Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B

Percentage Accurate: 98.0% → 98.0%
Time: 7.9s
Alternatives: 10
Speedup: 11.0×

Specification

?
\[x + y \cdot \frac{z - t}{a - t} \]

Your Program's Arguments

Results

Enter valid numbers for all inputs

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 10 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Alternative 1: 98.0% accurate, 1.0× speedup?

\[x + y \cdot \frac{z - t}{a - t} \]
Derivation

  1. Initial program 98.4%

    \[x + y \cdot \frac{z - t}{a - t} \]

  2. Final simplification98.4%

    \[\leadsto x + y \cdot \frac{z - t}{a - t} \]

Alternative 2: 76.4% accurate, 0.7× speedup?

\[\begin{array}{l} t_1 := x - z \cdot \frac{y}{t}\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{+58}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-77}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
Derivation
  1. Split input into 3 regimes

  2. if t < -8.8000000000000003e58 or 2.6999999999999999e-14 < t

    1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in t around inf 80.5%

      \[\leadsto \color{blue}{y + x} \]

    if -8.8000000000000003e58 < t < -8e-82 or 9.9999999999999993e-78 < t < 2.6999999999999999e-14

    1. Initial program 99.8%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in z around inf 78.6%

      \[\leadsto x + y \cdot \color{blue}{\frac{z}{a - t}} \]

    3. Taylor expanded in a around 0 68.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{t} + x} \]
    4. Step-by-step derivation

      1. +-commutative68.2%

        \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot z}{t}} \]

      2. mul-1-neg68.2%

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

      3. unsub-neg68.2%

        \[\leadsto \color{blue}{x - \frac{y \cdot z}{t}} \]

      4. *-commutative68.2%

        \[\leadsto x - \frac{\color{blue}{z \cdot y}}{t} \]

      5. associate-/l*66.4%

        \[\leadsto x - \color{blue}{\frac{z}{\frac{t}{y}}} \]

    5. Simplified66.4%

      \[\leadsto \color{blue}{x - \frac{z}{\frac{t}{y}}} \]
    6. Step-by-step derivation

      1. div-inv66.5%

        \[\leadsto x - \color{blue}{z \cdot \frac{1}{\frac{t}{y}}} \]

      2. clear-num68.0%

        \[\leadsto x - z \cdot \color{blue}{\frac{y}{t}} \]

    7. Applied egg-rr68.0%

      \[\leadsto x - \color{blue}{z \cdot \frac{y}{t}} \]

    if -8e-82 < t < 9.9999999999999993e-78

    1. Initial program 95.6%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in t around 0 86.4%

      \[\leadsto \color{blue}{\frac{y \cdot z}{a} + x} \]
    3. Step-by-step derivation

      1. associate-/l*86.3%

        \[\leadsto \color{blue}{\frac{y}{\frac{a}{z}}} + x \]

    4. Simplified86.3%

      \[\leadsto \color{blue}{\frac{y}{\frac{a}{z}} + x} \]
    5. Step-by-step derivation

      1. associate-/r/87.4%

        \[\leadsto \color{blue}{\frac{y}{a} \cdot z} + x \]

    6. Applied egg-rr87.4%

      \[\leadsto \color{blue}{\frac{y}{a} \cdot z} + x \]
  3. Recombined 3 regimes into one program.

  4. Final simplification80.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{+58}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-82}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \mathbf{elif}\;t \leq 10^{-77}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3: 76.4% accurate, 0.7× speedup?

\[\begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+58}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -4.85 \cdot 10^{-79}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-78}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-14}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
Derivation
  1. Split input into 4 regimes

  2. if t < -8.9999999999999996e58 or 5.9999999999999997e-14 < t

    1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in t around inf 80.5%

      \[\leadsto \color{blue}{y + x} \]

    if -8.9999999999999996e58 < t < -4.84999999999999978e-79

    1. Initial program 99.8%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in z around inf 75.2%

      \[\leadsto x + y \cdot \color{blue}{\frac{z}{a - t}} \]

    3. Taylor expanded in a around 0 64.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{t} + x} \]
    4. Step-by-step derivation

      1. +-commutative64.1%

        \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot z}{t}} \]

      2. mul-1-neg64.1%

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

      3. unsub-neg64.1%

        \[\leadsto \color{blue}{x - \frac{y \cdot z}{t}} \]

      4. *-commutative64.1%

        \[\leadsto x - \frac{\color{blue}{z \cdot y}}{t} \]

      5. associate-/l*61.6%

        \[\leadsto x - \color{blue}{\frac{z}{\frac{t}{y}}} \]

    5. Simplified61.6%

      \[\leadsto \color{blue}{x - \frac{z}{\frac{t}{y}}} \]
    6. Step-by-step derivation

      1. associate-/r/64.2%

        \[\leadsto x - \color{blue}{\frac{z}{t} \cdot y} \]

    7. Applied egg-rr64.2%

      \[\leadsto x - \color{blue}{\frac{z}{t} \cdot y} \]

    if -4.84999999999999978e-79 < t < 1.9500000000000001e-78

    1. Initial program 95.6%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in t around 0 86.4%

      \[\leadsto \color{blue}{\frac{y \cdot z}{a} + x} \]
    3. Step-by-step derivation

      1. associate-/l*86.3%

        \[\leadsto \color{blue}{\frac{y}{\frac{a}{z}}} + x \]

    4. Simplified86.3%

      \[\leadsto \color{blue}{\frac{y}{\frac{a}{z}} + x} \]
    5. Step-by-step derivation

      1. associate-/r/87.4%

        \[\leadsto \color{blue}{\frac{y}{a} \cdot z} + x \]

    6. Applied egg-rr87.4%

      \[\leadsto \color{blue}{\frac{y}{a} \cdot z} + x \]

    if 1.9500000000000001e-78 < t < 5.9999999999999997e-14

    1. Initial program 99.7%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in z around inf 87.2%

      \[\leadsto x + y \cdot \color{blue}{\frac{z}{a - t}} \]

    3. Taylor expanded in a around 0 78.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{t} + x} \]
    4. Step-by-step derivation

      1. +-commutative78.4%

        \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot z}{t}} \]

      2. mul-1-neg78.4%

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

      3. unsub-neg78.4%

        \[\leadsto \color{blue}{x - \frac{y \cdot z}{t}} \]

      4. *-commutative78.4%

        \[\leadsto x - \frac{\color{blue}{z \cdot y}}{t} \]

      5. associate-/l*78.3%

        \[\leadsto x - \color{blue}{\frac{z}{\frac{t}{y}}} \]

    5. Simplified78.3%

      \[\leadsto \color{blue}{x - \frac{z}{\frac{t}{y}}} \]
    6. Step-by-step derivation

      1. div-inv78.4%

        \[\leadsto x - \color{blue}{z \cdot \frac{1}{\frac{t}{y}}} \]

      2. clear-num78.4%

        \[\leadsto x - z \cdot \color{blue}{\frac{y}{t}} \]

    7. Applied egg-rr78.4%

      \[\leadsto x - \color{blue}{z \cdot \frac{y}{t}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification80.5%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+58}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq -4.85 \cdot 10^{-79}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-78}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-14}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4: 82.3% accurate, 0.8× speedup?

\[\begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+59} \lor \neg \left(t \leq 6.3 \cdot 10^{-14}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a - t}\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if t < -1.04999999999999992e59 or 6.29999999999999966e-14 < t

    1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in t around inf 80.5%

      \[\leadsto \color{blue}{y + x} \]

    if -1.04999999999999992e59 < t < 6.29999999999999966e-14

    1. Initial program 97.1%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in z around inf 87.3%

      \[\leadsto x + y \cdot \color{blue}{\frac{z}{a - t}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification84.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+59} \lor \neg \left(t \leq 6.3 \cdot 10^{-14}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a - t}\\ \end{array} \]

Alternative 5: 82.8% accurate, 0.8× speedup?

\[\begin{array}{l} \mathbf{if}\;t \leq -1.16 \cdot 10^{+59}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-14}:\\ \;\;\;\;x + z \cdot \frac{y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if t < -1.16000000000000001e59 or 5.3999999999999997e-14 < t

    1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in t around inf 80.5%

      \[\leadsto \color{blue}{y + x} \]

    if -1.16000000000000001e59 < t < 5.3999999999999997e-14

    1. Initial program 97.1%

      \[x + y \cdot \frac{z - t}{a - t} \]
    2. Step-by-step derivation

      1. associate-*r/97.7%

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

    3. Simplified97.7%

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(z - t\right)}{a - t}} \]
    4. Step-by-step derivation

      1. associate-/l*97.1%

        \[\leadsto x + \color{blue}{\frac{y}{\frac{a - t}{z - t}}} \]

      2. associate-/r/97.6%

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

    5. Applied egg-rr97.6%

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

    6. Taylor expanded in z around inf 87.9%

      \[\leadsto x + \color{blue}{\frac{y \cdot z}{a - t}} \]
    7. Step-by-step derivation

      1. associate-*l/87.8%

        \[\leadsto x + \color{blue}{\frac{y}{a - t} \cdot z} \]

      2. *-commutative87.8%

        \[\leadsto x + \color{blue}{z \cdot \frac{y}{a - t}} \]

    8. Simplified87.8%

      \[\leadsto x + \color{blue}{z \cdot \frac{y}{a - t}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification84.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.16 \cdot 10^{+59}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-14}:\\ \;\;\;\;x + z \cdot \frac{y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 6: 87.4% accurate, 0.8× speedup?

\[\begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+57}:\\ \;\;\;\;x + y \cdot \frac{z}{a - t}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-35}:\\ \;\;\;\;x - \frac{y}{\frac{a}{t} + -1}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{a - t}\\ \end{array} \]
Derivation
  1. Split input into 3 regimes

  2. if z < -4.19999999999999982e57

    1. Initial program 98.0%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in z around inf 93.2%

      \[\leadsto x + y \cdot \color{blue}{\frac{z}{a - t}} \]

    if -4.19999999999999982e57 < z < 3.10000000000000012e-35

    1. Initial program 99.3%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in z around 0 80.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot t}{a - t} + x} \]
    3. Step-by-step derivation

      1. +-commutative80.8%

        \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot t}{a - t}} \]

      2. mul-1-neg80.8%

        \[\leadsto x + \color{blue}{\left(-\frac{y \cdot t}{a - t}\right)} \]

      3. unsub-neg80.8%

        \[\leadsto \color{blue}{x - \frac{y \cdot t}{a - t}} \]

      4. associate-/l*93.9%

        \[\leadsto x - \color{blue}{\frac{y}{\frac{a - t}{t}}} \]

      5. div-sub93.9%

        \[\leadsto x - \frac{y}{\color{blue}{\frac{a}{t} - \frac{t}{t}}} \]

      6. *-inverses93.9%

        \[\leadsto x - \frac{y}{\frac{a}{t} - \color{blue}{1}} \]

    4. Simplified93.9%

      \[\leadsto \color{blue}{x - \frac{y}{\frac{a}{t} - 1}} \]

    if 3.10000000000000012e-35 < z

    1. Initial program 96.9%

      \[x + y \cdot \frac{z - t}{a - t} \]
    2. Step-by-step derivation

      1. associate-*r/87.7%

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

    3. Simplified87.7%

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(z - t\right)}{a - t}} \]
    4. Step-by-step derivation

      1. associate-/l*97.0%

        \[\leadsto x + \color{blue}{\frac{y}{\frac{a - t}{z - t}}} \]

      2. associate-/r/98.3%

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

    5. Applied egg-rr98.3%

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

    6. Taylor expanded in z around inf 82.9%

      \[\leadsto x + \color{blue}{\frac{y \cdot z}{a - t}} \]
    7. Step-by-step derivation

      1. associate-*l/87.7%

        \[\leadsto x + \color{blue}{\frac{y}{a - t} \cdot z} \]

      2. *-commutative87.7%

        \[\leadsto x + \color{blue}{z \cdot \frac{y}{a - t}} \]

    8. Simplified87.7%

      \[\leadsto x + \color{blue}{z \cdot \frac{y}{a - t}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification92.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+57}:\\ \;\;\;\;x + y \cdot \frac{z}{a - t}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-35}:\\ \;\;\;\;x - \frac{y}{\frac{a}{t} + -1}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{a - t}\\ \end{array} \]

Alternative 7: 75.7% accurate, 1.0× speedup?

\[\begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{-66}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-37}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if t < -1.8500000000000001e-66 or 2.90000000000000005e-37 < t

    1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in t around inf 72.0%

      \[\leadsto \color{blue}{y + x} \]

    if -1.8500000000000001e-66 < t < 2.90000000000000005e-37

    1. Initial program 96.0%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in t around 0 81.5%

      \[\leadsto x + y \cdot \color{blue}{\frac{z}{a}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification75.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{-66}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-37}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 8: 75.8% accurate, 1.0× speedup?

\[\begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-66}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-38}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if t < -5.80000000000000023e-66 or 9.00000000000000018e-38 < t

    1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in t around inf 72.0%

      \[\leadsto \color{blue}{y + x} \]

    if -5.80000000000000023e-66 < t < 9.00000000000000018e-38

    1. Initial program 96.0%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in t around 0 81.6%

      \[\leadsto \color{blue}{\frac{y \cdot z}{a} + x} \]
    3. Step-by-step derivation

      1. associate-/l*81.5%

        \[\leadsto \color{blue}{\frac{y}{\frac{a}{z}}} + x \]

    4. Simplified81.5%

      \[\leadsto \color{blue}{\frac{y}{\frac{a}{z}} + x} \]
    5. Step-by-step derivation

      1. associate-/r/82.5%

        \[\leadsto \color{blue}{\frac{y}{a} \cdot z} + x \]

    6. Applied egg-rr82.5%

      \[\leadsto \color{blue}{\frac{y}{a} \cdot z} + x \]
  3. Recombined 2 regimes into one program.

  4. Final simplification75.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-66}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-38}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 9: 63.2% accurate, 1.5× speedup?

\[\begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-129}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if t < -9.00000000000000061e-129 or 8.20000000000000028e-41 < t

    1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in t around inf 69.1%

      \[\leadsto \color{blue}{y + x} \]

    if -9.00000000000000061e-129 < t < 8.20000000000000028e-41

    1. Initial program 95.6%

      \[x + y \cdot \frac{z - t}{a - t} \]

    2. Taylor expanded in x around inf 63.8%

      \[\leadsto \color{blue}{x} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification67.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-129}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 10: 50.9% accurate, 11.0× speedup?

\[x \]
Derivation

  1. Initial program 98.4%

    \[x + y \cdot \frac{z - t}{a - t} \]

  2. Taylor expanded in x around inf 48.6%

    \[\leadsto \color{blue}{x} \]

  3. Final simplification48.6%

    \[\leadsto x \]

Developer target: 99.3% accurate, 0.6× speedup?

\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]

Reproduce

?
herbie shell --seed 2023167 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

  (+ x (* y (/ (- z t) (- a t)))))