Average Error: 11.0 → 0.4
Time: 3.3s
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 r578989 = x;
        double r578990 = y;
        double r578991 = z;
        double r578992 = t;
        double r578993 = r578991 - r578992;
        double r578994 = r578990 * r578993;
        double r578995 = a;
        double r578996 = r578991 - r578995;
        double r578997 = r578994 / r578996;
        double r578998 = r578989 + r578997;
        return r578998;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r578999 = y;
        double r579000 = z;
        double r579001 = t;
        double r579002 = r579000 - r579001;
        double r579003 = r578999 * r579002;
        double r579004 = a;
        double r579005 = r579000 - r579004;
        double r579006 = r579003 / r579005;
        double r579007 = -inf.0;
        bool r579008 = r579006 <= r579007;
        double r579009 = r579002 / r579005;
        double r579010 = r579009 * r578999;
        double r579011 = x;
        double r579012 = r579010 + r579011;
        double r579013 = 3.941060132304762e+276;
        bool r579014 = r579006 <= r579013;
        double r579015 = r579011 + r579006;
        double r579016 = 1.0;
        double r579017 = r579005 / r578999;
        double r579018 = r579017 / r579002;
        double r579019 = r579016 / r579018;
        double r579020 = r579019 + r579011;
        double r579021 = r579014 ? r579015 : r579020;
        double r579022 = r579008 ? r579012 : r579021;
        return r579022;
}

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))))