Average Error: 5.4 → 0.1
Time: 16.5s
Precision: 64
\[x \cdot x - \left(y \cdot 4.0\right) \cdot \left(z \cdot z - t\right)\]
↓
\[x \cdot x - \left(\left(z \cdot \left(y \cdot 4.0\right)\right) \cdot z + \left(-t \cdot \left(y \cdot 4.0\right)\right)\right)\]
x \cdot x - \left(y \cdot 4.0\right) \cdot \left(z \cdot z - t\right)
↓
x \cdot x - \left(\left(z \cdot \left(y \cdot 4.0\right)\right) \cdot z + \left(-t \cdot \left(y \cdot 4.0\right)\right)\right)
double f(double x, double y, double z, double t) {
        double r26095972 = x;
        double r26095973 = r26095972 * r26095972;
        double r26095974 = y;
        double r26095975 = 4.0;
        double r26095976 = r26095974 * r26095975;
        double r26095977 = z;
        double r26095978 = r26095977 * r26095977;
        double r26095979 = t;
        double r26095980 = r26095978 - r26095979;
        double r26095981 = r26095976 * r26095980;
        double r26095982 = r26095973 - r26095981;
        return r26095982;
}
↓
double f(double x, double y, double z, double t) {
        double r26095983 = x;
        double r26095984 = r26095983 * r26095983;
        double r26095985 = z;
        double r26095986 = y;
        double r26095987 = 4.0;
        double r26095988 = r26095986 * r26095987;
        double r26095989 = r26095985 * r26095988;
        double r26095990 = r26095989 * r26095985;
        double r26095991 = t;
        double r26095992 = r26095991 * r26095988;
        double r26095993 = -r26095992;
        double r26095994 = r26095990 + r26095993;
        double r26095995 = r26095984 - r26095994;
        return r26095995;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

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Results

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Target

Original5.4
Target5.4
Herbie0.1
\[x \cdot x - 4.0 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Initial program 5.4

    \[x \cdot x - \left(y \cdot 4.0\right) \cdot \left(z \cdot z - t\right)\]
  2. Using strategy rm

  3. Applied sub-neg5.4

    \[\leadsto x \cdot x - \left(y \cdot 4.0\right) \cdot \color{blue}{\left(z \cdot z + \left(-t\right)\right)}\]

  4. Applied distribute-rgt-in5.4

    \[\leadsto x \cdot x - \color{blue}{\left(\left(z \cdot z\right) \cdot \left(y \cdot 4.0\right) + \left(-t\right) \cdot \left(y \cdot 4.0\right)\right)}\]

  5. Taylor expanded around 0 5.4

    \[\leadsto x \cdot x - \left(\color{blue}{4.0 \cdot \left({z}^{2} \cdot y\right)} + \left(-t\right) \cdot \left(y \cdot 4.0\right)\right)\]

  6. Simplified0.1

    \[\leadsto x \cdot x - \left(\color{blue}{\left(z \cdot \left(y \cdot 4.0\right)\right) \cdot z} + \left(-t\right) \cdot \left(y \cdot 4.0\right)\right)\]

  7. Final simplification0.1

    \[\leadsto x \cdot x - \left(\left(z \cdot \left(y \cdot 4.0\right)\right) \cdot z + \left(-t \cdot \left(y \cdot 4.0\right)\right)\right)\]

Reproduce

herbie shell --seed 2019163 
(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))))