Result for Numeric.Histogram:binBounds from Chart-1.5.3

Original	93.2%
Target	97.7%
Herbie	96.6%

Alternative 1?

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+305} \lor \neg \left(t_1 \leq 10^{+308}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Derivation

Split input into 2 regimes
if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -1.9999999999999999e305 or 1e308 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))
1. Initial program 85.0%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in t around 0 85.0%
  \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} \]
3. Step-by-step derivation
  1. *-commutative85.0%
    \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{t} \]
  2. associate-*l/95.1%
    \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} \]
4. Applied egg-rr95.1%
  \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} \]
if -1.9999999999999999e305 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 1e308
1. Initial program 99.8%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
Recombined 2 regimes into one program.
Final simplification98.2%
\[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq -2 \cdot 10^{+305} \lor \neg \left(x + \frac{\left(y - x\right) \cdot z}{t} \leq 10^{+308}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \end{array} \]

Alternative 2?

\[\begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+66} \lor \neg \left(y \leq 5.8 \cdot 10^{-37} \lor \neg \left(y \leq 13.4\right) \land y \leq 6.2 \cdot 10^{+131}\right):\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Derivation

Split input into 2 regimes
if y < -6.6000000000000003e66 or 5.80000000000000009e-37 < y < 13.4000000000000004 or 6.20000000000000032e131 < y
1. Initial program 92.4%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in t around 0 73.5%
  \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} \]
3. Taylor expanded in y around inf 63.3%
  \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
4. Step-by-step derivation
  1. associate-*r/67.6%
    \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
5. Simplified67.6%
  \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
if -6.6000000000000003e66 < y < 5.80000000000000009e-37 or 13.4000000000000004 < y < 6.20000000000000032e131
1. Initial program 96.2%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in z around 0 52.2%
  \[\leadsto \color{blue}{x} \]
Recombined 2 regimes into one program.
Final simplification58.4%
\[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+66} \lor \neg \left(y \leq 5.8 \cdot 10^{-37} \lor \neg \left(y \leq 13.4\right) \land y \leq 6.2 \cdot 10^{+131}\right):\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3?

\[\begin{array}{l} t_1 := \frac{y}{\frac{t}{z}}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 14.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+134}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \end{array} \]

Derivation

Split input into 3 regimes
if y < -1.25e51 or 5.6000000000000002e-36 < y < 14.199999999999999
1. Initial program 94.3%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in t around 0 73.0%
  \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} \]
3. Taylor expanded in y around inf 60.2%
  \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
4. Step-by-step derivation
  1. associate-*r/64.0%
    \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
5. Simplified64.0%
  \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
6. Step-by-step derivation
  1. clear-num63.9%
    \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{t}{z}}} \]
  2. un-div-inv64.0%
    \[\leadsto \color{blue}{\frac{y}{\frac{t}{z}}} \]
7. Applied egg-rr64.0%
  \[\leadsto \color{blue}{\frac{y}{\frac{t}{z}}} \]
if -1.25e51 < y < 5.6000000000000002e-36 or 14.199999999999999 < y < 4.4e134
1. Initial program 96.7%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in z around 0 52.3%
  \[\leadsto \color{blue}{x} \]
if 4.4e134 < y
1. Initial program 87.3%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in t around 0 71.0%
  \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} \]
3. Taylor expanded in y around inf 65.7%
  \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
4. Step-by-step derivation
  1. associate-*r/73.0%
    \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
5. Simplified73.0%
  \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
Recombined 3 regimes into one program.
Final simplification58.4%
\[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+51}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 14.2:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+134}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 4?

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-42}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -4.9 \cdot 10^{-198}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-65}:\\ \;\;\;\;-z \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Derivation

Split input into 3 regimes
if t < -5.2e-42 or 7.4e-65 < t
1. Initial program 90.5%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in z around 0 60.5%
  \[\leadsto \color{blue}{x} \]
if -5.2e-42 < t < -4.9000000000000002e-198
1. Initial program 99.8%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in t around 0 81.3%
  \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} \]
3. Taylor expanded in y around inf 56.4%
  \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
4. Step-by-step derivation
  1. associate-*r/56.6%
    \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
5. Simplified56.6%
  \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
if -4.9000000000000002e-198 < t < 7.4e-65
1. Initial program 99.8%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in t around 0 90.0%
  \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} \]
3. Step-by-step derivation
  1. *-commutative90.0%
    \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{t} \]
  2. associate-*l/86.4%
    \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} \]
4. Applied egg-rr86.4%
  \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} \]
5. Taylor expanded in y around 0 63.5%
  \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot x}{t}} \]
6. Step-by-step derivation
  1. mul-1-neg63.5%
    \[\leadsto \color{blue}{-\frac{z \cdot x}{t}} \]
  2. associate-*r/59.9%
    \[\leadsto -\color{blue}{z \cdot \frac{x}{t}} \]
  3. distribute-rgt-neg-in59.9%
    \[\leadsto \color{blue}{z \cdot \left(-\frac{x}{t}\right)} \]
7. Simplified59.9%
  \[\leadsto \color{blue}{z \cdot \left(-\frac{x}{t}\right)} \]
Recombined 3 regimes into one program.
Final simplification59.7%
\[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-42}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -4.9 \cdot 10^{-198}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-65}:\\ \;\;\;\;-z \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5?

\[\begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-200}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \frac{-z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Derivation

Split input into 3 regimes
if t < -1.1e-41 or 4.9999999999999999e-61 < t
1. Initial program 90.5%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in z around 0 60.5%
  \[\leadsto \color{blue}{x} \]
if -1.1e-41 < t < -8.50000000000000014e-200
1. Initial program 99.8%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in t around 0 81.3%
  \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} \]
3. Taylor expanded in y around inf 56.4%
  \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
4. Step-by-step derivation
  1. associate-*r/56.6%
    \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
5. Simplified56.6%
  \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
if -8.50000000000000014e-200 < t < 4.9999999999999999e-61
1. Initial program 99.8%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in t around 0 90.0%
  \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} \]
3. Taylor expanded in y around 0 63.5%
  \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot x}{t}} \]
4. Step-by-step derivation
  1. mul-1-neg63.5%
    \[\leadsto \color{blue}{-\frac{z \cdot x}{t}} \]
  2. distribute-neg-frac63.5%
    \[\leadsto \color{blue}{\frac{-z \cdot x}{t}} \]
  3. distribute-rgt-neg-out63.5%
    \[\leadsto \frac{\color{blue}{z \cdot \left(-x\right)}}{t} \]
  4. associate-*l/64.6%
    \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(-x\right)} \]
5. Simplified64.6%
  \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(-x\right)} \]
Recombined 3 regimes into one program.
Final simplification61.1%
\[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-200}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \frac{-z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6?

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-101} \lor \neg \left(t \leq 4.5 \cdot 10^{-67}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \end{array} \]

Derivation

Split input into 2 regimes
if t < -2.0000000000000001e-101 or 4.50000000000000015e-67 < t
1. Initial program 91.4%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in y around inf 84.3%
  \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]
3. Step-by-step derivation
  1. associate-*r/32.0%
    \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
4. Simplified87.9%
  \[\leadsto x + \color{blue}{y \cdot \frac{z}{t}} \]
if -2.0000000000000001e-101 < t < 4.50000000000000015e-67
1. Initial program 99.8%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in t around 0 89.5%
  \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} \]
3. Step-by-step derivation
  1. *-commutative89.5%
    \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{t} \]
  2. associate-*l/85.8%
    \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} \]
4. Applied egg-rr85.8%
  \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} \]
Recombined 2 regimes into one program.
Final simplification87.1%
\[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-101} \lor \neg \left(t \leq 4.5 \cdot 10^{-67}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \end{array} \]

Alternative 7?

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+55}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+163}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Derivation

Split input into 2 regimes
if t < -7.50000000000000014e55 or 7.19999999999999955e163 < t
1. Initial program 87.0%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in z around 0 74.3%
  \[\leadsto \color{blue}{x} \]
if -7.50000000000000014e55 < t < 7.19999999999999955e163
1. Initial program 98.3%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in t around 0 76.2%
  \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} \]
3. Step-by-step derivation
  1. *-commutative76.2%
    \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{t} \]
  2. associate-*l/75.6%
    \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} \]
4. Applied egg-rr75.6%
  \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} \]
Recombined 2 regimes into one program.
Final simplification75.2%
\[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+55}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+163}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8?

\[\begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{-101}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \]

Derivation

Split input into 3 regimes
if t < -2.05000000000000013e-101
1. Initial program 92.0%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in y around inf 83.1%
  \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]
3. Step-by-step derivation
  1. associate-*r/29.5%
    \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
4. Simplified85.3%
  \[\leadsto x + \color{blue}{y \cdot \frac{z}{t}} \]
if -2.05000000000000013e-101 < t < 4.9999999999999998e-70
1. Initial program 99.8%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in t around 0 89.5%
  \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} \]
3. Step-by-step derivation
  1. *-commutative89.5%
    \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{t} \]
  2. associate-*l/85.8%
    \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} \]
4. Applied egg-rr85.8%
  \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} \]
if 4.9999999999999998e-70 < t
1. Initial program 90.7%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in y around inf 85.7%
  \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]
3. Step-by-step derivation
  1. associate-*l/91.3%
    \[\leadsto x + \color{blue}{\frac{y}{t} \cdot z} \]
  2. *-commutative91.3%
    \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]
4. Simplified91.3%
  \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]
Recombined 3 regimes into one program.
Final simplification87.2%
\[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{-101}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 9?

\[\begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-102}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-67}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \]

Derivation

Split input into 3 regimes
if t < -6.99999999999999973e-102
1. Initial program 92.0%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Step-by-step derivation
  1. +-commutative92.0%
    \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t} + x} \]
  2. *-commutative92.0%
    \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{t} + x \]
  3. associate-*l/99.9%
    \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} + x \]
  4. fma-def99.9%
    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)} \]
3. Simplified99.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)} \]
4. Step-by-step derivation
  1. fma-udef99.9%
    \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right) + x} \]
  2. associate-*l/92.0%
    \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{t}} + x \]
  3. *-commutative92.0%
    \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{t} + x \]
  4. associate-*r/99.9%
    \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{t}} + x \]
5. Applied egg-rr99.9%
  \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{t} + x} \]
6. Step-by-step derivation
  1. associate-*r/92.0%
    \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} + x \]
  2. *-commutative92.0%
    \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{t} + x \]
  3. associate-/l*99.9%
    \[\leadsto \color{blue}{\frac{z}{\frac{t}{y - x}}} + x \]
7. Applied egg-rr99.9%
  \[\leadsto \color{blue}{\frac{z}{\frac{t}{y - x}}} + x \]
8. Taylor expanded in y around inf 85.4%
  \[\leadsto \frac{z}{\color{blue}{\frac{t}{y}}} + x \]
if -6.99999999999999973e-102 < t < 2.4e-67
1. Initial program 99.8%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in t around 0 89.5%
  \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} \]
3. Step-by-step derivation
  1. *-commutative89.5%
    \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{t} \]
  2. associate-*l/85.8%
    \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} \]
4. Applied egg-rr85.8%
  \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} \]
if 2.4e-67 < t
1. Initial program 90.7%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in y around inf 85.7%
  \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]
3. Step-by-step derivation
  1. associate-*l/91.3%
    \[\leadsto x + \color{blue}{\frac{y}{t} \cdot z} \]
  2. *-commutative91.3%
    \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]
4. Simplified91.3%
  \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]
Recombined 3 regimes into one program.
Final simplification87.2%
\[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-102}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-67}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 10?

\[\begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+49}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]

Derivation

Split input into 3 regimes
if y < -1.5000000000000001e49
1. Initial program 93.1%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in y around inf 86.7%
  \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]
3. Step-by-step derivation
  1. associate-*r/67.0%
    \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
4. Simplified89.8%
  \[\leadsto x + \color{blue}{y \cdot \frac{z}{t}} \]
5. Step-by-step derivation
  1. clear-num66.9%
    \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{t}{z}}} \]
  2. un-div-inv67.1%
    \[\leadsto \color{blue}{\frac{y}{\frac{t}{z}}} \]
6. Applied egg-rr89.8%
  \[\leadsto x + \color{blue}{\frac{y}{\frac{t}{z}}} \]
if -1.5000000000000001e49 < y < 2.1000000000000001e-16
1. Initial program 97.9%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in x around inf 89.3%
  \[\leadsto \color{blue}{\left(1 + -1 \cdot \frac{z}{t}\right) \cdot x} \]
3. Step-by-step derivation
  1. *-commutative89.3%
    \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{z}{t}\right)} \]
  2. distribute-lft-in89.3%
    \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left(-1 \cdot \frac{z}{t}\right)} \]
  3. *-rgt-identity89.3%
    \[\leadsto \color{blue}{x} + x \cdot \left(-1 \cdot \frac{z}{t}\right) \]
  4. mul-1-neg89.3%
    \[\leadsto x + x \cdot \color{blue}{\left(-\frac{z}{t}\right)} \]
  5. distribute-rgt-neg-in89.3%
    \[\leadsto x + \color{blue}{\left(-x \cdot \frac{z}{t}\right)} \]
  6. unsub-neg89.3%
    \[\leadsto \color{blue}{x - x \cdot \frac{z}{t}} \]
4. Simplified89.3%
  \[\leadsto \color{blue}{x - x \cdot \frac{z}{t}} \]
if 2.1000000000000001e-16 < y
1. Initial program 88.3%
  \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Taylor expanded in y around inf 84.0%
  \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]
3. Step-by-step derivation
  1. associate-*r/58.6%
    \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
4. Simplified90.6%
  \[\leadsto x + \color{blue}{y \cdot \frac{z}{t}} \]
Recombined 3 regimes into one program.
Final simplification89.7%
\[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+49}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 11?

\[x + \left(y - x\right) \cdot \frac{z}{t} \]

Derivation

Initial program 94.7%
\[x + \frac{\left(y - x\right) \cdot z}{t} \]
Step-by-step derivation
1. +-commutative94.7%
  \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t} + x} \]
2. *-commutative94.7%
  \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{t} + x \]
3. associate-*l/98.0%
  \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} + x \]
4. fma-def98.0%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)} \]
Simplified98.0%
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)} \]
Step-by-step derivation
1. fma-udef98.0%
  \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right) + x} \]
2. associate-*l/94.7%
  \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{t}} + x \]
3. *-commutative94.7%
  \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{t} + x \]
4. associate-*r/98.0%
  \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{t}} + x \]
Applied egg-rr98.0%
\[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{t} + x} \]
Final simplification98.0%
\[\leadsto x + \left(y - x\right) \cdot \frac{z}{t} \]

In
Out

25.9% of points produce a very large (infinite) output. You may want to add a precondition. (more)

Specification

Local Percentage Accuracy vs ?

Accuracy vs Speed?

Target

Alternative 1?

if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -1.9999999999999999e305 or 1e308 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

if -1.9999999999999999e305 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 1e308

Alternative 2?

if y < -6.6000000000000003e66 or 5.80000000000000009e-37 < y < 13.4000000000000004 or 6.20000000000000032e131 < y

if -6.6000000000000003e66 < y < 5.80000000000000009e-37 or 13.4000000000000004 < y < 6.20000000000000032e131

Alternative 3?

if y < -1.25e51 or 5.6000000000000002e-36 < y < 14.199999999999999

if -1.25e51 < y < 5.6000000000000002e-36 or 14.199999999999999 < y < 4.4e134

if 4.4e134 < y

Alternative 4?

if t < -5.2e-42 or 7.4e-65 < t

if -5.2e-42 < t < -4.9000000000000002e-198

if -4.9000000000000002e-198 < t < 7.4e-65

Alternative 5?

if t < -1.1e-41 or 4.9999999999999999e-61 < t

if -1.1e-41 < t < -8.50000000000000014e-200

if -8.50000000000000014e-200 < t < 4.9999999999999999e-61

Alternative 6?

if t < -2.0000000000000001e-101 or 4.50000000000000015e-67 < t

if -2.0000000000000001e-101 < t < 4.50000000000000015e-67

Alternative 7?

if t < -7.50000000000000014e55 or 7.19999999999999955e163 < t

if -7.50000000000000014e55 < t < 7.19999999999999955e163

Alternative 8?

if t < -2.05000000000000013e-101

if -2.05000000000000013e-101 < t < 4.9999999999999998e-70

if 4.9999999999999998e-70 < t

Alternative 9?

if t < -6.99999999999999973e-102

if -6.99999999999999973e-102 < t < 2.4e-67

if 2.4e-67 < t

Alternative 10?

if y < -1.5000000000000001e49

if -1.5000000000000001e49 < y < 2.1000000000000001e-16

if 2.1000000000000001e-16 < y

Alternative 11?

Alternative 12?

Alternative 13?

Reproduce

`if (+.f64 x (/.f64 (.f64 (-.f64 y x) z) t)) < -1.9999999999999999e305 or 1e308 < (+.f64 x (/.f64 (.f64 (-.f64 y x) z) t))`

`if -1.9999999999999999e305 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 1e308`

`if y < -6.6000000000000003e66 or 5.80000000000000009e-37 < y < 13.4000000000000004 or 6.20000000000000032e131 < y`

`if -6.6000000000000003e66 < y < 5.80000000000000009e-37 or 13.4000000000000004 < y < 6.20000000000000032e131`

`if y < -1.25e51 or 5.6000000000000002e-36 < y < 14.199999999999999`

`if -1.25e51 < y < 5.6000000000000002e-36 or 14.199999999999999 < y < 4.4e134`

`if 4.4e134 < y`

`if t < -5.2e-42 or 7.4e-65 < t`

`if -5.2e-42 < t < -4.9000000000000002e-198`

`if -4.9000000000000002e-198 < t < 7.4e-65`

`if t < -1.1e-41 or 4.9999999999999999e-61 < t`

`if -1.1e-41 < t < -8.50000000000000014e-200`

`if -8.50000000000000014e-200 < t < 4.9999999999999999e-61`

`if t < -2.0000000000000001e-101 or 4.50000000000000015e-67 < t`

`if -2.0000000000000001e-101 < t < 4.50000000000000015e-67`

`if t < -7.50000000000000014e55 or 7.19999999999999955e163 < t`

`if -7.50000000000000014e55 < t < 7.19999999999999955e163`

`if t < -2.05000000000000013e-101`

`if -2.05000000000000013e-101 < t < 4.9999999999999998e-70`

`if 4.9999999999999998e-70 < t`

`if t < -6.99999999999999973e-102`

`if -6.99999999999999973e-102 < t < 2.4e-67`

`if 2.4e-67 < t`

`if y < -1.5000000000000001e49`

`if -1.5000000000000001e49 < y < 2.1000000000000001e-16`

`if 2.1000000000000001e-16 < y`