Average Error: 0.1 → 0.1
Time: 8.7s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) + \left(-y\right)\right)\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) + \left(-y\right)\right)
double f(double x, double y, double z) {
        double r323486 = x;
        double r323487 = y;
        double r323488 = 0.5;
        double r323489 = r323487 + r323488;
        double r323490 = log(r323487);
        double r323491 = r323489 * r323490;
        double r323492 = r323486 - r323491;
        double r323493 = r323492 + r323487;
        double r323494 = z;
        double r323495 = r323493 - r323494;
        return r323495;
}
↓
double f(double x, double y, double z) {
        double r323496 = x;
        double r323497 = y;
        double r323498 = log(r323497);
        double r323499 = 0.5;
        double r323500 = r323497 + r323499;
        double r323501 = z;
        double r323502 = fma(r323498, r323500, r323501);
        double r323503 = -r323497;
        double r323504 = r323502 + r323503;
        double r323505 = r323496 - r323504;
        return r323505;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

    \[\leadsto \color{blue}{x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)}\]
  3. Using strategy rm

  4. Applied sub-neg0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

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

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