Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B Percentage Accurate: 74.8% → 91.8%
Time: 14.7s
Alternatives: 17
Speedup: 17.0×
Specification ? \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
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? 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: 91.8% accurate, 0.2× speedup? \[\begin{array}{l}
t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\
t_2 := \frac{1}{\frac{b}{z} + \frac{t}{y} \cdot \left(\frac{a + 1}{z} - \frac{x \cdot b}{z \cdot z}\right)}\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;t_2\\
\mathbf{elif}\;t_1 \leq -5 \cdot 10^{-126}:\\
\;\;\;\;t_1\\
\mathbf{elif}\;t_1 \leq 0:\\
\;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\
\mathbf{elif}\;t_1 \leq 5 \cdot 10^{+300}:\\
\;\;\;\;t_1\\
\mathbf{elif}\;t_1 \leq \infty:\\
\;\;\;\;t_2\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b}\\
\end{array}
\]
Derivation Split input into 4 regimes if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -inf.0 or 5.00000000000000026e300 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < +inf.0 Initial program 21.4%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative21.4%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*52.9%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/45.9%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified45.9%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Step-by-step derivation clear-num45.8%
\[\leadsto \color{blue}{\frac{1}{\frac{\left(a + 1\right) + \frac{y}{t} \cdot b}{x + \frac{z}{\frac{t}{y}}}}}
\]
inv-pow45.8%
\[\leadsto \color{blue}{{\left(\frac{\left(a + 1\right) + \frac{y}{t} \cdot b}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}}
\]
+-commutative45.8%
\[\leadsto {\left(\frac{\color{blue}{\frac{y}{t} \cdot b + \left(a + 1\right)}}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}
\]
fma-def45.8%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}
\]
+-commutative45.8%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{\frac{z}{\frac{t}{y}} + x}}\right)}^{-1}
\]
div-inv45.9%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{z \cdot \frac{1}{\frac{t}{y}}} + x}\right)}^{-1}
\]
clear-num45.8%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{z \cdot \color{blue}{\frac{y}{t}} + x}\right)}^{-1}
\]
fma-udef45.8%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}\right)}^{-1}
\]
Applied egg-rr 45.8%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}\right)}^{-1}}
\]
Step-by-step derivation unpow-145.8%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}}
\]
fma-udef45.8%
\[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{t} \cdot b + \left(a + 1\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-+r+45.8%
\[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{y}{t} \cdot b + a\right) + 1}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-*l/52.8%
\[\leadsto \frac{1}{\frac{\left(\color{blue}{\frac{y \cdot b}{t}} + a\right) + 1}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
+-commutative52.8%
\[\leadsto \frac{1}{\frac{\color{blue}{1 + \left(\frac{y \cdot b}{t} + a\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-*r/52.8%
\[\leadsto \frac{1}{\frac{1 + \left(\color{blue}{y \cdot \frac{b}{t}} + a\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
fma-def52.8%
\[\leadsto \frac{1}{\frac{1 + \color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
fma-def52.8%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{z \cdot \frac{y}{t} + x}}}
\]
associate-*r/21.4%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\frac{z \cdot y}{t}} + x}}
\]
associate-*l/52.9%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\frac{z}{t} \cdot y} + x}}
\]
*-commutative52.9%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{y \cdot \frac{z}{t}} + x}}
\]
fma-def52.8%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}}
\]
Simplified52.8%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}}
\]
Taylor expanded in y around inf 51.4%
\[\leadsto \frac{1}{\color{blue}{\left(\frac{b}{z} + \frac{t \cdot \left(1 + a\right)}{y \cdot z}\right) - \frac{t \cdot \left(b \cdot x\right)}{y \cdot {z}^{2}}}}
\]
Step-by-step derivation associate--l+51.4%
\[\leadsto \frac{1}{\color{blue}{\frac{b}{z} + \left(\frac{t \cdot \left(1 + a\right)}{y \cdot z} - \frac{t \cdot \left(b \cdot x\right)}{y \cdot {z}^{2}}\right)}}
\]
times-frac83.1%
\[\leadsto \frac{1}{\frac{b}{z} + \left(\color{blue}{\frac{t}{y} \cdot \frac{1 + a}{z}} - \frac{t \cdot \left(b \cdot x\right)}{y \cdot {z}^{2}}\right)}
\]
times-frac86.4%
\[\leadsto \frac{1}{\frac{b}{z} + \left(\frac{t}{y} \cdot \frac{1 + a}{z} - \color{blue}{\frac{t}{y} \cdot \frac{b \cdot x}{{z}^{2}}}\right)}
\]
distribute-lft-out--86.4%
\[\leadsto \frac{1}{\frac{b}{z} + \color{blue}{\frac{t}{y} \cdot \left(\frac{1 + a}{z} - \frac{b \cdot x}{{z}^{2}}\right)}}
\]
*-commutative86.4%
\[\leadsto \frac{1}{\frac{b}{z} + \frac{t}{y} \cdot \left(\frac{1 + a}{z} - \frac{\color{blue}{x \cdot b}}{{z}^{2}}\right)}
\]
unpow286.4%
\[\leadsto \frac{1}{\frac{b}{z} + \frac{t}{y} \cdot \left(\frac{1 + a}{z} - \frac{x \cdot b}{\color{blue}{z \cdot z}}\right)}
\]
Simplified86.4%
\[\leadsto \frac{1}{\color{blue}{\frac{b}{z} + \frac{t}{y} \cdot \left(\frac{1 + a}{z} - \frac{x \cdot b}{z \cdot z}\right)}}
\]
if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -5.00000000000000006e-126 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 5.00000000000000026e300 Initial program 99.8%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
if -5.00000000000000006e-126 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 0.0 Initial program 67.9%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative67.9%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*67.1%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/79.5%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified79.5%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Step-by-step derivation associate-/r/83.0%
\[\leadsto \frac{x + \color{blue}{\frac{z}{t} \cdot y}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
Applied egg-rr 83.0%
\[\leadsto \frac{x + \color{blue}{\frac{z}{t} \cdot y}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) Initial program 0.0%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative0.0%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*0.5%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/15.3%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified15.3%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in t around 0 97.2%
\[\leadsto \color{blue}{\frac{z}{b}}
\]
Recombined 4 regimes into one program. Final simplification93.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\
\;\;\;\;\frac{1}{\frac{b}{z} + \frac{t}{y} \cdot \left(\frac{a + 1}{z} - \frac{x \cdot b}{z \cdot z}\right)}\\
\mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -5 \cdot 10^{-126}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\
\mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\
\;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\
\mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 5 \cdot 10^{+300}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\
\mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\
\;\;\;\;\frac{1}{\frac{b}{z} + \frac{t}{y} \cdot \left(\frac{a + 1}{z} - \frac{x \cdot b}{z \cdot z}\right)}\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b}\\
\end{array}
\]
Alternative 2: 91.5% accurate, 0.2× speedup? \[\begin{array}{l}
t_1 := \frac{y \cdot b}{t}\\
t_2 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + t_1}\\
t_3 := \frac{y}{t} \cdot \frac{z}{1 + \left(a + t_1\right)}\\
\mathbf{if}\;t_2 \leq -\infty:\\
\;\;\;\;t_3\\
\mathbf{elif}\;t_2 \leq -5 \cdot 10^{-126}:\\
\;\;\;\;t_2\\
\mathbf{elif}\;t_2 \leq 0:\\
\;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\
\mathbf{elif}\;t_2 \leq 5 \cdot 10^{+300}:\\
\;\;\;\;t_2\\
\mathbf{elif}\;t_2 \leq \infty:\\
\;\;\;\;t_3\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b}\\
\end{array}
\]
Derivation Split input into 4 regimes if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -inf.0 or 5.00000000000000026e300 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < +inf.0 Initial program 21.4%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative21.4%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*52.9%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/45.9%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified45.9%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in x around 0 45.6%
\[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(\frac{y \cdot b}{t} + a\right)\right)}}
\]
Step-by-step derivation times-frac83.1%
\[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(\frac{y \cdot b}{t} + a\right)}}
\]
+-commutative83.1%
\[\leadsto \frac{y}{t} \cdot \frac{z}{1 + \color{blue}{\left(a + \frac{y \cdot b}{t}\right)}}
\]
Applied egg-rr 83.1%
\[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{y \cdot b}{t}\right)}}
\]
if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -5.00000000000000006e-126 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 5.00000000000000026e300 Initial program 99.8%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
if -5.00000000000000006e-126 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 0.0 Initial program 67.9%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative67.9%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*67.1%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/79.5%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified79.5%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Step-by-step derivation associate-/r/83.0%
\[\leadsto \frac{x + \color{blue}{\frac{z}{t} \cdot y}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
Applied egg-rr 83.0%
\[\leadsto \frac{x + \color{blue}{\frac{z}{t} \cdot y}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) Initial program 0.0%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative0.0%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*0.5%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/15.3%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified15.3%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in t around 0 97.2%
\[\leadsto \color{blue}{\frac{z}{b}}
\]
Recombined 4 regimes into one program. Final simplification92.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\
\;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\
\mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -5 \cdot 10^{-126}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\
\mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\
\;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\
\mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 5 \cdot 10^{+300}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\
\mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\
\;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b}\\
\end{array}
\]
Alternative 3: 66.8% accurate, 0.7× speedup? \[\begin{array}{l}
t_1 := x + \frac{y \cdot z}{t}\\
t_2 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\
\mathbf{if}\;y \leq -2.2 \cdot 10^{+81}:\\
\;\;\;\;t_2\\
\mathbf{elif}\;y \leq -5.3 \cdot 10^{+39}:\\
\;\;\;\;\frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\
\mathbf{elif}\;y \leq -8 \cdot 10^{-77}:\\
\;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\
\mathbf{elif}\;y \leq 1.5 \cdot 10^{-59}:\\
\;\;\;\;\frac{t_1}{a + 1}\\
\mathbf{elif}\;y \leq 1.6 \cdot 10^{+122}:\\
\;\;\;\;\frac{t_1}{1 + \frac{y \cdot b}{t}}\\
\mathbf{else}:\\
\;\;\;\;t_2\\
\end{array}
\]
Derivation Split input into 5 regimes if y < -2.19999999999999987e81 or 1.60000000000000006e122 < y Initial program 37.4%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative37.4%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*38.9%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/48.1%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified48.1%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in b around inf 28.4%
\[\leadsto \color{blue}{\frac{t \cdot \left(\frac{y \cdot z}{t} + x\right)}{y \cdot b}}
\]
Taylor expanded in t around 0 62.7%
\[\leadsto \color{blue}{\frac{t \cdot x}{y \cdot b} + \frac{z}{b}}
\]
Step-by-step derivation +-commutative62.7%
\[\leadsto \color{blue}{\frac{z}{b} + \frac{t \cdot x}{y \cdot b}}
\]
times-frac67.7%
\[\leadsto \frac{z}{b} + \color{blue}{\frac{t}{y} \cdot \frac{x}{b}}
\]
Simplified67.7%
\[\leadsto \color{blue}{\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}}
\]
if -2.19999999999999987e81 < y < -5.29999999999999979e39 Initial program 52.1%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative52.1%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*64.0%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/87.3%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified87.3%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in x around inf 88.0%
\[\leadsto \frac{\color{blue}{x}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
if -5.29999999999999979e39 < y < -7.9999999999999994e-77 Initial program 82.0%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative82.0%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*82.1%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/82.0%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified82.0%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in x around 0 56.2%
\[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(\frac{y \cdot b}{t} + a\right)\right)}}
\]
Taylor expanded in t around 0 73.5%
\[\leadsto \frac{y \cdot z}{\color{blue}{y \cdot b + t \cdot \left(1 + a\right)}}
\]
if -7.9999999999999994e-77 < y < 1.5e-59 Initial program 96.9%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative96.9%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*97.8%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/97.8%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified97.8%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in b around 0 80.0%
\[\leadsto \color{blue}{\frac{\frac{y \cdot z}{t} + x}{1 + a}}
\]
if 1.5e-59 < y < 1.60000000000000006e122 Initial program 71.8%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation +-commutative71.8%
\[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
*-commutative71.8%
\[\leadsto \frac{\frac{\color{blue}{z \cdot y}}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*r/74.4%
\[\leadsto \frac{\color{blue}{z \cdot \frac{y}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
fma-def74.4%
\[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-+l+74.4%
\[\leadsto \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}}
\]
+-commutative74.4%
\[\leadsto \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \color{blue}{\left(\frac{y \cdot b}{t} + 1\right)}}
\]
associate-*r/74.4%
\[\leadsto \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \left(\color{blue}{y \cdot \frac{b}{t}} + 1\right)}
\]
fma-def74.4%
\[\leadsto \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, 1\right)}}
\]
Simplified74.4%
\[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \mathsf{fma}\left(y, \frac{b}{t}, 1\right)}}
\]
Taylor expanded in a around 0 66.1%
\[\leadsto \color{blue}{\frac{\frac{y \cdot z}{t} + x}{\frac{y \cdot b}{t} + 1}}
\]
Recombined 5 regimes into one program. Final simplification73.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;y \leq -2.2 \cdot 10^{+81}:\\
\;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\
\mathbf{elif}\;y \leq -5.3 \cdot 10^{+39}:\\
\;\;\;\;\frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\
\mathbf{elif}\;y \leq -8 \cdot 10^{-77}:\\
\;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\
\mathbf{elif}\;y \leq 1.5 \cdot 10^{-59}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\
\mathbf{elif}\;y \leq 1.6 \cdot 10^{+122}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + \frac{y \cdot b}{t}}\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\
\end{array}
\]
Alternative 4: 68.3% accurate, 0.7× speedup? \[\begin{array}{l}
\mathbf{if}\;a + 1 \leq -1:\\
\;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a}\\
\mathbf{elif}\;a + 1 \leq 5 \cdot 10^{+27}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + b \cdot \frac{y}{t}}\\
\mathbf{else}:\\
\;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\
\end{array}
\]
Derivation Split input into 3 regimes if (+.f64 a 1) < -1 Initial program 62.6%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative62.6%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*66.5%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/67.7%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified67.7%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in a around inf 51.1%
\[\leadsto \color{blue}{\frac{\frac{y \cdot z}{t} + x}{a}}
\]
Step-by-step derivation *-un-lft-identity51.1%
\[\leadsto \frac{\color{blue}{1 \cdot \frac{y \cdot z}{t}} + x}{a}
\]
associate-/l*59.5%
\[\leadsto \frac{1 \cdot \color{blue}{\frac{y}{\frac{t}{z}}} + x}{a}
\]
Applied egg-rr 59.5%
\[\leadsto \frac{\color{blue}{1 \cdot \frac{y}{\frac{t}{z}}} + x}{a}
\]
if -1 < (+.f64 a 1) < 4.99999999999999979e27 Initial program 72.7%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation +-commutative72.7%
\[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
*-commutative72.7%
\[\leadsto \frac{\frac{\color{blue}{z \cdot y}}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*r/73.3%
\[\leadsto \frac{\color{blue}{z \cdot \frac{y}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
fma-def73.3%
\[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-+l+73.3%
\[\leadsto \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}}
\]
+-commutative73.3%
\[\leadsto \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \color{blue}{\left(\frac{y \cdot b}{t} + 1\right)}}
\]
associate-*r/76.1%
\[\leadsto \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \left(\color{blue}{y \cdot \frac{b}{t}} + 1\right)}
\]
fma-def76.1%
\[\leadsto \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, 1\right)}}
\]
Simplified76.1%
\[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \mathsf{fma}\left(y, \frac{b}{t}, 1\right)}}
\]
Taylor expanded in a around 0 71.6%
\[\leadsto \color{blue}{\frac{\frac{y \cdot z}{t} + x}{\frac{y \cdot b}{t} + 1}}
\]
Step-by-step derivation associate-*l/74.5%
\[\leadsto \frac{\frac{y \cdot z}{t} + x}{\color{blue}{\frac{y}{t} \cdot b} + 1}
\]
Applied egg-rr 74.5%
\[\leadsto \frac{\frac{y \cdot z}{t} + x}{\color{blue}{\frac{y}{t} \cdot b} + 1}
\]
if 4.99999999999999979e27 < (+.f64 a 1) Initial program 67.5%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative67.5%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*69.0%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/72.7%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified72.7%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in b around 0 60.0%
\[\leadsto \color{blue}{\frac{\frac{y \cdot z}{t} + x}{1 + a}}
\]
Taylor expanded in a around inf 60.0%
\[\leadsto \color{blue}{\frac{\frac{y \cdot z}{t} + x}{a}}
\]
Step-by-step derivation associate-/l*61.7%
\[\leadsto \frac{\color{blue}{\frac{y}{\frac{t}{z}}} + x}{a}
\]
+-commutative61.7%
\[\leadsto \frac{\color{blue}{x + \frac{y}{\frac{t}{z}}}}{a}
\]
associate-/r/63.4%
\[\leadsto \frac{x + \color{blue}{\frac{y}{t} \cdot z}}{a}
\]
Simplified63.4%
\[\leadsto \color{blue}{\frac{x + \frac{y}{t} \cdot z}{a}}
\]
Recombined 3 regimes into one program. Final simplification68.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;a + 1 \leq -1:\\
\;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a}\\
\mathbf{elif}\;a + 1 \leq 5 \cdot 10^{+27}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + b \cdot \frac{y}{t}}\\
\mathbf{else}:\\
\;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\
\end{array}
\]
Alternative 5: 81.0% accurate, 0.8× speedup? \[\begin{array}{l}
\mathbf{if}\;y \leq -1.5 \cdot 10^{+83} \lor \neg \left(y \leq 4.4 \cdot 10^{+140}\right):\\
\;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\
\mathbf{else}:\\
\;\;\;\;\frac{x + z \cdot \frac{y}{t}}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\
\end{array}
\]
Derivation Split input into 2 regimes if y < -1.5e83 or 4.3999999999999997e140 < y Initial program 36.8%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative36.8%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*38.3%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/46.9%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified46.9%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in b around inf 28.5%
\[\leadsto \color{blue}{\frac{t \cdot \left(\frac{y \cdot z}{t} + x\right)}{y \cdot b}}
\]
Taylor expanded in t around 0 63.3%
\[\leadsto \color{blue}{\frac{t \cdot x}{y \cdot b} + \frac{z}{b}}
\]
Step-by-step derivation +-commutative63.3%
\[\leadsto \color{blue}{\frac{z}{b} + \frac{t \cdot x}{y \cdot b}}
\]
times-frac67.4%
\[\leadsto \frac{z}{b} + \color{blue}{\frac{t}{y} \cdot \frac{x}{b}}
\]
Simplified67.4%
\[\leadsto \color{blue}{\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}}
\]
if -1.5e83 < y < 4.3999999999999997e140 Initial program 86.2%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative86.2%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*87.8%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/90.1%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified90.1%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Step-by-step derivation div-inv90.1%
\[\leadsto \frac{x + \color{blue}{z \cdot \frac{1}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
clear-num90.1%
\[\leadsto \frac{x + z \cdot \color{blue}{\frac{y}{t}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
Applied egg-rr 90.1%
\[\leadsto \frac{x + \color{blue}{z \cdot \frac{y}{t}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
Recombined 2 regimes into one program. Final simplification82.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;y \leq -1.5 \cdot 10^{+83} \lor \neg \left(y \leq 4.4 \cdot 10^{+140}\right):\\
\;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\
\mathbf{else}:\\
\;\;\;\;\frac{x + z \cdot \frac{y}{t}}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\
\end{array}
\]
Alternative 6: 53.2% accurate, 0.9× speedup? \[\begin{array}{l}
t_1 := \frac{x}{a + 1}\\
\mathbf{if}\;t \leq -1.35 \cdot 10^{-19}:\\
\;\;\;\;t_1\\
\mathbf{elif}\;t \leq -6.2 \cdot 10^{-189}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{elif}\;t \leq -3.2 \cdot 10^{-217}:\\
\;\;\;\;\frac{x}{1 + \frac{y \cdot b}{t}}\\
\mathbf{elif}\;t \leq 8.2 \cdot 10^{-31}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{elif}\;t \leq 2.1 \cdot 10^{+151}:\\
\;\;\;\;\frac{x}{1 + \frac{y}{\frac{t}{b}}}\\
\mathbf{else}:\\
\;\;\;\;t_1\\
\end{array}
\]
Derivation Split input into 4 regimes if t < -1.35e-19 or 2.1000000000000001e151 < t Initial program 75.5%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative75.5%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*82.4%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/94.6%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified94.6%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in t around inf 65.1%
\[\leadsto \color{blue}{\frac{x}{1 + a}}
\]
if -1.35e-19 < t < -6.2000000000000001e-189 or -3.2000000000000001e-217 < t < 8.1999999999999993e-31 Initial program 56.0%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative56.0%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*50.7%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/48.5%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified48.5%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in t around 0 59.5%
\[\leadsto \color{blue}{\frac{z}{b}}
\]
if -6.2000000000000001e-189 < t < -3.2000000000000001e-217 Initial program 89.2%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation +-commutative89.2%
\[\leadsto \frac{\color{blue}{\frac{y \cdot z}{t} + x}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
*-commutative89.2%
\[\leadsto \frac{\frac{\color{blue}{z \cdot y}}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*r/99.7%
\[\leadsto \frac{\color{blue}{z \cdot \frac{y}{t}} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
fma-def99.7%
\[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-+l+99.7%
\[\leadsto \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}}
\]
+-commutative99.7%
\[\leadsto \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \color{blue}{\left(\frac{y \cdot b}{t} + 1\right)}}
\]
associate-*r/89.0%
\[\leadsto \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \left(\color{blue}{y \cdot \frac{b}{t}} + 1\right)}
\]
fma-def89.0%
\[\leadsto \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, 1\right)}}
\]
Simplified89.0%
\[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \mathsf{fma}\left(y, \frac{b}{t}, 1\right)}}
\]
Taylor expanded in a around 0 78.5%
\[\leadsto \color{blue}{\frac{\frac{y \cdot z}{t} + x}{\frac{y \cdot b}{t} + 1}}
\]
Taylor expanded in z around 0 67.9%
\[\leadsto \color{blue}{\frac{x}{\frac{y \cdot b}{t} + 1}}
\]
if 8.1999999999999993e-31 < t < 2.1000000000000001e151 Initial program 80.7%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative80.7%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*83.4%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/83.4%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified83.4%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in x around inf 63.3%
\[\leadsto \frac{\color{blue}{x}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
Step-by-step derivation div-inv63.2%
\[\leadsto \color{blue}{x \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
associate-+l+63.2%
\[\leadsto x \cdot \frac{1}{\color{blue}{a + \left(1 + \frac{y}{t} \cdot b\right)}}
\]
associate-*l/63.1%
\[\leadsto x \cdot \frac{1}{a + \left(1 + \color{blue}{\frac{y \cdot b}{t}}\right)}
\]
+-commutative63.1%
\[\leadsto x \cdot \frac{1}{a + \color{blue}{\left(\frac{y \cdot b}{t} + 1\right)}}
\]
associate-*l/63.2%
\[\leadsto x \cdot \frac{1}{a + \left(\color{blue}{\frac{y}{t} \cdot b} + 1\right)}
\]
fma-def63.2%
\[\leadsto x \cdot \frac{1}{a + \color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, 1\right)}}
\]
Applied egg-rr 63.2%
\[\leadsto \color{blue}{x \cdot \frac{1}{a + \mathsf{fma}\left(\frac{y}{t}, b, 1\right)}}
\]
Taylor expanded in a around 0 51.0%
\[\leadsto \color{blue}{\frac{x}{\frac{y \cdot b}{t} + 1}}
\]
Step-by-step derivation +-commutative51.0%
\[\leadsto \frac{x}{\color{blue}{1 + \frac{y \cdot b}{t}}}
\]
associate-/l*51.0%
\[\leadsto \frac{x}{1 + \color{blue}{\frac{y}{\frac{t}{b}}}}
\]
Simplified51.0%
\[\leadsto \color{blue}{\frac{x}{1 + \frac{y}{\frac{t}{b}}}}
\]
Recombined 4 regimes into one program. Final simplification61.1%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -1.35 \cdot 10^{-19}:\\
\;\;\;\;\frac{x}{a + 1}\\
\mathbf{elif}\;t \leq -6.2 \cdot 10^{-189}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{elif}\;t \leq -3.2 \cdot 10^{-217}:\\
\;\;\;\;\frac{x}{1 + \frac{y \cdot b}{t}}\\
\mathbf{elif}\;t \leq 8.2 \cdot 10^{-31}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{elif}\;t \leq 2.1 \cdot 10^{+151}:\\
\;\;\;\;\frac{x}{1 + \frac{y}{\frac{t}{b}}}\\
\mathbf{else}:\\
\;\;\;\;\frac{x}{a + 1}\\
\end{array}
\]
Alternative 7: 64.1% accurate, 0.9× speedup? \[\begin{array}{l}
t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\
t_2 := \frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\
\mathbf{if}\;t \leq -1.6 \cdot 10^{-20}:\\
\;\;\;\;t_2\\
\mathbf{elif}\;t \leq -5.6 \cdot 10^{-102}:\\
\;\;\;\;t_1\\
\mathbf{elif}\;t \leq -2.6 \cdot 10^{-164}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\
\mathbf{elif}\;t \leq 1.8 \cdot 10^{-130}:\\
\;\;\;\;t_1\\
\mathbf{else}:\\
\;\;\;\;t_2\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -1.59999999999999985e-20 or 1.8000000000000001e-130 < t Initial program 75.4%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative75.4%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*80.0%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/88.2%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified88.2%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in x around inf 69.0%
\[\leadsto \frac{\color{blue}{x}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
if -1.59999999999999985e-20 < t < -5.60000000000000026e-102 or -2.6000000000000002e-164 < t < 1.8000000000000001e-130 Initial program 53.9%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative53.9%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*49.9%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/47.3%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified47.3%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in b around inf 44.5%
\[\leadsto \color{blue}{\frac{t \cdot \left(\frac{y \cdot z}{t} + x\right)}{y \cdot b}}
\]
Taylor expanded in t around 0 78.5%
\[\leadsto \color{blue}{\frac{t \cdot x}{y \cdot b} + \frac{z}{b}}
\]
Step-by-step derivation +-commutative78.5%
\[\leadsto \color{blue}{\frac{z}{b} + \frac{t \cdot x}{y \cdot b}}
\]
times-frac72.3%
\[\leadsto \frac{z}{b} + \color{blue}{\frac{t}{y} \cdot \frac{x}{b}}
\]
Simplified72.3%
\[\leadsto \color{blue}{\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}}
\]
if -5.60000000000000026e-102 < t < -2.6000000000000002e-164 Initial program 92.4%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative92.4%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*92.2%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/92.2%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified92.2%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in b around 0 76.7%
\[\leadsto \color{blue}{\frac{\frac{y \cdot z}{t} + x}{1 + a}}
\]
Recombined 3 regimes into one program. Final simplification70.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -1.6 \cdot 10^{-20}:\\
\;\;\;\;\frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\
\mathbf{elif}\;t \leq -5.6 \cdot 10^{-102}:\\
\;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\
\mathbf{elif}\;t \leq -2.6 \cdot 10^{-164}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\
\mathbf{elif}\;t \leq 1.8 \cdot 10^{-130}:\\
\;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\
\mathbf{else}:\\
\;\;\;\;\frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\
\end{array}
\]
Alternative 8: 54.0% accurate, 1.1× speedup? \[\begin{array}{l}
t_1 := \frac{x}{a + 1}\\
\mathbf{if}\;t \leq -1.02 \cdot 10^{-19}:\\
\;\;\;\;t_1\\
\mathbf{elif}\;t \leq 6.5 \cdot 10^{-29}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{elif}\;t \leq 3.4 \cdot 10^{+151}:\\
\;\;\;\;\frac{x}{1 + \frac{y}{\frac{t}{b}}}\\
\mathbf{else}:\\
\;\;\;\;t_1\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -1.02000000000000004e-19 or 3.4e151 < t Initial program 75.5%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative75.5%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*82.4%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/94.6%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified94.6%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in t around inf 65.1%
\[\leadsto \color{blue}{\frac{x}{1 + a}}
\]
if -1.02000000000000004e-19 < t < 6.5e-29 Initial program 58.7%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative58.7%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*54.7%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/52.7%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified52.7%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in t around 0 56.0%
\[\leadsto \color{blue}{\frac{z}{b}}
\]
if 6.5e-29 < t < 3.4e151 Initial program 80.7%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative80.7%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*83.4%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/83.4%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified83.4%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in x around inf 63.3%
\[\leadsto \frac{\color{blue}{x}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
Step-by-step derivation div-inv63.2%
\[\leadsto \color{blue}{x \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
associate-+l+63.2%
\[\leadsto x \cdot \frac{1}{\color{blue}{a + \left(1 + \frac{y}{t} \cdot b\right)}}
\]
associate-*l/63.1%
\[\leadsto x \cdot \frac{1}{a + \left(1 + \color{blue}{\frac{y \cdot b}{t}}\right)}
\]
+-commutative63.1%
\[\leadsto x \cdot \frac{1}{a + \color{blue}{\left(\frac{y \cdot b}{t} + 1\right)}}
\]
associate-*l/63.2%
\[\leadsto x \cdot \frac{1}{a + \left(\color{blue}{\frac{y}{t} \cdot b} + 1\right)}
\]
fma-def63.2%
\[\leadsto x \cdot \frac{1}{a + \color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, 1\right)}}
\]
Applied egg-rr 63.2%
\[\leadsto \color{blue}{x \cdot \frac{1}{a + \mathsf{fma}\left(\frac{y}{t}, b, 1\right)}}
\]
Taylor expanded in a around 0 51.0%
\[\leadsto \color{blue}{\frac{x}{\frac{y \cdot b}{t} + 1}}
\]
Step-by-step derivation +-commutative51.0%
\[\leadsto \frac{x}{\color{blue}{1 + \frac{y \cdot b}{t}}}
\]
associate-/l*51.0%
\[\leadsto \frac{x}{1 + \color{blue}{\frac{y}{\frac{t}{b}}}}
\]
Simplified51.0%
\[\leadsto \color{blue}{\frac{x}{1 + \frac{y}{\frac{t}{b}}}}
\]
Recombined 3 regimes into one program. Final simplification59.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -1.02 \cdot 10^{-19}:\\
\;\;\;\;\frac{x}{a + 1}\\
\mathbf{elif}\;t \leq 6.5 \cdot 10^{-29}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{elif}\;t \leq 3.4 \cdot 10^{+151}:\\
\;\;\;\;\frac{x}{1 + \frac{y}{\frac{t}{b}}}\\
\mathbf{else}:\\
\;\;\;\;\frac{x}{a + 1}\\
\end{array}
\]
Alternative 9: 54.6% accurate, 1.1× speedup? \[\begin{array}{l}
t_1 := \frac{x + z \cdot \frac{y}{t}}{a}\\
\mathbf{if}\;a \leq -2.75:\\
\;\;\;\;t_1\\
\mathbf{elif}\;a \leq -2.05 \cdot 10^{-190}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{elif}\;a \leq 4 \cdot 10^{+27}:\\
\;\;\;\;\frac{x}{1 + \frac{y}{\frac{t}{b}}}\\
\mathbf{else}:\\
\;\;\;\;t_1\\
\end{array}
\]
Derivation Split input into 3 regimes if a < -2.75 or 4.0000000000000001e27 < a Initial program 64.8%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative64.8%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*67.6%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/69.9%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified69.9%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in b around 0 55.9%
\[\leadsto \color{blue}{\frac{\frac{y \cdot z}{t} + x}{1 + a}}
\]
Taylor expanded in a around inf 55.0%
\[\leadsto \color{blue}{\frac{\frac{y \cdot z}{t} + x}{a}}
\]
Step-by-step derivation associate-/l*60.5%
\[\leadsto \frac{\color{blue}{\frac{y}{\frac{t}{z}}} + x}{a}
\]
+-commutative60.5%
\[\leadsto \frac{\color{blue}{x + \frac{y}{\frac{t}{z}}}}{a}
\]
associate-/r/61.2%
\[\leadsto \frac{x + \color{blue}{\frac{y}{t} \cdot z}}{a}
\]
Simplified61.2%
\[\leadsto \color{blue}{\frac{x + \frac{y}{t} \cdot z}{a}}
\]
if -2.75 < a < -2.0500000000000001e-190 Initial program 75.4%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative75.4%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*73.0%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/77.6%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified77.6%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in t around 0 54.1%
\[\leadsto \color{blue}{\frac{z}{b}}
\]
if -2.0500000000000001e-190 < a < 4.0000000000000001e27 Initial program 71.6%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative71.6%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*73.4%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/80.4%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified80.4%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in x around inf 63.2%
\[\leadsto \frac{\color{blue}{x}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
Step-by-step derivation div-inv63.1%
\[\leadsto \color{blue}{x \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
associate-+l+63.1%
\[\leadsto x \cdot \frac{1}{\color{blue}{a + \left(1 + \frac{y}{t} \cdot b\right)}}
\]
associate-*l/59.8%
\[\leadsto x \cdot \frac{1}{a + \left(1 + \color{blue}{\frac{y \cdot b}{t}}\right)}
\]
+-commutative59.8%
\[\leadsto x \cdot \frac{1}{a + \color{blue}{\left(\frac{y \cdot b}{t} + 1\right)}}
\]
associate-*l/63.1%
\[\leadsto x \cdot \frac{1}{a + \left(\color{blue}{\frac{y}{t} \cdot b} + 1\right)}
\]
fma-def63.1%
\[\leadsto x \cdot \frac{1}{a + \color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, 1\right)}}
\]
Applied egg-rr 63.1%
\[\leadsto \color{blue}{x \cdot \frac{1}{a + \mathsf{fma}\left(\frac{y}{t}, b, 1\right)}}
\]
Taylor expanded in a around 0 59.4%
\[\leadsto \color{blue}{\frac{x}{\frac{y \cdot b}{t} + 1}}
\]
Step-by-step derivation +-commutative59.4%
\[\leadsto \frac{x}{\color{blue}{1 + \frac{y \cdot b}{t}}}
\]
associate-/l*59.7%
\[\leadsto \frac{x}{1 + \color{blue}{\frac{y}{\frac{t}{b}}}}
\]
Simplified59.7%
\[\leadsto \color{blue}{\frac{x}{1 + \frac{y}{\frac{t}{b}}}}
\]
Recombined 3 regimes into one program. Final simplification59.5%
\[\leadsto \begin{array}{l}
\mathbf{if}\;a \leq -2.75:\\
\;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\
\mathbf{elif}\;a \leq -2.05 \cdot 10^{-190}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{elif}\;a \leq 4 \cdot 10^{+27}:\\
\;\;\;\;\frac{x}{1 + \frac{y}{\frac{t}{b}}}\\
\mathbf{else}:\\
\;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\
\end{array}
\]
Alternative 10: 57.9% accurate, 1.1× speedup? \[\begin{array}{l}
\mathbf{if}\;t \leq -6.5 \cdot 10^{-19} \lor \neg \left(t \leq 8.6 \cdot 10^{+114}\right):\\
\;\;\;\;\frac{x}{a + 1}\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\
\end{array}
\]
Derivation Split input into 2 regimes if t < -6.5000000000000001e-19 or 8.6000000000000001e114 < t Initial program 77.3%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative77.3%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*83.7%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/95.0%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified95.0%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in t around inf 63.8%
\[\leadsto \color{blue}{\frac{x}{1 + a}}
\]
if -6.5000000000000001e-19 < t < 8.6000000000000001e114 Initial program 61.7%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative61.7%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*59.1%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/57.5%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified57.5%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in b around inf 37.5%
\[\leadsto \color{blue}{\frac{t \cdot \left(\frac{y \cdot z}{t} + x\right)}{y \cdot b}}
\]
Taylor expanded in t around 0 62.7%
\[\leadsto \color{blue}{\frac{t \cdot x}{y \cdot b} + \frac{z}{b}}
\]
Step-by-step derivation +-commutative62.7%
\[\leadsto \color{blue}{\frac{z}{b} + \frac{t \cdot x}{y \cdot b}}
\]
times-frac58.9%
\[\leadsto \frac{z}{b} + \color{blue}{\frac{t}{y} \cdot \frac{x}{b}}
\]
Simplified58.9%
\[\leadsto \color{blue}{\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}}
\]
Recombined 2 regimes into one program. Final simplification61.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -6.5 \cdot 10^{-19} \lor \neg \left(t \leq 8.6 \cdot 10^{+114}\right):\\
\;\;\;\;\frac{x}{a + 1}\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\
\end{array}
\]
Alternative 11: 65.2% accurate, 1.1× speedup? \[\begin{array}{l}
\mathbf{if}\;y \leq -8.4 \cdot 10^{+21} \lor \neg \left(y \leq 1.45 \cdot 10^{+93}\right):\\
\;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\
\mathbf{else}:\\
\;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\
\end{array}
\]
Derivation Split input into 2 regimes if y < -8.4e21 or 1.4499999999999999e93 < y Initial program 41.1%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative41.1%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*43.2%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/52.6%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified52.6%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in b around inf 28.5%
\[\leadsto \color{blue}{\frac{t \cdot \left(\frac{y \cdot z}{t} + x\right)}{y \cdot b}}
\]
Taylor expanded in t around 0 61.3%
\[\leadsto \color{blue}{\frac{t \cdot x}{y \cdot b} + \frac{z}{b}}
\]
Step-by-step derivation +-commutative61.3%
\[\leadsto \color{blue}{\frac{z}{b} + \frac{t \cdot x}{y \cdot b}}
\]
times-frac65.5%
\[\leadsto \frac{z}{b} + \color{blue}{\frac{t}{y} \cdot \frac{x}{b}}
\]
Simplified65.5%
\[\leadsto \color{blue}{\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}}
\]
if -8.4e21 < y < 1.4499999999999999e93 Initial program 90.7%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative90.7%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*91.9%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/92.6%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified92.6%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in x around inf 70.2%
\[\leadsto \color{blue}{\frac{x}{1 + \left(\frac{y \cdot b}{t} + a\right)}}
\]
Recombined 2 regimes into one program. Final simplification68.1%
\[\leadsto \begin{array}{l}
\mathbf{if}\;y \leq -8.4 \cdot 10^{+21} \lor \neg \left(y \leq 1.45 \cdot 10^{+93}\right):\\
\;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\
\mathbf{else}:\\
\;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\
\end{array}
\]
Alternative 12: 64.5% accurate, 1.1× speedup? \[\begin{array}{l}
\mathbf{if}\;t \leq -2.5 \cdot 10^{-20} \lor \neg \left(t \leq 1.8 \cdot 10^{-130}\right):\\
\;\;\;\;\frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\
\end{array}
\]
Derivation Split input into 2 regimes if t < -2.4999999999999999e-20 or 1.8000000000000001e-130 < t Initial program 75.4%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative75.4%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*80.0%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/88.2%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified88.2%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in x around inf 69.0%
\[\leadsto \frac{\color{blue}{x}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
if -2.4999999999999999e-20 < t < 1.8000000000000001e-130 Initial program 57.3%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative57.3%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*53.6%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/51.2%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified51.2%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in b around inf 42.9%
\[\leadsto \color{blue}{\frac{t \cdot \left(\frac{y \cdot z}{t} + x\right)}{y \cdot b}}
\]
Taylor expanded in t around 0 73.9%
\[\leadsto \color{blue}{\frac{t \cdot x}{y \cdot b} + \frac{z}{b}}
\]
Step-by-step derivation +-commutative73.9%
\[\leadsto \color{blue}{\frac{z}{b} + \frac{t \cdot x}{y \cdot b}}
\]
times-frac68.2%
\[\leadsto \frac{z}{b} + \color{blue}{\frac{t}{y} \cdot \frac{x}{b}}
\]
Simplified68.2%
\[\leadsto \color{blue}{\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}}
\]
Recombined 2 regimes into one program. Final simplification68.7%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -2.5 \cdot 10^{-20} \lor \neg \left(t \leq 1.8 \cdot 10^{-130}\right):\\
\;\;\;\;\frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\
\end{array}
\]
Alternative 13: 40.8% accurate, 1.5× speedup? \[\begin{array}{l}
\mathbf{if}\;y \leq -6.4 \cdot 10^{-77}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{elif}\;y \leq 7 \cdot 10^{-257}:\\
\;\;\;\;\frac{x}{a}\\
\mathbf{elif}\;y \leq 1.05 \cdot 10^{+45}:\\
\;\;\;\;x - x \cdot a\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b}\\
\end{array}
\]
Derivation Split input into 3 regimes if y < -6.39999999999999999e-77 or 1.04999999999999997e45 < y Initial program 48.2%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative48.2%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*50.5%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/58.4%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified58.4%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in t around 0 50.5%
\[\leadsto \color{blue}{\frac{z}{b}}
\]
if -6.39999999999999999e-77 < y < 7.00000000000000058e-257 Initial program 96.9%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative96.9%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*98.2%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/98.2%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified98.2%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in x around inf 77.3%
\[\leadsto \frac{\color{blue}{x}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
Taylor expanded in a around inf 44.0%
\[\leadsto \color{blue}{\frac{x}{a}}
\]
if 7.00000000000000058e-257 < y < 1.04999999999999997e45 Initial program 94.2%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative94.2%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*94.2%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/94.2%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified94.2%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Step-by-step derivation clear-num94.0%
\[\leadsto \color{blue}{\frac{1}{\frac{\left(a + 1\right) + \frac{y}{t} \cdot b}{x + \frac{z}{\frac{t}{y}}}}}
\]
inv-pow94.0%
\[\leadsto \color{blue}{{\left(\frac{\left(a + 1\right) + \frac{y}{t} \cdot b}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}}
\]
+-commutative94.0%
\[\leadsto {\left(\frac{\color{blue}{\frac{y}{t} \cdot b + \left(a + 1\right)}}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}
\]
fma-def94.0%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}
\]
+-commutative94.0%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{\frac{z}{\frac{t}{y}} + x}}\right)}^{-1}
\]
div-inv94.0%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{z \cdot \frac{1}{\frac{t}{y}}} + x}\right)}^{-1}
\]
clear-num94.0%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{z \cdot \color{blue}{\frac{y}{t}} + x}\right)}^{-1}
\]
fma-udef94.0%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}\right)}^{-1}
\]
Applied egg-rr 94.0%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}\right)}^{-1}}
\]
Step-by-step derivation unpow-194.0%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}}
\]
fma-udef94.0%
\[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{t} \cdot b + \left(a + 1\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-+r+94.0%
\[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{y}{t} \cdot b + a\right) + 1}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-*l/94.0%
\[\leadsto \frac{1}{\frac{\left(\color{blue}{\frac{y \cdot b}{t}} + a\right) + 1}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
+-commutative94.0%
\[\leadsto \frac{1}{\frac{\color{blue}{1 + \left(\frac{y \cdot b}{t} + a\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-*r/88.2%
\[\leadsto \frac{1}{\frac{1 + \left(\color{blue}{y \cdot \frac{b}{t}} + a\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
fma-def88.2%
\[\leadsto \frac{1}{\frac{1 + \color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
fma-def88.2%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{z \cdot \frac{y}{t} + x}}}
\]
associate-*r/88.2%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\frac{z \cdot y}{t}} + x}}
\]
associate-*l/84.2%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\frac{z}{t} \cdot y} + x}}
\]
*-commutative84.2%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{y \cdot \frac{z}{t}} + x}}
\]
fma-def84.2%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}}
\]
Simplified84.2%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}}
\]
Taylor expanded in y around 0 56.3%
\[\leadsto \frac{1}{\color{blue}{\frac{1 + a}{x}}}
\]
Taylor expanded in a around 0 38.4%
\[\leadsto \color{blue}{-1 \cdot \left(a \cdot x\right) + x}
\]
Step-by-step derivation +-commutative38.4%
\[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot x\right)}
\]
mul-1-neg38.4%
\[\leadsto x + \color{blue}{\left(-a \cdot x\right)}
\]
unsub-neg38.4%
\[\leadsto \color{blue}{x - a \cdot x}
\]
*-commutative38.4%
\[\leadsto x - \color{blue}{x \cdot a}
\]
Simplified38.4%
\[\leadsto \color{blue}{x - x \cdot a}
\]
Recombined 3 regimes into one program. Final simplification46.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;y \leq -6.4 \cdot 10^{-77}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{elif}\;y \leq 7 \cdot 10^{-257}:\\
\;\;\;\;\frac{x}{a}\\
\mathbf{elif}\;y \leq 1.05 \cdot 10^{+45}:\\
\;\;\;\;x - x \cdot a\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b}\\
\end{array}
\]
Alternative 14: 41.0% accurate, 1.9× speedup? \[\begin{array}{l}
\mathbf{if}\;y \leq -8 \cdot 10^{-77}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{elif}\;y \leq 4.5 \cdot 10^{-253}:\\
\;\;\;\;\frac{x}{a}\\
\mathbf{elif}\;y \leq 5.5 \cdot 10^{+43}:\\
\;\;\;\;x\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b}\\
\end{array}
\]
Derivation Split input into 3 regimes if y < -7.9999999999999994e-77 or 5.49999999999999989e43 < y Initial program 48.2%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative48.2%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*50.5%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/58.4%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified58.4%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in t around 0 50.5%
\[\leadsto \color{blue}{\frac{z}{b}}
\]
if -7.9999999999999994e-77 < y < 4.50000000000000029e-253 Initial program 96.9%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative96.9%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*98.2%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/98.2%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified98.2%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in x around inf 77.3%
\[\leadsto \frac{\color{blue}{x}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
Taylor expanded in a around inf 44.0%
\[\leadsto \color{blue}{\frac{x}{a}}
\]
if 4.50000000000000029e-253 < y < 5.49999999999999989e43 Initial program 94.2%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative94.2%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*94.2%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/94.2%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified94.2%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Step-by-step derivation clear-num94.0%
\[\leadsto \color{blue}{\frac{1}{\frac{\left(a + 1\right) + \frac{y}{t} \cdot b}{x + \frac{z}{\frac{t}{y}}}}}
\]
inv-pow94.0%
\[\leadsto \color{blue}{{\left(\frac{\left(a + 1\right) + \frac{y}{t} \cdot b}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}}
\]
+-commutative94.0%
\[\leadsto {\left(\frac{\color{blue}{\frac{y}{t} \cdot b + \left(a + 1\right)}}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}
\]
fma-def94.0%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}
\]
+-commutative94.0%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{\frac{z}{\frac{t}{y}} + x}}\right)}^{-1}
\]
div-inv94.0%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{z \cdot \frac{1}{\frac{t}{y}}} + x}\right)}^{-1}
\]
clear-num94.0%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{z \cdot \color{blue}{\frac{y}{t}} + x}\right)}^{-1}
\]
fma-udef94.0%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}\right)}^{-1}
\]
Applied egg-rr 94.0%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}\right)}^{-1}}
\]
Step-by-step derivation unpow-194.0%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}}
\]
fma-udef94.0%
\[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{t} \cdot b + \left(a + 1\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-+r+94.0%
\[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{y}{t} \cdot b + a\right) + 1}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-*l/94.0%
\[\leadsto \frac{1}{\frac{\left(\color{blue}{\frac{y \cdot b}{t}} + a\right) + 1}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
+-commutative94.0%
\[\leadsto \frac{1}{\frac{\color{blue}{1 + \left(\frac{y \cdot b}{t} + a\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-*r/88.2%
\[\leadsto \frac{1}{\frac{1 + \left(\color{blue}{y \cdot \frac{b}{t}} + a\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
fma-def88.2%
\[\leadsto \frac{1}{\frac{1 + \color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
fma-def88.2%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{z \cdot \frac{y}{t} + x}}}
\]
associate-*r/88.2%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\frac{z \cdot y}{t}} + x}}
\]
associate-*l/84.2%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\frac{z}{t} \cdot y} + x}}
\]
*-commutative84.2%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{y \cdot \frac{z}{t}} + x}}
\]
fma-def84.2%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}}
\]
Simplified84.2%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}}
\]
Taylor expanded in y around 0 56.3%
\[\leadsto \frac{1}{\color{blue}{\frac{1 + a}{x}}}
\]
Taylor expanded in a around 0 38.3%
\[\leadsto \color{blue}{x}
\]
Recombined 3 regimes into one program. Final simplification46.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;y \leq -8 \cdot 10^{-77}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{elif}\;y \leq 4.5 \cdot 10^{-253}:\\
\;\;\;\;\frac{x}{a}\\
\mathbf{elif}\;y \leq 5.5 \cdot 10^{+43}:\\
\;\;\;\;x\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b}\\
\end{array}
\]
Alternative 15: 56.0% accurate, 1.9× speedup? \[\begin{array}{l}
\mathbf{if}\;y \leq -1.15 \cdot 10^{+81}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{elif}\;y \leq 3.2 \cdot 10^{+97}:\\
\;\;\;\;\frac{x}{a + 1}\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b}\\
\end{array}
\]
Derivation Split input into 2 regimes if y < -1.1499999999999999e81 or 3.20000000000000016e97 < y Initial program 38.4%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative38.4%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*39.8%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/49.4%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified49.4%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in t around 0 55.4%
\[\leadsto \color{blue}{\frac{z}{b}}
\]
if -1.1499999999999999e81 < y < 3.20000000000000016e97 Initial program 88.3%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative88.3%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*90.1%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/91.2%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified91.2%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in t around inf 56.3%
\[\leadsto \color{blue}{\frac{x}{1 + a}}
\]
Recombined 2 regimes into one program. Final simplification56.0%
\[\leadsto \begin{array}{l}
\mathbf{if}\;y \leq -1.15 \cdot 10^{+81}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{elif}\;y \leq 3.2 \cdot 10^{+97}:\\
\;\;\;\;\frac{x}{a + 1}\\
\mathbf{else}:\\
\;\;\;\;\frac{z}{b}\\
\end{array}
\]
Alternative 16: 41.7% accurate, 2.4× speedup? \[\begin{array}{l}
\mathbf{if}\;a \leq -1:\\
\;\;\;\;\frac{x}{a}\\
\mathbf{elif}\;a \leq 1:\\
\;\;\;\;x\\
\mathbf{else}:\\
\;\;\;\;\frac{x}{a}\\
\end{array}
\]
Derivation Split input into 2 regimes if a < -1 or 1 < a Initial program 63.1%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative63.1%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*65.7%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/69.5%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified69.5%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Taylor expanded in x around inf 54.6%
\[\leadsto \frac{\color{blue}{x}}{\left(a + 1\right) + \frac{y}{t} \cdot b}
\]
Taylor expanded in a around inf 44.6%
\[\leadsto \color{blue}{\frac{x}{a}}
\]
if -1 < a < 1 Initial program 74.6%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative74.6%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*75.3%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/80.4%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified80.4%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Step-by-step derivation clear-num80.3%
\[\leadsto \color{blue}{\frac{1}{\frac{\left(a + 1\right) + \frac{y}{t} \cdot b}{x + \frac{z}{\frac{t}{y}}}}}
\]
inv-pow80.3%
\[\leadsto \color{blue}{{\left(\frac{\left(a + 1\right) + \frac{y}{t} \cdot b}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}}
\]
+-commutative80.3%
\[\leadsto {\left(\frac{\color{blue}{\frac{y}{t} \cdot b + \left(a + 1\right)}}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}
\]
fma-def80.3%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}
\]
+-commutative80.3%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{\frac{z}{\frac{t}{y}} + x}}\right)}^{-1}
\]
div-inv80.3%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{z \cdot \frac{1}{\frac{t}{y}}} + x}\right)}^{-1}
\]
clear-num80.3%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{z \cdot \color{blue}{\frac{y}{t}} + x}\right)}^{-1}
\]
fma-udef80.3%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}\right)}^{-1}
\]
Applied egg-rr 80.3%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}\right)}^{-1}}
\]
Step-by-step derivation unpow-180.3%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}}
\]
fma-udef80.3%
\[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{t} \cdot b + \left(a + 1\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-+r+80.3%
\[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{y}{t} \cdot b + a\right) + 1}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-*l/75.2%
\[\leadsto \frac{1}{\frac{\left(\color{blue}{\frac{y \cdot b}{t}} + a\right) + 1}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
+-commutative75.2%
\[\leadsto \frac{1}{\frac{\color{blue}{1 + \left(\frac{y \cdot b}{t} + a\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-*r/76.7%
\[\leadsto \frac{1}{\frac{1 + \left(\color{blue}{y \cdot \frac{b}{t}} + a\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
fma-def76.7%
\[\leadsto \frac{1}{\frac{1 + \color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
fma-def76.7%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{z \cdot \frac{y}{t} + x}}}
\]
associate-*r/73.1%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\frac{z \cdot y}{t}} + x}}
\]
associate-*l/74.5%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\frac{z}{t} \cdot y} + x}}
\]
*-commutative74.5%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{y \cdot \frac{z}{t}} + x}}
\]
fma-def74.5%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}}
\]
Simplified74.5%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}}
\]
Taylor expanded in y around 0 35.1%
\[\leadsto \frac{1}{\color{blue}{\frac{1 + a}{x}}}
\]
Taylor expanded in a around 0 34.7%
\[\leadsto \color{blue}{x}
\]
Recombined 2 regimes into one program. Final simplification39.5%
\[\leadsto \begin{array}{l}
\mathbf{if}\;a \leq -1:\\
\;\;\;\;\frac{x}{a}\\
\mathbf{elif}\;a \leq 1:\\
\;\;\;\;x\\
\mathbf{else}:\\
\;\;\;\;\frac{x}{a}\\
\end{array}
\]
Alternative 17: 20.2% accurate, 17.0× speedup? \[x
\]
Derivation Initial program 69.0%
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
Step-by-step derivation *-commutative69.0%
\[\leadsto \frac{x + \frac{\color{blue}{z \cdot y}}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-/l*70.6%
\[\leadsto \frac{x + \color{blue}{\frac{z}{\frac{t}{y}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\]
associate-*l/75.1%
\[\leadsto \frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \color{blue}{\frac{y}{t} \cdot b}}
\]
Simplified75.1%
\[\leadsto \color{blue}{\frac{x + \frac{z}{\frac{t}{y}}}{\left(a + 1\right) + \frac{y}{t} \cdot b}}
\]
Step-by-step derivation clear-num74.9%
\[\leadsto \color{blue}{\frac{1}{\frac{\left(a + 1\right) + \frac{y}{t} \cdot b}{x + \frac{z}{\frac{t}{y}}}}}
\]
inv-pow74.9%
\[\leadsto \color{blue}{{\left(\frac{\left(a + 1\right) + \frac{y}{t} \cdot b}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}}
\]
+-commutative74.9%
\[\leadsto {\left(\frac{\color{blue}{\frac{y}{t} \cdot b + \left(a + 1\right)}}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}
\]
fma-def74.9%
\[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}}{x + \frac{z}{\frac{t}{y}}}\right)}^{-1}
\]
+-commutative74.9%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{\frac{z}{\frac{t}{y}} + x}}\right)}^{-1}
\]
div-inv74.9%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{z \cdot \frac{1}{\frac{t}{y}}} + x}\right)}^{-1}
\]
clear-num74.9%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{z \cdot \color{blue}{\frac{y}{t}} + x}\right)}^{-1}
\]
fma-udef74.9%
\[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\color{blue}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}\right)}^{-1}
\]
Applied egg-rr 74.9%
\[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}\right)}^{-1}}
\]
Step-by-step derivation unpow-174.9%
\[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}}
\]
fma-udef74.9%
\[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{t} \cdot b + \left(a + 1\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-+r+74.9%
\[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{y}{t} \cdot b + a\right) + 1}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-*l/70.5%
\[\leadsto \frac{1}{\frac{\left(\color{blue}{\frac{y \cdot b}{t}} + a\right) + 1}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
+-commutative70.5%
\[\leadsto \frac{1}{\frac{\color{blue}{1 + \left(\frac{y \cdot b}{t} + a\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
associate-*r/72.4%
\[\leadsto \frac{1}{\frac{1 + \left(\color{blue}{y \cdot \frac{b}{t}} + a\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
fma-def72.4%
\[\leadsto \frac{1}{\frac{1 + \color{blue}{\mathsf{fma}\left(y, \frac{b}{t}, a\right)}}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}
\]
fma-def72.4%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{z \cdot \frac{y}{t} + x}}}
\]
associate-*r/68.9%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\frac{z \cdot y}{t}} + x}}
\]
associate-*l/72.9%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\frac{z}{t} \cdot y} + x}}
\]
*-commutative72.9%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{y \cdot \frac{z}{t}} + x}}
\]
fma-def72.9%
\[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}}
\]
Simplified72.9%
\[\leadsto \color{blue}{\frac{1}{\frac{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}}
\]
Taylor expanded in y around 0 40.1%
\[\leadsto \frac{1}{\color{blue}{\frac{1 + a}{x}}}
\]
Taylor expanded in a around 0 19.5%
\[\leadsto \color{blue}{x}
\]
Final simplification19.5%
\[\leadsto x
\]
Developer target: 78.8% accurate, 0.7× speedup? \[\begin{array}{l}
\mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\
\;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\
\mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\
\;\;\;\;\frac{z}{b}\\
\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\
\end{array}
\]
Reproduce ? herbie shell --seed 2023167
(FPCore (x y z t a b)
:name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
:precision binary64
:herbie-target
(if (< t -1.3659085366310088e-271) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))
(/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
