Average Error: 0.1 → 0.1
Time: 6.8s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\left(x + \mathsf{fma}\left(-\left(y + 0.5\right), \log y, y\right)\right) - z\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(x + \mathsf{fma}\left(-\left(y + 0.5\right), \log y, y\right)\right) - z
double f(double x, double y, double z) {
        double r316552 = x;
        double r316553 = y;
        double r316554 = 0.5;
        double r316555 = r316553 + r316554;
        double r316556 = log(r316553);
        double r316557 = r316555 * r316556;
        double r316558 = r316552 - r316557;
        double r316559 = r316558 + r316553;
        double r316560 = z;
        double r316561 = r316559 - r316560;
        return r316561;
}
↓
double f(double x, double y, double z) {
        double r316562 = x;
        double r316563 = y;
        double r316564 = 0.5;
        double r316565 = r316563 + r316564;
        double r316566 = -r316565;
        double r316567 = log(r316563);
        double r316568 = fma(r316566, r316567, r316563);
        double r316569 = r316562 + r316568;
        double r316570 = z;
        double r316571 = r316569 - r316570;
        return r316571;
}

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. Using strategy rm

  3. Applied sub-neg0.1

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

  4. Applied associate-+l+0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019356 +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))