Average Error: 22.3 → 0.2
Time: 23.8s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -159699341.562465727329254150390625:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 115501974.66236674785614013671875:\\ \;\;\;\;\mathsf{fma}\left(1, 1, -\left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\right) + \mathsf{fma}\left(-\left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}, \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -159699341.562465727329254150390625:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\

\mathbf{elif}\;y \le 115501974.66236674785614013671875:\\
\;\;\;\;\mathsf{fma}\left(1, 1, -\left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\right) + \mathsf{fma}\left(-\left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}, \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\

\end{array}
double f(double x, double y) {
        double r120591780 = 1.0;
        double r120591781 = x;
        double r120591782 = r120591780 - r120591781;
        double r120591783 = y;
        double r120591784 = r120591782 * r120591783;
        double r120591785 = r120591783 + r120591780;
        double r120591786 = r120591784 / r120591785;
        double r120591787 = r120591780 - r120591786;
        return r120591787;
}
↓
double f(double x, double y) {
        double r120591788 = y;
        double r120591789 = -159699341.56246573;
        bool r120591790 = r120591788 <= r120591789;
        double r120591791 = 1.0;
        double r120591792 = 1.0;
        double r120591793 = r120591792 / r120591788;
        double r120591794 = x;
        double r120591795 = r120591794 / r120591788;
        double r120591796 = r120591793 - r120591795;
        double r120591797 = fma(r120591791, r120591796, r120591794);
        double r120591798 = 115501974.66236675;
        bool r120591799 = r120591788 <= r120591798;
        double r120591800 = r120591788 * r120591788;
        double r120591801 = r120591791 * r120591791;
        double r120591802 = r120591788 * r120591791;
        double r120591803 = r120591801 - r120591802;
        double r120591804 = r120591800 + r120591803;
        double r120591805 = r120591791 - r120591794;
        double r120591806 = r120591805 * r120591788;
        double r120591807 = 3.0;
        double r120591808 = pow(r120591788, r120591807);
        double r120591809 = pow(r120591791, r120591807);
        double r120591810 = r120591808 + r120591809;
        double r120591811 = r120591806 / r120591810;
        double r120591812 = r120591804 * r120591811;
        double r120591813 = -r120591812;
        double r120591814 = fma(r120591792, r120591791, r120591813);
        double r120591815 = -r120591804;
        double r120591816 = fma(r120591815, r120591811, r120591812);
        double r120591817 = r120591814 + r120591816;
        double r120591818 = r120591799 ? r120591817 : r120591797;
        double r120591819 = r120591790 ? r120591797 : r120591818;
        return r120591819;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.3
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -159699341.56246573 or 115501974.66236675 < y

    1. Initial program 46.0

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

    2. Simplified29.5

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

    3. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}}\]

    4. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)}\]

    if -159699341.56246573 < y < 115501974.66236675

    1. Initial program 0.2

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
    2. Using strategy rm

    3. Applied flip3-+0.2

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

    4. Applied associate-/r/0.2

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

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

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

    6. Applied prod-diff0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, 1, -\left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\right) + \mathsf{fma}\left(-\left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}, \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -159699341.562465727329254150390625:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 115501974.66236674785614013671875:\\ \;\;\;\;\mathsf{fma}\left(1, 1, -\left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\right) + \mathsf{fma}\left(-\left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right), \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}, \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))