Average Error: 11.0 → 0.4
Time: 3.1s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} = -\infty:\\ \;\;\;\;\frac{z - t}{z - a} \cdot y + x\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 3.941060132304761944440726014815691553449 \cdot 10^{276}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z - a}{y}}{z - t}} + x\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} = -\infty:\\
\;\;\;\;\frac{z - t}{z - a} \cdot y + x\\

\mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 3.941060132304761944440726014815691553449 \cdot 10^{276}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\frac{z - a}{y}}{z - t}} + x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r558230 = x;
        double r558231 = y;
        double r558232 = z;
        double r558233 = t;
        double r558234 = r558232 - r558233;
        double r558235 = r558231 * r558234;
        double r558236 = a;
        double r558237 = r558232 - r558236;
        double r558238 = r558235 / r558237;
        double r558239 = r558230 + r558238;
        return r558239;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r558240 = y;
        double r558241 = z;
        double r558242 = t;
        double r558243 = r558241 - r558242;
        double r558244 = r558240 * r558243;
        double r558245 = a;
        double r558246 = r558241 - r558245;
        double r558247 = r558244 / r558246;
        double r558248 = -inf.0;
        bool r558249 = r558247 <= r558248;
        double r558250 = r558243 / r558246;
        double r558251 = r558250 * r558240;
        double r558252 = x;
        double r558253 = r558251 + r558252;
        double r558254 = 3.941060132304762e+276;
        bool r558255 = r558247 <= r558254;
        double r558256 = r558252 + r558247;
        double r558257 = 1.0;
        double r558258 = r558246 / r558240;
        double r558259 = r558258 / r558243;
        double r558260 = r558257 / r558259;
        double r558261 = r558260 + r558252;
        double r558262 = r558255 ? r558256 : r558261;
        double r558263 = r558249 ? r558253 : r558262;
        return r558263;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.0
Target1.3
Herbie0.4
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* y (- z t)) (- z a)) < -inf.0

    1. Initial program 64.0

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

    2. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)}\]
    3. Using strategy rm

    4. Applied clear-num0.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{z - a}{y}}}, z - t, x\right)\]
    5. Using strategy rm

    6. Applied fma-udef0.2

      \[\leadsto \color{blue}{\frac{1}{\frac{z - a}{y}} \cdot \left(z - t\right) + x}\]

    7. Simplified0.1

      \[\leadsto \color{blue}{\frac{z - t}{\frac{z - a}{y}}} + x\]
    8. Using strategy rm

    9. Applied associate-/r/0.1

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

    if -inf.0 < (/ (* y (- z t)) (- z a)) < 3.941060132304762e+276

    1. Initial program 0.2

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

    if 3.941060132304762e+276 < (/ (* y (- z t)) (- z a))

    1. Initial program 59.1

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

    2. Simplified1.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)}\]
    3. Using strategy rm

    4. Applied clear-num2.0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{z - a}{y}}}, z - t, x\right)\]
    5. Using strategy rm

    6. Applied fma-udef2.0

      \[\leadsto \color{blue}{\frac{1}{\frac{z - a}{y}} \cdot \left(z - t\right) + x}\]

    7. Simplified1.8

      \[\leadsto \color{blue}{\frac{z - t}{\frac{z - a}{y}}} + x\]
    8. Using strategy rm

    9. Applied clear-num1.9

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{z - a}{y}}{z - t}}} + x\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} = -\infty:\\ \;\;\;\;\frac{z - t}{z - a} \cdot y + x\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 3.941060132304761944440726014815691553449 \cdot 10^{276}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z - a}{y}}{z - t}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

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