Average Error: 7.6 → 3.5
Time: 25.9s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.478650958825779255055896013297103728471 \cdot 10^{-93} \lor \neg \left(x \le 1.896379000670417166121402150992566419893 \cdot 10^{-147}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{1}{\mathsf{fma}\left(\frac{z}{x}, t, -1\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{{x}^{3} + {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.478650958825779255055896013297103728471 \cdot 10^{-93} \lor \neg \left(x \le 1.896379000670417166121402150992566419893 \cdot 10^{-147}\right):\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{1}{\mathsf{fma}\left(\frac{z}{x}, t, -1\right)}}{x + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{{x}^{3} + {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r396755 = x;
        double r396756 = y;
        double r396757 = z;
        double r396758 = r396756 * r396757;
        double r396759 = r396758 - r396755;
        double r396760 = t;
        double r396761 = r396760 * r396757;
        double r396762 = r396761 - r396755;
        double r396763 = r396759 / r396762;
        double r396764 = r396755 + r396763;
        double r396765 = 1.0;
        double r396766 = r396755 + r396765;
        double r396767 = r396764 / r396766;
        return r396767;
}
↓
double f(double x, double y, double z, double t) {
        double r396768 = x;
        double r396769 = -3.4786509588257793e-93;
        bool r396770 = r396768 <= r396769;
        double r396771 = 1.8963790006704172e-147;
        bool r396772 = r396768 <= r396771;
        double r396773 = !r396772;
        bool r396774 = r396770 || r396773;
        double r396775 = y;
        double r396776 = t;
        double r396777 = z;
        double r396778 = r396776 * r396777;
        double r396779 = r396778 - r396768;
        double r396780 = r396775 / r396779;
        double r396781 = fma(r396780, r396777, r396768);
        double r396782 = 1.0;
        double r396783 = r396777 / r396768;
        double r396784 = -1.0;
        double r396785 = fma(r396783, r396776, r396784);
        double r396786 = r396782 / r396785;
        double r396787 = r396781 - r396786;
        double r396788 = 1.0;
        double r396789 = r396768 + r396788;
        double r396790 = r396787 / r396789;
        double r396791 = r396775 * r396777;
        double r396792 = r396791 - r396768;
        double r396793 = r396792 / r396779;
        double r396794 = r396768 + r396793;
        double r396795 = 3.0;
        double r396796 = pow(r396768, r396795);
        double r396797 = pow(r396788, r396795);
        double r396798 = r396796 + r396797;
        double r396799 = r396794 / r396798;
        double r396800 = r396768 * r396768;
        double r396801 = r396788 * r396788;
        double r396802 = r396768 * r396788;
        double r396803 = r396801 - r396802;
        double r396804 = r396800 + r396803;
        double r396805 = r396799 * r396804;
        double r396806 = r396774 ? r396790 : r396805;
        return r396806;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.6
Target0.3
Herbie3.5
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.4786509588257793e-93 or 1.8963790006704172e-147 < x

    1. Initial program 7.6

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
    2. Using strategy rm

    3. Applied div-sub7.6

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

    4. Applied associate-+r-7.6

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

    5. Simplified1.9

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)} - \frac{x}{t \cdot z - x}}{x + 1}\]
    6. Using strategy rm

    7. Applied clear-num1.9

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

    8. Simplified1.9

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

    if -3.4786509588257793e-93 < x < 1.8963790006704172e-147

    1. Initial program 7.5

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
    2. Using strategy rm

    3. Applied flip3-+7.5

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

    4. Applied associate-/r/7.5

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

  4. Final simplification3.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.478650958825779255055896013297103728471 \cdot 10^{-93} \lor \neg \left(x \le 1.896379000670417166121402150992566419893 \cdot 10^{-147}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{1}{\mathsf{fma}\left(\frac{z}{x}, t, -1\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{{x}^{3} + {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1)))