Average Error: 0.1 → 0.1
Time: 5.4s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[\mathsf{fma}\left(x, 0.5, y \cdot \left(\left(\log z + 1\right) - z\right)\right)\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
\mathsf{fma}\left(x, 0.5, y \cdot \left(\left(\log z + 1\right) - z\right)\right)
double f(double x, double y, double z) {
        double r283304 = x;
        double r283305 = 0.5;
        double r283306 = r283304 * r283305;
        double r283307 = y;
        double r283308 = 1.0;
        double r283309 = z;
        double r283310 = r283308 - r283309;
        double r283311 = log(r283309);
        double r283312 = r283310 + r283311;
        double r283313 = r283307 * r283312;
        double r283314 = r283306 + r283313;
        return r283314;
}
↓
double f(double x, double y, double z) {
        double r283315 = x;
        double r283316 = 0.5;
        double r283317 = y;
        double r283318 = z;
        double r283319 = log(r283318);
        double r283320 = 1.0;
        double r283321 = r283319 + r283320;
        double r283322 = r283321 - r283318;
        double r283323 = r283317 * r283322;
        double r283324 = fma(r283315, r283316, r283323);
        return r283324;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.1
\[\left(y + 0.5 \cdot x\right) - y \cdot \left(z - \log z\right)\]

Derivation

  1. Initial program 0.1

    \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\]

  3. Taylor expanded around 0 0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, y \cdot \color{blue}{\left(\left(\log z + 1\right) - z\right)}\right)\]

  4. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, y \cdot \left(\left(\log z + 1\right) - z\right)\right)\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

  (+ (* x 0.5) (* y (+ (- 1 z) (log z)))))