Average Error: 6.8 → 1.7
Time: 22.0s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[x + \left(y \cdot \frac{\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{t}}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
x + \left(y \cdot \frac{\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{z - x}}{\sqrt[3]{t}}
double f(double x, double y, double z, double t) {
        double r325957 = x;
        double r325958 = y;
        double r325959 = z;
        double r325960 = r325959 - r325957;
        double r325961 = r325958 * r325960;
        double r325962 = t;
        double r325963 = r325961 / r325962;
        double r325964 = r325957 + r325963;
        return r325964;
}
↓
double f(double x, double y, double z, double t) {
        double r325965 = x;
        double r325966 = y;
        double r325967 = z;
        double r325968 = r325967 - r325965;
        double r325969 = cbrt(r325968);
        double r325970 = r325969 * r325969;
        double r325971 = t;
        double r325972 = cbrt(r325971);
        double r325973 = r325972 * r325972;
        double r325974 = r325970 / r325973;
        double r325975 = r325966 * r325974;
        double r325976 = r325969 / r325972;
        double r325977 = r325975 * r325976;
        double r325978 = r325965 + r325977;
        return r325978;
}

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

Original6.8
Target2.1
Herbie1.7
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Initial program 6.8

    \[x + \frac{y \cdot \left(z - x\right)}{t}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity6.8

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

  4. Applied times-frac6.2

    \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{z - x}{t}}\]

  5. Simplified6.2

    \[\leadsto x + \color{blue}{y} \cdot \frac{z - x}{t}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt6.7

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

  8. Applied add-cube-cbrt6.8

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

  9. Applied times-frac6.8

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

  10. Applied associate-*r*1.7

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

  11. Final simplification1.7

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

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