Average Error: 9.6 → 0.1
Time: 4.1s
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 r894198 = x;
        double r894199 = y;
        double r894200 = r894198 / r894199;
        double r894201 = 2.0;
        double r894202 = z;
        double r894203 = r894202 * r894201;
        double r894204 = 1.0;
        double r894205 = t;
        double r894206 = r894204 - r894205;
        double r894207 = r894203 * r894206;
        double r894208 = r894201 + r894207;
        double r894209 = r894205 * r894202;
        double r894210 = r894208 / r894209;
        double r894211 = r894200 + r894210;
        return r894211;
}
↓
double f(double x, double y, double z, double t) {
        double r894212 = x;
        double r894213 = y;
        double r894214 = r894212 / r894213;
        double r894215 = 2.0;
        double r894216 = 1.0;
        double r894217 = t;
        double r894218 = r894216 / r894217;
        double r894219 = z;
        double r894220 = r894218 / r894219;
        double r894221 = r894215 * r894218;
        double r894222 = r894221 - r894215;
        double r894223 = fma(r894215, r894220, r894222);
        double r894224 = r894214 + r894223;
        return r894224;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.6
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.6

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

  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 2020081 +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))))