Average Error: 16.5 → 8.7
Time: 19.1s
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 r637318 = x;
        double r637319 = y;
        double r637320 = r637318 + r637319;
        double r637321 = z;
        double r637322 = t;
        double r637323 = r637321 - r637322;
        double r637324 = r637323 * r637319;
        double r637325 = a;
        double r637326 = r637325 - r637322;
        double r637327 = r637324 / r637326;
        double r637328 = r637320 - r637327;
        return r637328;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r637329 = a;
        double r637330 = -8.94712349400521e-168;
        bool r637331 = r637329 <= r637330;
        double r637332 = y;
        double r637333 = t;
        double r637334 = z;
        double r637335 = r637333 - r637334;
        double r637336 = r637329 - r637333;
        double r637337 = r637335 / r637336;
        double r637338 = r637332 * r637337;
        double r637339 = cbrt(r637338);
        double r637340 = r637339 * r637339;
        double r637341 = x;
        double r637342 = r637341 + r637332;
        double r637343 = fma(r637340, r637339, r637342);
        double r637344 = 8.232213477554133e-92;
        bool r637345 = r637329 <= r637344;
        double r637346 = r637334 / r637333;
        double r637347 = fma(r637346, r637332, r637341);
        double r637348 = r637340 * r637339;
        double r637349 = r637348 + r637342;
        double r637350 = r637345 ? r637347 : r637349;
        double r637351 = r637331 ? r637343 : r637350;
        return r637351;
}

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))))