Average Error: 7.5 → 3.8
Time: 4.7s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2146516340209799649034240 \lor \neg \left(x \le 1.736185251937085420450408841627359572649 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{1}{\frac{t \cdot z - x}{x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right)\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2146516340209799649034240 \lor \neg \left(x \le 1.736185251937085420450408841627359572649 \cdot 10^{-75}\right):\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{1}{\frac{t \cdot z - x}{x}}}{x + 1}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r666154 = x;
        double r666155 = y;
        double r666156 = z;
        double r666157 = r666155 * r666156;
        double r666158 = r666157 - r666154;
        double r666159 = t;
        double r666160 = r666159 * r666156;
        double r666161 = r666160 - r666154;
        double r666162 = r666158 / r666161;
        double r666163 = r666154 + r666162;
        double r666164 = 1.0;
        double r666165 = r666154 + r666164;
        double r666166 = r666163 / r666165;
        return r666166;
}
↓
double f(double x, double y, double z, double t) {
        double r666167 = x;
        double r666168 = -2.1465163402097996e+24;
        bool r666169 = r666167 <= r666168;
        double r666170 = 1.7361852519370854e-75;
        bool r666171 = r666167 <= r666170;
        double r666172 = !r666171;
        bool r666173 = r666169 || r666172;
        double r666174 = y;
        double r666175 = t;
        double r666176 = z;
        double r666177 = r666175 * r666176;
        double r666178 = r666177 - r666167;
        double r666179 = r666174 / r666178;
        double r666180 = fma(r666179, r666176, r666167);
        double r666181 = 1.0;
        double r666182 = r666167 + r666181;
        double r666183 = 1.0;
        double r666184 = r666182 * r666183;
        double r666185 = r666180 / r666184;
        double r666186 = r666178 / r666167;
        double r666187 = r666183 / r666186;
        double r666188 = r666187 / r666182;
        double r666189 = r666185 - r666188;
        double r666190 = r666174 * r666176;
        double r666191 = r666190 - r666167;
        double r666192 = r666191 / r666178;
        double r666193 = r666167 + r666192;
        double r666194 = r666167 * r666167;
        double r666195 = r666181 * r666181;
        double r666196 = r666194 - r666195;
        double r666197 = r666193 / r666196;
        double r666198 = r666167 - r666181;
        double r666199 = r666197 * r666198;
        double r666200 = r666173 ? r666189 : r666199;
        return r666200;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.5
Target0.3
Herbie3.8
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2.1465163402097996e+24 or 1.7361852519370854e-75 < x

    1. Initial program 7.9

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

    3. Applied div-sub7.9

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

    4. Applied associate-+r-7.9

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

    5. Applied div-sub7.9

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

    6. Simplified1.2

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

    8. Applied clear-num1.2

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

    if -2.1465163402097996e+24 < x < 1.7361852519370854e-75

    1. Initial program 7.0

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

    3. Applied flip-+7.0

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

    4. Applied associate-/r/7.0

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

  4. Final simplification3.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2146516340209799649034240 \lor \neg \left(x \le 1.736185251937085420450408841627359572649 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{1}{\frac{t \cdot z - x}{x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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)))