Average Error: 16.5 → 9.7
Time: 4.2s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -3.9804113722305575 \cdot 10^{-241} \lor \neg \left(\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0\right):\\ \;\;\;\;y \cdot \frac{t - z}{a - t} + \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -3.9804113722305575 \cdot 10^{-241} \lor \neg \left(\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0\right):\\
\;\;\;\;y \cdot \frac{t - z}{a - t} + \left(x + y\right)\\

\mathbf{else}:\\
\;\;\;\;x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r525749 = x;
        double r525750 = y;
        double r525751 = r525749 + r525750;
        double r525752 = z;
        double r525753 = t;
        double r525754 = r525752 - r525753;
        double r525755 = r525754 * r525750;
        double r525756 = a;
        double r525757 = r525756 - r525753;
        double r525758 = r525755 / r525757;
        double r525759 = r525751 - r525758;
        return r525759;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r525760 = x;
        double r525761 = y;
        double r525762 = r525760 + r525761;
        double r525763 = z;
        double r525764 = t;
        double r525765 = r525763 - r525764;
        double r525766 = r525765 * r525761;
        double r525767 = a;
        double r525768 = r525767 - r525764;
        double r525769 = r525766 / r525768;
        double r525770 = r525762 - r525769;
        double r525771 = -3.9804113722305575e-241;
        bool r525772 = r525770 <= r525771;
        double r525773 = 0.0;
        bool r525774 = r525770 <= r525773;
        double r525775 = !r525774;
        bool r525776 = r525772 || r525775;
        double r525777 = r525764 - r525763;
        double r525778 = r525777 / r525768;
        double r525779 = r525761 * r525778;
        double r525780 = r525779 + r525762;
        double r525781 = r525776 ? r525780 : r525760;
        return r525781;
}

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

Original16.5
Target8.6
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (+ x y) (/ (* (- z t) y) (- a t))) < -3.9804113722305575e-241 or 0.0 < (- (+ x y) (/ (* (- z t) y) (- a t)))

    1. Initial program 13.0

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

    2. Simplified8.0

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

    4. Applied clear-num8.1

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

    6. Applied div-inv8.1

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

    8. Applied fma-udef8.2

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

    9. Simplified7.7

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

    if -3.9804113722305575e-241 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 0.0

    1. Initial program 57.9

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

    2. Simplified57.9

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

    3. Taylor expanded around 0 33.5

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -3.9804113722305575 \cdot 10^{-241} \lor \neg \left(\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0\right):\\ \;\;\;\;y \cdot \frac{t - z}{a - t} + \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

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