Average Error: 0.2 → 0.3
Time: 35.5s
Precision: 64
\[\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]
↓
\[\mathsf{fma}\left(1, 1.0 - \frac{1.0}{x \cdot 9.0}, \frac{\frac{y}{\sqrt[3]{3.0}}}{\sqrt{x}} \cdot \frac{-1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}}\right) + \mathsf{fma}\left(-\frac{\frac{y}{\sqrt[3]{3.0}}}{\sqrt{x}}, \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}}, \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \frac{\frac{y}{\sqrt[3]{3.0}}}{\sqrt{x}}\right)\]
\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}
↓
\mathsf{fma}\left(1, 1.0 - \frac{1.0}{x \cdot 9.0}, \frac{\frac{y}{\sqrt[3]{3.0}}}{\sqrt{x}} \cdot \frac{-1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}}\right) + \mathsf{fma}\left(-\frac{\frac{y}{\sqrt[3]{3.0}}}{\sqrt{x}}, \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}}, \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \frac{\frac{y}{\sqrt[3]{3.0}}}{\sqrt{x}}\right)
double f(double x, double y) {
        double r17460157 = 1.0;
        double r17460158 = x;
        double r17460159 = 9.0;
        double r17460160 = r17460158 * r17460159;
        double r17460161 = r17460157 / r17460160;
        double r17460162 = r17460157 - r17460161;
        double r17460163 = y;
        double r17460164 = 3.0;
        double r17460165 = sqrt(r17460158);
        double r17460166 = r17460164 * r17460165;
        double r17460167 = r17460163 / r17460166;
        double r17460168 = r17460162 - r17460167;
        return r17460168;
}
↓
double f(double x, double y) {
        double r17460169 = 1.0;
        double r17460170 = 1.0;
        double r17460171 = x;
        double r17460172 = 9.0;
        double r17460173 = r17460171 * r17460172;
        double r17460174 = r17460170 / r17460173;
        double r17460175 = r17460170 - r17460174;
        double r17460176 = y;
        double r17460177 = 3.0;
        double r17460178 = cbrt(r17460177);
        double r17460179 = r17460176 / r17460178;
        double r17460180 = sqrt(r17460171);
        double r17460181 = r17460179 / r17460180;
        double r17460182 = -1.0;
        double r17460183 = r17460178 * r17460178;
        double r17460184 = r17460182 / r17460183;
        double r17460185 = r17460181 * r17460184;
        double r17460186 = fma(r17460169, r17460175, r17460185);
        double r17460187 = -r17460181;
        double r17460188 = r17460169 / r17460183;
        double r17460189 = r17460188 * r17460181;
        double r17460190 = fma(r17460187, r17460188, r17460189);
        double r17460191 = r17460186 + r17460190;
        return r17460191;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.2
Target0.2
Herbie0.3
\[\left(1.0 - \frac{\frac{1.0}{x}}{9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

    \[\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]
  2. Using strategy rm

  3. Applied associate-/r*0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \color{blue}{\frac{\frac{y}{3.0}}{\sqrt{x}}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{\frac{y}{3.0}}{\color{blue}{1 \cdot \sqrt{x}}}\]

  6. Applied add-cube-cbrt0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{\frac{y}{\color{blue}{\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \sqrt[3]{3.0}}}}{1 \cdot \sqrt{x}}\]

  7. Applied *-un-lft-identity0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{\frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \sqrt[3]{3.0}}}{1 \cdot \sqrt{x}}\]

  8. Applied times-frac0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{\color{blue}{\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \frac{y}{\sqrt[3]{3.0}}}}{1 \cdot \sqrt{x}}\]

  9. Applied times-frac0.3

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \color{blue}{\frac{\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}}}{1} \cdot \frac{\frac{y}{\sqrt[3]{3.0}}}{\sqrt{x}}}\]

  10. Applied *-un-lft-identity0.3

    \[\leadsto \color{blue}{1 \cdot \left(1.0 - \frac{1.0}{x \cdot 9.0}\right)} - \frac{\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}}}{1} \cdot \frac{\frac{y}{\sqrt[3]{3.0}}}{\sqrt{x}}\]

  11. Applied prod-diff0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, 1.0 - \frac{1.0}{x \cdot 9.0}, -\frac{\frac{y}{\sqrt[3]{3.0}}}{\sqrt{x}} \cdot \frac{\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}}}{1}\right) + \mathsf{fma}\left(-\frac{\frac{y}{\sqrt[3]{3.0}}}{\sqrt{x}}, \frac{\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}}}{1}, \frac{\frac{y}{\sqrt[3]{3.0}}}{\sqrt{x}} \cdot \frac{\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}}}{1}\right)}\]

  12. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(1, 1.0 - \frac{1.0}{x \cdot 9.0}, \frac{\frac{y}{\sqrt[3]{3.0}}}{\sqrt{x}} \cdot \frac{-1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}}\right) + \mathsf{fma}\left(-\frac{\frac{y}{\sqrt[3]{3.0}}}{\sqrt{x}}, \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}}, \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \frac{\frac{y}{\sqrt[3]{3.0}}}{\sqrt{x}}\right)\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"

  :herbie-target
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))