Average Error: 9.7 → 0.1
Time: 4.6s
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 r1399197 = x;
        double r1399198 = y;
        double r1399199 = r1399197 / r1399198;
        double r1399200 = 2.0;
        double r1399201 = z;
        double r1399202 = r1399201 * r1399200;
        double r1399203 = 1.0;
        double r1399204 = t;
        double r1399205 = r1399203 - r1399204;
        double r1399206 = r1399202 * r1399205;
        double r1399207 = r1399200 + r1399206;
        double r1399208 = r1399204 * r1399201;
        double r1399209 = r1399207 / r1399208;
        double r1399210 = r1399199 + r1399209;
        return r1399210;
}
↓
double f(double x, double y, double z, double t) {
        double r1399211 = x;
        double r1399212 = y;
        double r1399213 = r1399211 / r1399212;
        double r1399214 = 2.0;
        double r1399215 = 1.0;
        double r1399216 = t;
        double r1399217 = r1399215 / r1399216;
        double r1399218 = z;
        double r1399219 = r1399217 / r1399218;
        double r1399220 = r1399214 * r1399217;
        double r1399221 = r1399220 - r1399214;
        double r1399222 = fma(r1399214, r1399219, r1399221);
        double r1399223 = r1399213 + r1399222;
        return r1399223;
}

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))))