Average Error: 9.6 → 0.1
Time: 17.2s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(\frac{2}{t}, \frac{\mathsf{fma}\left(z, 1, 1\right)}{z}, \frac{x}{y}\right) - 2\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\mathsf{fma}\left(\frac{2}{t}, \frac{\mathsf{fma}\left(z, 1, 1\right)}{z}, \frac{x}{y}\right) - 2
double f(double x, double y, double z, double t) {
        double r29950383 = x;
        double r29950384 = y;
        double r29950385 = r29950383 / r29950384;
        double r29950386 = 2.0;
        double r29950387 = z;
        double r29950388 = r29950387 * r29950386;
        double r29950389 = 1.0;
        double r29950390 = t;
        double r29950391 = r29950389 - r29950390;
        double r29950392 = r29950388 * r29950391;
        double r29950393 = r29950386 + r29950392;
        double r29950394 = r29950390 * r29950387;
        double r29950395 = r29950393 / r29950394;
        double r29950396 = r29950385 + r29950395;
        return r29950396;
}
↓
double f(double x, double y, double z, double t) {
        double r29950397 = 2.0;
        double r29950398 = t;
        double r29950399 = r29950397 / r29950398;
        double r29950400 = z;
        double r29950401 = 1.0;
        double r29950402 = 1.0;
        double r29950403 = fma(r29950400, r29950401, r29950402);
        double r29950404 = r29950403 / r29950400;
        double r29950405 = x;
        double r29950406 = y;
        double r29950407 = r29950405 / r29950406;
        double r29950408 = fma(r29950399, r29950404, r29950407);
        double r29950409 = r29950408 - r29950397;
        return r29950409;
}

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. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019170 +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))))