Average Error: 6.0 → 0.1
Time: 13.5s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[x \cdot x - \left(\left(4 \cdot y\right) \cdot \left(-t\right) + \left(\left(z \cdot 4\right) \cdot y\right) \cdot z\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
x \cdot x - \left(\left(4 \cdot y\right) \cdot \left(-t\right) + \left(\left(z \cdot 4\right) \cdot y\right) \cdot z\right)
double f(double x, double y, double z, double t) {
        double r570304 = x;
        double r570305 = r570304 * r570304;
        double r570306 = y;
        double r570307 = 4.0;
        double r570308 = r570306 * r570307;
        double r570309 = z;
        double r570310 = r570309 * r570309;
        double r570311 = t;
        double r570312 = r570310 - r570311;
        double r570313 = r570308 * r570312;
        double r570314 = r570305 - r570313;
        return r570314;
}
↓
double f(double x, double y, double z, double t) {
        double r570315 = x;
        double r570316 = r570315 * r570315;
        double r570317 = 4.0;
        double r570318 = y;
        double r570319 = r570317 * r570318;
        double r570320 = t;
        double r570321 = -r570320;
        double r570322 = r570319 * r570321;
        double r570323 = z;
        double r570324 = r570323 * r570317;
        double r570325 = r570324 * r570318;
        double r570326 = r570325 * r570323;
        double r570327 = r570322 + r570326;
        double r570328 = r570316 - r570327;
        return r570328;
}

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

Original6.0
Target6.0
Herbie0.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Initial program 6.0

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

  3. Applied sub-neg6.0

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

  4. Applied distribute-lft-in6.0

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

  5. Simplified6.0

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

  6. Simplified6.0

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

  7. Taylor expanded around 0 6.0

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019194 
(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))))