Average Error: 16.7 → 8.1
Time: 6.4s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -9.252090162115357006237294731896797375812 \cdot 10^{100} \lor \neg \left(t \le 3.516092148404992652607455309589265621051 \cdot 10^{107}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}, x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -9.252090162115357006237294731896797375812 \cdot 10^{100} \lor \neg \left(t \le 3.516092148404992652607455309589265621051 \cdot 10^{107}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}, x + y\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r586064 = x;
        double r586065 = y;
        double r586066 = r586064 + r586065;
        double r586067 = z;
        double r586068 = t;
        double r586069 = r586067 - r586068;
        double r586070 = r586069 * r586065;
        double r586071 = a;
        double r586072 = r586071 - r586068;
        double r586073 = r586070 / r586072;
        double r586074 = r586066 - r586073;
        return r586074;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r586075 = t;
        double r586076 = -9.252090162115357e+100;
        bool r586077 = r586075 <= r586076;
        double r586078 = 3.5160921484049927e+107;
        bool r586079 = r586075 <= r586078;
        double r586080 = !r586079;
        bool r586081 = r586077 || r586080;
        double r586082 = z;
        double r586083 = r586082 / r586075;
        double r586084 = y;
        double r586085 = x;
        double r586086 = fma(r586083, r586084, r586085);
        double r586087 = r586075 - r586082;
        double r586088 = cbrt(r586087);
        double r586089 = r586088 * r586088;
        double r586090 = a;
        double r586091 = r586090 - r586075;
        double r586092 = cbrt(r586091);
        double r586093 = r586092 * r586092;
        double r586094 = cbrt(r586084);
        double r586095 = r586094 * r586094;
        double r586096 = r586093 / r586095;
        double r586097 = r586089 / r586096;
        double r586098 = r586092 / r586094;
        double r586099 = r586088 / r586098;
        double r586100 = r586085 + r586084;
        double r586101 = fma(r586097, r586099, r586100);
        double r586102 = r586081 ? r586086 : r586101;
        return r586102;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.7
Target8.3
Herbie8.1
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -9.252090162115357e+100 or 3.5160921484049927e+107 < t

    1. Initial program 30.2

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

    2. Simplified21.0

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

    4. Applied clear-num21.1

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

    6. Applied fma-udef21.2

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

    7. Simplified21.2

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

    8. Taylor expanded around inf 17.5

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

    9. Simplified12.1

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

    if -9.252090162115357e+100 < t < 3.5160921484049927e+107

    1. Initial program 9.1

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

    2. Simplified6.9

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

    4. Applied clear-num7.0

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

    6. Applied fma-udef7.0

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

    7. Simplified6.8

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

    9. Applied add-cube-cbrt7.0

      \[\leadsto \frac{t - z}{\frac{a - t}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}} + \left(x + y\right)\]

    10. Applied add-cube-cbrt7.1

      \[\leadsto \frac{t - z}{\frac{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}} + \left(x + y\right)\]

    11. Applied times-frac7.1

      \[\leadsto \frac{t - z}{\color{blue}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}} + \left(x + y\right)\]

    12. Applied add-cube-cbrt7.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}\right) \cdot \sqrt[3]{t - z}}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}} + \left(x + y\right)\]

    13. Applied times-frac5.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{\sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}} + \left(x + y\right)\]

    14. Applied fma-def5.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}, x + y\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.252090162115357006237294731896797375812 \cdot 10^{100} \lor \neg \left(t \le 3.516092148404992652607455309589265621051 \cdot 10^{107}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}, x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))))