Average Error: 9.3 → 0.1
Time: 3.5s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \mathsf{fma}\left(2, \frac{1}{t} \cdot \frac{1}{z}, 2 \cdot \frac{1}{t} - 2\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\frac{x}{y} + \mathsf{fma}\left(2, \frac{1}{t} \cdot \frac{1}{z}, 2 \cdot \frac{1}{t} - 2\right)
double f(double x, double y, double z, double t) {
        double r759140 = x;
        double r759141 = y;
        double r759142 = r759140 / r759141;
        double r759143 = 2.0;
        double r759144 = z;
        double r759145 = r759144 * r759143;
        double r759146 = 1.0;
        double r759147 = t;
        double r759148 = r759146 - r759147;
        double r759149 = r759145 * r759148;
        double r759150 = r759143 + r759149;
        double r759151 = r759147 * r759144;
        double r759152 = r759150 / r759151;
        double r759153 = r759142 + r759152;
        return r759153;
}
↓
double f(double x, double y, double z, double t) {
        double r759154 = x;
        double r759155 = y;
        double r759156 = r759154 / r759155;
        double r759157 = 2.0;
        double r759158 = 1.0;
        double r759159 = t;
        double r759160 = r759158 / r759159;
        double r759161 = z;
        double r759162 = r759158 / r759161;
        double r759163 = r759160 * r759162;
        double r759164 = r759157 * r759160;
        double r759165 = r759164 - r759157;
        double r759166 = fma(r759157, r759163, r759165);
        double r759167 = r759156 + r759166;
        return r759167;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.3
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.3

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

  2. Taylor expanded around 0 0.1

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

  3. Simplified0.1

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

  5. Applied add-cube-cbrt0.1

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

  6. Applied times-frac0.1

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

  7. Simplified0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020033 +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))))