Average Error: 6.2 → 0.1
Time: 16.2s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[x \cdot x - \left(4 \cdot \left({z}^{\left(\frac{2}{2}\right)} \cdot \left(z \cdot y\right)\right) + \left(y \cdot 4\right) \cdot \left(-t\right)\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
x \cdot x - \left(4 \cdot \left({z}^{\left(\frac{2}{2}\right)} \cdot \left(z \cdot y\right)\right) + \left(y \cdot 4\right) \cdot \left(-t\right)\right)
double f(double x, double y, double z, double t) {
        double r668380 = x;
        double r668381 = r668380 * r668380;
        double r668382 = y;
        double r668383 = 4.0;
        double r668384 = r668382 * r668383;
        double r668385 = z;
        double r668386 = r668385 * r668385;
        double r668387 = t;
        double r668388 = r668386 - r668387;
        double r668389 = r668384 * r668388;
        double r668390 = r668381 - r668389;
        return r668390;
}
↓
double f(double x, double y, double z, double t) {
        double r668391 = x;
        double r668392 = r668391 * r668391;
        double r668393 = 4.0;
        double r668394 = z;
        double r668395 = 2.0;
        double r668396 = r668395 / r668395;
        double r668397 = pow(r668394, r668396);
        double r668398 = y;
        double r668399 = r668394 * r668398;
        double r668400 = r668397 * r668399;
        double r668401 = r668393 * r668400;
        double r668402 = r668398 * r668393;
        double r668403 = t;
        double r668404 = -r668403;
        double r668405 = r668402 * r668404;
        double r668406 = r668401 + r668405;
        double r668407 = r668392 - r668406;
        return r668407;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Results

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Target

Original6.2
Target6.2
Herbie0.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Initial program 6.2

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

  3. Applied sub-neg6.2

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

  4. Applied distribute-lft-in6.2

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

  5. Simplified6.2

    \[\leadsto x \cdot x - \left(\color{blue}{4 \cdot \left({z}^{2} \cdot y\right)} + \left(y \cdot 4\right) \cdot \left(-t\right)\right)\]
  6. Using strategy rm

  7. Applied sqr-pow6.2

    \[\leadsto x \cdot x - \left(4 \cdot \left(\color{blue}{\left({z}^{\left(\frac{2}{2}\right)} \cdot {z}^{\left(\frac{2}{2}\right)}\right)} \cdot y\right) + \left(y \cdot 4\right) \cdot \left(-t\right)\right)\]

  8. Applied associate-*l*0.1

    \[\leadsto x \cdot x - \left(4 \cdot \color{blue}{\left({z}^{\left(\frac{2}{2}\right)} \cdot \left({z}^{\left(\frac{2}{2}\right)} \cdot y\right)\right)} + \left(y \cdot 4\right) \cdot \left(-t\right)\right)\]

  9. Simplified0.1

    \[\leadsto x \cdot x - \left(4 \cdot \left({z}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{\left(z \cdot y\right)}\right) + \left(y \cdot 4\right) \cdot \left(-t\right)\right)\]

  10. Final simplification0.1

    \[\leadsto x \cdot x - \left(4 \cdot \left({z}^{\left(\frac{2}{2}\right)} \cdot \left(z \cdot y\right)\right) + \left(y \cdot 4\right) \cdot \left(-t\right)\right)\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4) (- (* z z) t))))