Average Error: 9.7 → 0.1
Time: 7.9s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \mathsf{fma}\left(2, \frac{\frac{1}{t}}{z}, 2 \cdot \frac{1}{t} - 2\right)\]
\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{\frac{1}{t}}{z}, 2 \cdot \frac{1}{t} - 2\right)
double f(double x, double y, double z, double t) {
        double r3920 = x;
        double r3921 = y;
        double r3922 = r3920 / r3921;
        double r3923 = 2.0;
        double r3924 = z;
        double r3925 = r3924 * r3923;
        double r3926 = 1.0;
        double r3927 = t;
        double r3928 = r3926 - r3927;
        double r3929 = r3925 * r3928;
        double r3930 = r3923 + r3929;
        double r3931 = r3927 * r3924;
        double r3932 = r3930 / r3931;
        double r3933 = r3922 + r3932;
        return r3933;
}
↓
double f(double x, double y, double z, double t) {
        double r3934 = x;
        double r3935 = y;
        double r3936 = r3934 / r3935;
        double r3937 = 2.0;
        double r3938 = 1.0;
        double r3939 = t;
        double r3940 = r3938 / r3939;
        double r3941 = z;
        double r3942 = r3940 / r3941;
        double r3943 = r3937 * r3940;
        double r3944 = r3943 - r3937;
        double r3945 = fma(r3937, r3942, r3944);
        double r3946 = r3936 + r3945;
        return r3946;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.7
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.7

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

  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 associate-/r*0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020025 +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))))