Average Error: 6.8 → 3.2
Time: 3.8s
Precision: 64
\[\left(x \cdot y - z \cdot y\right) \cdot t\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.2510353792524261 \cdot 10^{97} \lor \neg \left(y \le 2.38863567504029856 \cdot 10^{103}\right):\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right) + \left(y \cdot \mathsf{fma}\left(-z, 1, z\right)\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array}\]
\left(x \cdot y - z \cdot y\right) \cdot t
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.2510353792524261 \cdot 10^{97} \lor \neg \left(y \le 2.38863567504029856 \cdot 10^{103}\right):\\
\;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right) + \left(y \cdot \mathsf{fma}\left(-z, 1, z\right)\right) \cdot t\\

\mathbf{else}:\\
\;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r654341 = x;
        double r654342 = y;
        double r654343 = r654341 * r654342;
        double r654344 = z;
        double r654345 = r654344 * r654342;
        double r654346 = r654343 - r654345;
        double r654347 = t;
        double r654348 = r654346 * r654347;
        return r654348;
}
↓
double f(double x, double y, double z, double t) {
        double r654349 = y;
        double r654350 = -1.2510353792524261e+97;
        bool r654351 = r654349 <= r654350;
        double r654352 = 2.3886356750402986e+103;
        bool r654353 = r654349 <= r654352;
        double r654354 = !r654353;
        bool r654355 = r654351 || r654354;
        double r654356 = x;
        double r654357 = z;
        double r654358 = r654356 - r654357;
        double r654359 = t;
        double r654360 = r654359 * r654349;
        double r654361 = r654358 * r654360;
        double r654362 = -r654357;
        double r654363 = 1.0;
        double r654364 = fma(r654362, r654363, r654357);
        double r654365 = r654349 * r654364;
        double r654366 = r654365 * r654359;
        double r654367 = r654361 + r654366;
        double r654368 = r654349 * r654358;
        double r654369 = r654359 * r654368;
        double r654370 = r654355 ? r654367 : r654369;
        return r654370;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.8
Target3.0
Herbie3.2
\[\begin{array}{l} \mathbf{if}\;t \lt -9.2318795828867769 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t \lt 2.5430670515648771 \cdot 10^{83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.2510353792524261e+97 or 2.3886356750402986e+103 < y

    1. Initial program 21.4

      \[\left(x \cdot y - z \cdot y\right) \cdot t\]

    2. Simplified21.4

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

    4. Applied add-cube-cbrt21.8

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

    5. Applied add-sqr-sqrt43.7

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

    6. Applied prod-diff43.7

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

    7. Applied distribute-lft-in43.7

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

    8. Applied distribute-lft-in43.7

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

    9. Simplified16.2

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

    10. Simplified4.9

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

    if -1.2510353792524261e+97 < y < 2.3886356750402986e+103

    1. Initial program 2.7

      \[\left(x \cdot y - z \cdot y\right) \cdot t\]

    2. Simplified2.7

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.2510353792524261 \cdot 10^{97} \lor \neg \left(y \le 2.38863567504029856 \cdot 10^{103}\right):\\ \;\;\;\;\left(x - z\right) \cdot \left(t \cdot y\right) + \left(y \cdot \mathsf{fma}\left(-z, 1, z\right)\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -9.231879582886777e-80) (* (* y t) (- x z)) (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t)))

  (* (- (* x y) (* z y)) t))