Average Error: 10.4 → 0.8
Time: 17.7s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.012545261316454786172528157872615913891 \cdot 10^{-96} \lor \neg \left(t \le 2.588908446285731015618837526142587793708 \cdot 10^{-201}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z} - \frac{z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \left(\sqrt[3]{y - z} \cdot t\right)}{a - z}\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;t \le -6.012545261316454786172528157872615913891 \cdot 10^{-96} \lor \neg \left(t \le 2.588908446285731015618837526142587793708 \cdot 10^{-201}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a - z} - \frac{z}{a - z}, t, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r404201 = x;
        double r404202 = y;
        double r404203 = z;
        double r404204 = r404202 - r404203;
        double r404205 = t;
        double r404206 = r404204 * r404205;
        double r404207 = a;
        double r404208 = r404207 - r404203;
        double r404209 = r404206 / r404208;
        double r404210 = r404201 + r404209;
        return r404210;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r404211 = t;
        double r404212 = -6.012545261316455e-96;
        bool r404213 = r404211 <= r404212;
        double r404214 = 2.588908446285731e-201;
        bool r404215 = r404211 <= r404214;
        double r404216 = !r404215;
        bool r404217 = r404213 || r404216;
        double r404218 = y;
        double r404219 = a;
        double r404220 = z;
        double r404221 = r404219 - r404220;
        double r404222 = r404218 / r404221;
        double r404223 = r404220 / r404221;
        double r404224 = r404222 - r404223;
        double r404225 = x;
        double r404226 = fma(r404224, r404211, r404225);
        double r404227 = r404218 - r404220;
        double r404228 = cbrt(r404227);
        double r404229 = r404228 * r404228;
        double r404230 = r404228 * r404211;
        double r404231 = r404229 * r404230;
        double r404232 = r404231 / r404221;
        double r404233 = r404225 + r404232;
        double r404234 = r404217 ? r404226 : r404233;
        return r404234;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.4
Target0.6
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;t \lt -1.068297449017406694366747246993994850729 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.911094988758637497591020599238553861375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -6.012545261316455e-96 or 2.588908446285731e-201 < t

    1. Initial program 14.3

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

    2. Simplified0.9

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

    4. Applied div-sub0.9

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

    if -6.012545261316455e-96 < t < 2.588908446285731e-201

    1. Initial program 0.4

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

    3. Applied add-cube-cbrt0.6

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

    4. Applied associate-*l*0.6

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.012545261316454786172528157872615913891 \cdot 10^{-96} \lor \neg \left(t \le 2.588908446285731015618837526142587793708 \cdot 10^{-201}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z} - \frac{z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \left(\sqrt[3]{y - z} \cdot t\right)}{a - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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