Average Error: 0.1 → 0.2
Time: 26.1s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\left(\left(x - \mathsf{fma}\left(3, y, 1.5\right) \cdot \left(\log y \cdot \frac{1}{3}\right)\right) + y\right) - z\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(\left(x - \mathsf{fma}\left(3, y, 1.5\right) \cdot \left(\log y \cdot \frac{1}{3}\right)\right) + y\right) - z
double f(double x, double y, double z) {
        double r16798811 = x;
        double r16798812 = y;
        double r16798813 = 0.5;
        double r16798814 = r16798812 + r16798813;
        double r16798815 = log(r16798812);
        double r16798816 = r16798814 * r16798815;
        double r16798817 = r16798811 - r16798816;
        double r16798818 = r16798817 + r16798812;
        double r16798819 = z;
        double r16798820 = r16798818 - r16798819;
        return r16798820;
}
↓
double f(double x, double y, double z) {
        double r16798821 = x;
        double r16798822 = 3.0;
        double r16798823 = y;
        double r16798824 = 1.5;
        double r16798825 = fma(r16798822, r16798823, r16798824);
        double r16798826 = log(r16798823);
        double r16798827 = 0.3333333333333333;
        double r16798828 = r16798826 * r16798827;
        double r16798829 = r16798825 * r16798828;
        double r16798830 = r16798821 - r16798829;
        double r16798831 = r16798830 + r16798823;
        double r16798832 = z;
        double r16798833 = r16798831 - r16798832;
        return r16798833;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.2
\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y\]

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 add-cube-cbrt0.1

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

  4. Applied log-prod0.2

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

  5. Applied distribute-lft-in0.2

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

  6. Simplified0.2

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

  7. Simplified0.2

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

  8. Taylor expanded around inf 0.2

    \[\leadsto \left(\left(x - \color{blue}{\left(1.5 \cdot \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) + 3 \cdot \left(y \cdot \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right)\right)\right)}\right) + y\right) - z\]

  9. Simplified0.2

    \[\leadsto \left(\left(x - \color{blue}{\left(\frac{-1}{3} \cdot \left(-\log y\right)\right) \cdot \mathsf{fma}\left(3, y, 1.5\right)}\right) + y\right) - z\]

  10. Final simplification0.2

    \[\leadsto \left(\left(x - \mathsf{fma}\left(3, y, 1.5\right) \cdot \left(\log y \cdot \frac{1}{3}\right)\right) + y\right) - z\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"

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

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