Average Error: 11.0 → 0.4
Time: 3.4s
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 r505696 = x;
        double r505697 = y;
        double r505698 = z;
        double r505699 = t;
        double r505700 = r505698 - r505699;
        double r505701 = r505697 * r505700;
        double r505702 = a;
        double r505703 = r505698 - r505702;
        double r505704 = r505701 / r505703;
        double r505705 = r505696 + r505704;
        return r505705;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r505706 = y;
        double r505707 = z;
        double r505708 = t;
        double r505709 = r505707 - r505708;
        double r505710 = r505706 * r505709;
        double r505711 = a;
        double r505712 = r505707 - r505711;
        double r505713 = r505710 / r505712;
        double r505714 = -inf.0;
        bool r505715 = r505713 <= r505714;
        double r505716 = r505709 / r505712;
        double r505717 = r505716 * r505706;
        double r505718 = x;
        double r505719 = r505717 + r505718;
        double r505720 = 3.941060132304762e+276;
        bool r505721 = r505713 <= r505720;
        double r505722 = r505718 + r505713;
        double r505723 = 1.0;
        double r505724 = r505712 / r505706;
        double r505725 = r505724 / r505709;
        double r505726 = r505723 / r505725;
        double r505727 = r505726 + r505718;
        double r505728 = r505721 ? r505722 : r505727;
        double r505729 = r505715 ? r505719 : r505728;
        return r505729;
}

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))))