Average Error: 9.3 → 0.1
Time: 3.2s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \mathsf{fma}\left(2, \frac{\frac{1}{z}}{t}, 2 \cdot \frac{1}{t} - 2\right)\]
\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{\frac{1}{z}}{t}, 2 \cdot \frac{1}{t} - 2\right)
double f(double x, double y, double z, double t) {
        double r837076 = x;
        double r837077 = y;
        double r837078 = r837076 / r837077;
        double r837079 = 2.0;
        double r837080 = z;
        double r837081 = r837080 * r837079;
        double r837082 = 1.0;
        double r837083 = t;
        double r837084 = r837082 - r837083;
        double r837085 = r837081 * r837084;
        double r837086 = r837079 + r837085;
        double r837087 = r837083 * r837080;
        double r837088 = r837086 / r837087;
        double r837089 = r837078 + r837088;
        return r837089;
}
↓
double f(double x, double y, double z, double t) {
        double r837090 = x;
        double r837091 = y;
        double r837092 = r837090 / r837091;
        double r837093 = 2.0;
        double r837094 = 1.0;
        double r837095 = z;
        double r837096 = r837094 / r837095;
        double r837097 = t;
        double r837098 = r837096 / r837097;
        double r837099 = r837094 / r837097;
        double r837100 = r837093 * r837099;
        double r837101 = r837100 - r837093;
        double r837102 = fma(r837093, r837098, r837101);
        double r837103 = r837092 + r837102;
        return r837103;
}

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. Using strategy rm

  10. Applied associate-*l/0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020064 +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))))