Average Error: 6.0 → 3.0
Time: 20.5s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 2.03849011016511565786308164633915074586 \cdot 10^{302}:\\ \;\;\;\;\mathsf{fma}\left(-\left(z \cdot z - t\right), y \cdot 4, \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right) + \mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 2.03849011016511565786308164633915074586 \cdot 10^{302}:\\
\;\;\;\;\mathsf{fma}\left(-\left(z \cdot z - t\right), y \cdot 4, \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right) + \mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(z - \sqrt{t}\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r26696472 = x;
        double r26696473 = r26696472 * r26696472;
        double r26696474 = y;
        double r26696475 = 4.0;
        double r26696476 = r26696474 * r26696475;
        double r26696477 = z;
        double r26696478 = r26696477 * r26696477;
        double r26696479 = t;
        double r26696480 = r26696478 - r26696479;
        double r26696481 = r26696476 * r26696480;
        double r26696482 = r26696473 - r26696481;
        return r26696482;
}
↓
double f(double x, double y, double z, double t) {
        double r26696483 = z;
        double r26696484 = r26696483 * r26696483;
        double r26696485 = 2.0384901101651157e+302;
        bool r26696486 = r26696484 <= r26696485;
        double r26696487 = t;
        double r26696488 = r26696484 - r26696487;
        double r26696489 = -r26696488;
        double r26696490 = y;
        double r26696491 = 4.0;
        double r26696492 = r26696490 * r26696491;
        double r26696493 = r26696492 * r26696488;
        double r26696494 = fma(r26696489, r26696492, r26696493);
        double r26696495 = x;
        double r26696496 = -r26696493;
        double r26696497 = fma(r26696495, r26696495, r26696496);
        double r26696498 = r26696494 + r26696497;
        double r26696499 = r26696495 * r26696495;
        double r26696500 = sqrt(r26696487);
        double r26696501 = r26696500 + r26696483;
        double r26696502 = r26696492 * r26696501;
        double r26696503 = r26696483 - r26696500;
        double r26696504 = r26696502 * r26696503;
        double r26696505 = r26696499 - r26696504;
        double r26696506 = r26696486 ? r26696498 : r26696505;
        return r26696506;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.0
Target6.0
Herbie3.0
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 2.0384901101651157e+302

    1. Initial program 0.1

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
    2. Using strategy rm

    3. Applied prod-diff0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -\left(z \cdot z - t\right) \cdot \left(y \cdot 4\right)\right) + \mathsf{fma}\left(-\left(z \cdot z - t\right), y \cdot 4, \left(z \cdot z - t\right) \cdot \left(y \cdot 4\right)\right)}\]

    if 2.0384901101651157e+302 < (* z z)

    1. Initial program 61.2

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt62.8

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - \color{blue}{\sqrt{t} \cdot \sqrt{t}}\right)\]

    4. Applied difference-of-squares62.8

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(\left(z + \sqrt{t}\right) \cdot \left(z - \sqrt{t}\right)\right)}\]

    5. Applied associate-*r*30.3

      \[\leadsto x \cdot x - \color{blue}{\left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 2.03849011016511565786308164633915074586 \cdot 10^{302}:\\ \;\;\;\;\mathsf{fma}\left(-\left(z \cdot z - t\right), y \cdot 4, \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right) + \mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))