Average Error: 9.4 → 0.1
Time: 16.5s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\left(\left(\frac{2}{t} + \frac{2}{t \cdot z}\right) - 2\right) + \frac{x}{y}\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\left(\left(\frac{2}{t} + \frac{2}{t \cdot z}\right) - 2\right) + \frac{x}{y}
double f(double x, double y, double z, double t) {
        double r532278 = x;
        double r532279 = y;
        double r532280 = r532278 / r532279;
        double r532281 = 2.0;
        double r532282 = z;
        double r532283 = r532282 * r532281;
        double r532284 = 1.0;
        double r532285 = t;
        double r532286 = r532284 - r532285;
        double r532287 = r532283 * r532286;
        double r532288 = r532281 + r532287;
        double r532289 = r532285 * r532282;
        double r532290 = r532288 / r532289;
        double r532291 = r532280 + r532290;
        return r532291;
}
↓
double f(double x, double y, double z, double t) {
        double r532292 = 2.0;
        double r532293 = t;
        double r532294 = r532292 / r532293;
        double r532295 = z;
        double r532296 = r532293 * r532295;
        double r532297 = r532292 / r532296;
        double r532298 = r532294 + r532297;
        double r532299 = r532298 - r532292;
        double r532300 = x;
        double r532301 = y;
        double r532302 = r532300 / r532301;
        double r532303 = r532299 + r532302;
        return r532303;
}

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

Original9.4
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.4

    \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]

  2. Simplified9.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, 1 - t, 1\right)}{z}, \frac{2}{t}, \frac{x}{y}\right)}\]

  3. Taylor expanded around 0 0.1

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{z} + 1\right) - t}, \frac{2}{t}, \frac{x}{y}\right)\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.4

    \[\leadsto \mathsf{fma}\left(\left(\frac{1}{z} + 1\right) - \color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}, \frac{2}{t}, \frac{x}{y}\right)\]

  6. Applied add-cube-cbrt0.6

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{\frac{1}{z} + 1} \cdot \sqrt[3]{\frac{1}{z} + 1}\right) \cdot \sqrt[3]{\frac{1}{z} + 1}} - \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}, \frac{2}{t}, \frac{x}{y}\right)\]

  7. Applied prod-diff0.6

    \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{1}{z} + 1} \cdot \sqrt[3]{\frac{1}{z} + 1}, \sqrt[3]{\frac{1}{z} + 1}, -\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{t}, \sqrt[3]{t} \cdot \sqrt[3]{t}, \sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)}, \frac{2}{t}, \frac{x}{y}\right)\]

  8. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left({\left(\sqrt[3]{\frac{1}{z} + 1}\right)}^{3} - t\right)} + \mathsf{fma}\left(-\sqrt[3]{t}, \sqrt[3]{t} \cdot \sqrt[3]{t}, \sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right), \frac{2}{t}, \frac{x}{y}\right)\]

  9. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\left({\left(\sqrt[3]{\frac{1}{z} + 1}\right)}^{3} - t\right) + \color{blue}{\left(t - t\right)}, \frac{2}{t}, \frac{x}{y}\right)\]
  10. Using strategy rm

  11. Applied fma-udef0.4

    \[\leadsto \color{blue}{\left(\left({\left(\sqrt[3]{\frac{1}{z} + 1}\right)}^{3} - t\right) + \left(t - t\right)\right) \cdot \frac{2}{t} + \frac{x}{y}}\]

  12. Simplified0.1

    \[\leadsto \color{blue}{\frac{2}{t} \cdot \left(\frac{1}{z} + \left(1 - t\right)\right)} + \frac{x}{y}\]

  13. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{\left(\left(2 \cdot \frac{1}{t \cdot z} + 2 \cdot \frac{1}{t}\right) - 2\right)} + \frac{x}{y}\]

  14. Simplified0.1

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

  15. Final simplification0.1

    \[\leadsto \left(\left(\frac{2}{t} + \frac{2}{t \cdot z}\right) - 2\right) + \frac{x}{y}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"
  :precision binary64

  :herbie-target
  (- (/ (+ (/ 2 z) 2) t) (- 2 (/ x y)))

  (+ (/ x y) (/ (+ 2 (* (* z 2) (- 1 t))) (* t z))))