Average Error: 6.1 → 2.0
Time: 3.1s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{t}} \cdot z\right) + \mathsf{fma}\left(-x, \frac{y}{t}, x\right)\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{t}} \cdot z\right) + \mathsf{fma}\left(-x, \frac{y}{t}, x\right)
double f(double x, double y, double z, double t) {
        double r279485 = x;
        double r279486 = y;
        double r279487 = z;
        double r279488 = r279487 - r279485;
        double r279489 = r279486 * r279488;
        double r279490 = t;
        double r279491 = r279489 / r279490;
        double r279492 = r279485 + r279491;
        return r279492;
}
↓
double f(double x, double y, double z, double t) {
        double r279493 = y;
        double r279494 = cbrt(r279493);
        double r279495 = r279494 * r279494;
        double r279496 = t;
        double r279497 = cbrt(r279496);
        double r279498 = r279497 * r279497;
        double r279499 = r279495 / r279498;
        double r279500 = r279494 / r279497;
        double r279501 = z;
        double r279502 = r279500 * r279501;
        double r279503 = r279499 * r279502;
        double r279504 = x;
        double r279505 = -r279504;
        double r279506 = r279493 / r279496;
        double r279507 = fma(r279505, r279506, r279504);
        double r279508 = r279503 + r279507;
        return r279508;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.1
Target2.2
Herbie2.0
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Initial program 6.1

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

  2. Simplified2.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)}\]
  3. Using strategy rm

  4. Applied fma-udef2.2

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

  6. Applied sub-neg2.2

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

  7. Applied distribute-lft-in2.2

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

  8. Applied associate-+l+2.2

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

  9. Simplified2.2

    \[\leadsto \frac{y}{t} \cdot z + \color{blue}{\mathsf{fma}\left(-x, \frac{y}{t}, x\right)}\]
  10. Using strategy rm

  11. Applied add-cube-cbrt2.5

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

  12. Applied add-cube-cbrt2.6

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

  13. Applied times-frac2.6

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

  14. Applied associate-*l*2.0

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

  15. Final simplification2.0

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(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)))