified99.7%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5}{x} + \left(x + x\right)\right)}, x\right)
\]
Recombined 3 regimes into one program. Final simplification99.4%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -1.25:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\
\mathbf{elif}\;x \leq 0.95:\\
\;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\
\end{array}
\]
Alternative 7: 99.4% accurate, 1.9× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq -1.25:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\
\mathbf{elif}\;x \leq 1.26:\\
\;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if x < -1.25 Initial program 51.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity51.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative51.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod51.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt0.0%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr0.0%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around -inf 100.0%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)}, x\right)
\]
if -1.25 < x < 1.26000000000000001 Initial program 9.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity9.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative9.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt3.5%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr3.5%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around 0 98.9%
\[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + x}, x\right)
\]
if 1.26000000000000001 < x Initial program 47.7%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Taylor expanded in x around inf 99.2%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + x\right)}, x\right)
\]
Step-by-step derivation rem-square-sqrt99.2%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + x\right), x\right)
\]
fabs-sqr99.2%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + x\right), x\right)
\]
rem-square-sqrt99.2%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + x\right), x\right)
\]
Simplified99.2%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x + x\right)}, x\right)
\]
Recombined 3 regimes into one program. Final simplification99.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -1.25:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\
\mathbf{elif}\;x \leq 1.26:\\
\;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\
\end{array}
\]
Alternative 8: 99.3% accurate, 2.0× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq -1.25:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\
\mathbf{elif}\;x \leq 1.26:\\
\;\;\;\;\mathsf{copysign}\left(x, x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if x < -1.25 Initial program 51.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity51.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative51.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod51.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt0.0%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr0.0%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval3.1%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified3.1%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around -inf 100.0%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)}, x\right)
\]
if -1.25 < x < 1.26000000000000001 Initial program 9.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity9.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative9.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt3.5%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr3.5%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval9.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified9.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around 0 98.5%
\[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right)
\]
if 1.26000000000000001 < x Initial program 47.7%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Taylor expanded in x around inf 99.2%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + x\right)}, x\right)
\]
Step-by-step derivation rem-square-sqrt99.2%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + x\right), x\right)
\]
fabs-sqr99.2%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + x\right), x\right)
\]
rem-square-sqrt99.2%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + x\right), x\right)
\]
Simplified99.2%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x + x\right)}, x\right)
\]
Recombined 3 regimes into one program. Final simplification99.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq -1.25:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\
\mathbf{elif}\;x \leq 1.26:\\
\;\;\;\;\mathsf{copysign}\left(x, x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\
\end{array}
\]
Alternative 9: 58.6% accurate, 2.0× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq 1.58:\\
\;\;\;\;\mathsf{copysign}\left(x, x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + 1\right), x\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if x < 1.5800000000000001 Initial program 22.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity22.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative22.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod22.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt2.4%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr2.4%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt7.6%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative7.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def7.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval7.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 7.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity7.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified7.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around 0 69.5%
\[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right)
\]
if 1.5800000000000001 < x Initial program 47.7%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Taylor expanded in x around 0 31.4%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left|x\right|\right)}, x\right)
\]
Step-by-step derivation rem-square-sqrt31.4%
\[\leadsto \mathsf{copysign}\left(\log \left(1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right), x\right)
\]
fabs-sqr31.4%
\[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right), x\right)
\]
rem-square-sqrt31.4%
\[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{x}\right), x\right)
\]
Simplified31.4%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + x\right)}, x\right)
\]
Recombined 2 regimes into one program. Final simplification60.1%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq 1.58:\\
\;\;\;\;\mathsf{copysign}\left(x, x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + 1\right), x\right)\\
\end{array}
\]
Alternative 10: 76.4% accurate, 2.0× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq 1.26:\\
\;\;\;\;\mathsf{copysign}\left(x, x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if x < 1.26000000000000001 Initial program 22.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity22.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative22.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod22.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt2.4%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr2.4%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt7.6%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative7.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def7.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval7.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 7.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity7.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified7.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around 0 69.5%
\[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right)
\]
if 1.26000000000000001 < x Initial program 47.7%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Taylor expanded in x around inf 99.2%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + x\right)}, x\right)
\]
Step-by-step derivation rem-square-sqrt99.2%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + x\right), x\right)
\]
fabs-sqr99.2%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + x\right), x\right)
\]
rem-square-sqrt99.2%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + x\right), x\right)
\]
Simplified99.2%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x + x\right)}, x\right)
\]
Recombined 2 regimes into one program. Final simplification76.8%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq 1.26:\\
\;\;\;\;\mathsf{copysign}\left(x, x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\
\end{array}
\]
Alternative 11: 58.6% accurate, 2.0× speedup? \[\begin{array}{l}
\mathbf{if}\;x \leq 1.58:\\
\;\;\;\;\mathsf{copysign}\left(x, x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if x < 1.5800000000000001 Initial program 22.6%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity22.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative22.6%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod22.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt2.4%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr2.4%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt7.6%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative7.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def7.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval7.6%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 7.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity7.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified7.6%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around 0 69.5%
\[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right)
\]
if 1.5800000000000001 < x Initial program 47.7%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Taylor expanded in x around 0 31.4%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right)
\]
Step-by-step derivation log1p-def31.4%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right)
\]
rem-square-sqrt31.4%
\[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right), x\right)
\]
fabs-sqr31.4%
\[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right), x\right)
\]
rem-square-sqrt31.4%
\[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x}\right), x\right)
\]
Simplified31.4%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right)
\]
Recombined 2 regimes into one program. Final simplification60.1%
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \leq 1.58:\\
\;\;\;\;\mathsf{copysign}\left(x, x\right)\\
\mathbf{else}:\\
\;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\
\end{array}
\]
Alternative 12: 51.8% accurate, 4.0× speedup? \[\mathsf{copysign}\left(x, x\right)
\]
Derivation Initial program 28.8%
\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)
\]
Step-by-step derivation *-un-lft-identity28.8%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right)
\]
*-commutative28.8%
\[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right)
\]
log-prod28.8%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right)
\]
add-sqr-sqrt13.6%
\[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
fabs-sqr13.6%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
add-sqr-sqrt17.5%
\[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right)
\]
+-commutative17.5%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right)
\]
hypot-1-def30.3%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right)
\]
metadata-eval30.3%
\[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right)
\]
Applied egg-rr 30.3%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right)
\]
Step-by-step derivation +-rgt-identity30.3%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Simplified30.3%
\[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right)
\]
Taylor expanded in x around 0 53.7%
\[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right)
\]
Final simplification53.7%
\[\leadsto \mathsf{copysign}\left(x, x\right)
\]
Developer target: 99.9% accurate, 0.6× speedup? \[\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, \frac{1}{\left|x\right|}\right) + \frac{1}{\left|x\right|}}\right), x\right)
\]
Reproduce ? herbie shell --seed 2023167
(FPCore (x)
:name "Rust f64::asinh"
:precision binary64
:herbie-target
(copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)
(copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
