Average Error: 0.1 → 0.1
Time: 9.6s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[x - 1 \cdot \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
x - 1 \cdot \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)
double f(double x, double y, double z) {
        double r285471 = x;
        double r285472 = y;
        double r285473 = 0.5;
        double r285474 = r285472 + r285473;
        double r285475 = log(r285472);
        double r285476 = r285474 * r285475;
        double r285477 = r285471 - r285476;
        double r285478 = r285477 + r285472;
        double r285479 = z;
        double r285480 = r285478 - r285479;
        return r285480;
}
↓
double f(double x, double y, double z) {
        double r285481 = x;
        double r285482 = 1.0;
        double r285483 = y;
        double r285484 = log(r285483);
        double r285485 = 0.5;
        double r285486 = r285483 + r285485;
        double r285487 = z;
        double r285488 = fma(r285484, r285486, r285487);
        double r285489 = r285488 - r285483;
        double r285490 = r285482 * r285489;
        double r285491 = r285481 - r285490;
        return r285491;
}

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 *-un-lft-identity0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020036 +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))