Average Error: 16.5 → 8.7
Time: 12.9s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -8.94712349400520929 \cdot 10^{-168}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{y \cdot \frac{t - z}{a - t}} \cdot \sqrt[3]{y \cdot \frac{t - z}{a - t}}, \sqrt[3]{y \cdot \frac{t - z}{a - t}}, x + y\right)\\ \mathbf{elif}\;a \le 8.2322134775541335 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{y \cdot \frac{t - z}{a - t}} \cdot \sqrt[3]{y \cdot \frac{t - z}{a - t}}\right) \cdot \sqrt[3]{y \cdot \frac{t - z}{a - t}} + \left(x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -8.94712349400520929 \cdot 10^{-168}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt[3]{y \cdot \frac{t - z}{a - t}} \cdot \sqrt[3]{y \cdot \frac{t - z}{a - t}}, \sqrt[3]{y \cdot \frac{t - z}{a - t}}, x + y\right)\\

\mathbf{elif}\;a \le 8.2322134775541335 \cdot 10^{-92}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r704148 = x;
        double r704149 = y;
        double r704150 = r704148 + r704149;
        double r704151 = z;
        double r704152 = t;
        double r704153 = r704151 - r704152;
        double r704154 = r704153 * r704149;
        double r704155 = a;
        double r704156 = r704155 - r704152;
        double r704157 = r704154 / r704156;
        double r704158 = r704150 - r704157;
        return r704158;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r704159 = a;
        double r704160 = -8.94712349400521e-168;
        bool r704161 = r704159 <= r704160;
        double r704162 = y;
        double r704163 = t;
        double r704164 = z;
        double r704165 = r704163 - r704164;
        double r704166 = r704159 - r704163;
        double r704167 = r704165 / r704166;
        double r704168 = r704162 * r704167;
        double r704169 = cbrt(r704168);
        double r704170 = r704169 * r704169;
        double r704171 = x;
        double r704172 = r704171 + r704162;
        double r704173 = fma(r704170, r704169, r704172);
        double r704174 = 8.232213477554133e-92;
        bool r704175 = r704159 <= r704174;
        double r704176 = r704164 / r704163;
        double r704177 = fma(r704176, r704162, r704171);
        double r704178 = r704170 * r704169;
        double r704179 = r704178 + r704172;
        double r704180 = r704175 ? r704177 : r704179;
        double r704181 = r704161 ? r704173 : r704180;
        return r704181;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.5
Target8.5
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -8.94712349400521e-168

    1. Initial program 15.3

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

    2. Simplified9.1

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

    4. Applied fma-udef9.1

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

    6. Applied div-inv9.1

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

    8. Applied add-cube-cbrt9.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\left(t - z\right) \cdot \frac{1}{a - t}\right) \cdot y} \cdot \sqrt[3]{\left(\left(t - z\right) \cdot \frac{1}{a - t}\right) \cdot y}\right) \cdot \sqrt[3]{\left(\left(t - z\right) \cdot \frac{1}{a - t}\right) \cdot y}} + \left(x + y\right)\]

    9. Simplified9.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{y \cdot \frac{t - z}{a - t}} \cdot \sqrt[3]{y \cdot \frac{t - z}{a - t}}\right)} \cdot \sqrt[3]{\left(\left(t - z\right) \cdot \frac{1}{a - t}\right) \cdot y} + \left(x + y\right)\]

    10. Simplified9.3

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

    12. Applied fma-def9.3

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

    if -8.94712349400521e-168 < a < 8.232213477554133e-92

    1. Initial program 20.7

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

    2. Simplified19.5

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

    3. Taylor expanded around inf 10.1

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

    4. Simplified8.9

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

    if 8.232213477554133e-92 < a

    1. Initial program 14.5

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

    2. Simplified7.6

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

    4. Applied fma-udef7.6

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

    6. Applied div-inv7.6

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

    8. Applied add-cube-cbrt7.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\left(t - z\right) \cdot \frac{1}{a - t}\right) \cdot y} \cdot \sqrt[3]{\left(\left(t - z\right) \cdot \frac{1}{a - t}\right) \cdot y}\right) \cdot \sqrt[3]{\left(\left(t - z\right) \cdot \frac{1}{a - t}\right) \cdot y}} + \left(x + y\right)\]

    9. Simplified7.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{y \cdot \frac{t - z}{a - t}} \cdot \sqrt[3]{y \cdot \frac{t - z}{a - t}}\right)} \cdot \sqrt[3]{\left(\left(t - z\right) \cdot \frac{1}{a - t}\right) \cdot y} + \left(x + y\right)\]

    10. Simplified7.8

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -8.94712349400520929 \cdot 10^{-168}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{y \cdot \frac{t - z}{a - t}} \cdot \sqrt[3]{y \cdot \frac{t - z}{a - t}}, \sqrt[3]{y \cdot \frac{t - z}{a - t}}, x + y\right)\\ \mathbf{elif}\;a \le 8.2322134775541335 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{y \cdot \frac{t - z}{a - t}} \cdot \sqrt[3]{y \cdot \frac{t - z}{a - t}}\right) \cdot \sqrt[3]{y \cdot \frac{t - z}{a - t}} + \left(x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))