Average Error: 6.7 → 0.9
Time: 10.5s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[x + \frac{\frac{z - x}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{\sqrt[3]{t}}{\sqrt[3]{y}}}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
x + \frac{\frac{z - x}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{\sqrt[3]{t}}{\sqrt[3]{y}}}
double f(double x, double y, double z, double t) {
        double r396516 = x;
        double r396517 = y;
        double r396518 = z;
        double r396519 = r396518 - r396516;
        double r396520 = r396517 * r396519;
        double r396521 = t;
        double r396522 = r396520 / r396521;
        double r396523 = r396516 + r396522;
        return r396523;
}
↓
double f(double x, double y, double z, double t) {
        double r396524 = x;
        double r396525 = z;
        double r396526 = r396525 - r396524;
        double r396527 = t;
        double r396528 = cbrt(r396527);
        double r396529 = r396528 * r396528;
        double r396530 = y;
        double r396531 = cbrt(r396530);
        double r396532 = r396531 * r396531;
        double r396533 = r396529 / r396532;
        double r396534 = r396526 / r396533;
        double r396535 = r396528 / r396531;
        double r396536 = r396534 / r396535;
        double r396537 = r396524 + r396536;
        return r396537;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.7
Target2.0
Herbie0.9
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Initial program 6.7

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

  2. Taylor expanded around 0 6.7

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

  3. Simplified1.9

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

  5. Applied add-cube-cbrt2.4

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

  6. Applied add-cube-cbrt2.6

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

  7. Applied times-frac2.6

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

  8. Applied associate-/r*0.9

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

  9. Final simplification0.9

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

Reproduce

herbie shell --seed 2020047 
(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)))