Average Error: 8.9 → 0.1
Time: 19.8s
Precision: 64
\[\frac{x}{y} + \frac{2.0 + \left(z \cdot 2.0\right) \cdot \left(1.0 - t\right)}{t \cdot z}\]
↓
\[\left(\frac{\frac{\mathsf{fma}\left(2.0, 1.0 \cdot z, 2.0\right)}{z}}{t} + \frac{x}{y}\right) - 2.0\]
\frac{x}{y} + \frac{2.0 + \left(z \cdot 2.0\right) \cdot \left(1.0 - t\right)}{t \cdot z}
↓
\left(\frac{\frac{\mathsf{fma}\left(2.0, 1.0 \cdot z, 2.0\right)}{z}}{t} + \frac{x}{y}\right) - 2.0
double f(double x, double y, double z, double t) {
        double r32806044 = x;
        double r32806045 = y;
        double r32806046 = r32806044 / r32806045;
        double r32806047 = 2.0;
        double r32806048 = z;
        double r32806049 = r32806048 * r32806047;
        double r32806050 = 1.0;
        double r32806051 = t;
        double r32806052 = r32806050 - r32806051;
        double r32806053 = r32806049 * r32806052;
        double r32806054 = r32806047 + r32806053;
        double r32806055 = r32806051 * r32806048;
        double r32806056 = r32806054 / r32806055;
        double r32806057 = r32806046 + r32806056;
        return r32806057;
}
↓
double f(double x, double y, double z, double t) {
        double r32806058 = 2.0;
        double r32806059 = 1.0;
        double r32806060 = z;
        double r32806061 = r32806059 * r32806060;
        double r32806062 = fma(r32806058, r32806061, r32806058);
        double r32806063 = r32806062 / r32806060;
        double r32806064 = t;
        double r32806065 = r32806063 / r32806064;
        double r32806066 = x;
        double r32806067 = y;
        double r32806068 = r32806066 / r32806067;
        double r32806069 = r32806065 + r32806068;
        double r32806070 = r32806069 - r32806058;
        return r32806070;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original8.9
Target0.1
Herbie0.1
\[\frac{\frac{2.0}{z} + 2.0}{t} - \left(2.0 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 8.9

    \[\frac{x}{y} + \frac{2.0 + \left(z \cdot 2.0\right) \cdot \left(1.0 - t\right)}{t \cdot z}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2.0}{t}, \frac{\mathsf{fma}\left(z, 1.0, 1\right)}{z}, \frac{x}{y}\right) - 2.0}\]
  3. Using strategy rm

  4. Applied fma-udef0.1

    \[\leadsto \color{blue}{\left(\frac{2.0}{t} \cdot \frac{\mathsf{fma}\left(z, 1.0, 1\right)}{z} + \frac{x}{y}\right)} - 2.0\]

  5. Simplified0.1

    \[\leadsto \left(\color{blue}{\frac{\frac{\mathsf{fma}\left(2.0, z \cdot 1.0, 2.0\right)}{z}}{t}} + \frac{x}{y}\right) - 2.0\]

  6. Final simplification0.1

    \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(2.0, 1.0 \cdot z, 2.0\right)}{z}}{t} + \frac{x}{y}\right) - 2.0\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"

  :herbie-target
  (- (/ (+ (/ 2.0 z) 2.0) t) (- 2.0 (/ x y)))

  (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))