Average Error: 0.1 → 0.1
Time: 6.4s
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 r326648 = x;
        double r326649 = y;
        double r326650 = 0.5;
        double r326651 = r326649 + r326650;
        double r326652 = log(r326649);
        double r326653 = r326651 * r326652;
        double r326654 = r326648 - r326653;
        double r326655 = r326654 + r326649;
        double r326656 = z;
        double r326657 = r326655 - r326656;
        return r326657;
}
↓
double f(double x, double y, double z) {
        double r326658 = x;
        double r326659 = y;
        double r326660 = 0.5;
        double r326661 = r326659 + r326660;
        double r326662 = -r326661;
        double r326663 = log(r326659);
        double r326664 = fma(r326662, r326663, r326659);
        double r326665 = r326658 + r326664;
        double r326666 = z;
        double r326667 = r326665 - r326666;
        return r326667;
}

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 2020064 +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))