Average Error: 0.1 → 0.9
Time: 10.5s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\left(x - \mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, -y \cdot 1\right)\right) - \mathsf{fma}\left(-\sqrt{y}, \sqrt{y}, \sqrt{y} \cdot \sqrt{y}\right)\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(x - \mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, -y \cdot 1\right)\right) - \mathsf{fma}\left(-\sqrt{y}, \sqrt{y}, \sqrt{y} \cdot \sqrt{y}\right)
double f(double x, double y, double z) {
        double r332029 = x;
        double r332030 = y;
        double r332031 = 0.5;
        double r332032 = r332030 + r332031;
        double r332033 = log(r332030);
        double r332034 = r332032 * r332033;
        double r332035 = r332029 - r332034;
        double r332036 = r332035 + r332030;
        double r332037 = z;
        double r332038 = r332036 - r332037;
        return r332038;
}
↓
double f(double x, double y, double z) {
        double r332039 = x;
        double r332040 = y;
        double r332041 = log(r332040);
        double r332042 = 0.5;
        double r332043 = r332040 + r332042;
        double r332044 = z;
        double r332045 = fma(r332041, r332043, r332044);
        double r332046 = cbrt(r332045);
        double r332047 = r332046 * r332046;
        double r332048 = 1.0;
        double r332049 = r332040 * r332048;
        double r332050 = -r332049;
        double r332051 = fma(r332047, r332046, r332050);
        double r332052 = r332039 - r332051;
        double r332053 = sqrt(r332040);
        double r332054 = -r332053;
        double r332055 = r332053 * r332053;
        double r332056 = fma(r332054, r332053, r332055);
        double r332057 = r332052 - r332056;
        return r332057;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.9
\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y\]

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 add-sqr-sqrt0.1

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

  5. Applied add-cube-cbrt0.9

    \[\leadsto x - \left(\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}} - \sqrt{y} \cdot \sqrt{y}\right)\]

  6. Applied prod-diff0.9

    \[\leadsto x - \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, -\sqrt{y} \cdot \sqrt{y}\right) + \mathsf{fma}\left(-\sqrt{y}, \sqrt{y}, \sqrt{y} \cdot \sqrt{y}\right)\right)}\]

  7. Applied associate--r+0.9

    \[\leadsto \color{blue}{\left(x - \mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, -\sqrt{y} \cdot \sqrt{y}\right)\right) - \mathsf{fma}\left(-\sqrt{y}, \sqrt{y}, \sqrt{y} \cdot \sqrt{y}\right)}\]

  8. Simplified0.9

    \[\leadsto \color{blue}{\left(x - \mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, -y \cdot 1\right)\right)} - \mathsf{fma}\left(-\sqrt{y}, \sqrt{y}, \sqrt{y} \cdot \sqrt{y}\right)\]

  9. Final simplification0.9

    \[\leadsto \left(x - \mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, \sqrt[3]{\mathsf{fma}\left(\log y, y + 0.5, z\right)}, -y \cdot 1\right)\right) - \mathsf{fma}\left(-\sqrt{y}, \sqrt{y}, \sqrt{y} \cdot \sqrt{y}\right)\]

Reproduce

herbie shell --seed 2020083 +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))