Average Error: 7.7 → 3.2
Time: 5.6s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.4707659967563265 \cdot 10^{138}:\\ \;\;\;\;\left(x + \frac{y}{t}\right) \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;z \le 8.81173598267756785 \cdot 10^{-23}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;z \le 1.9006882053443136 \cdot 10^{254}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;z \le 4.3010845694693669 \cdot 10^{271}:\\ \;\;\;\;\left(x + \frac{y}{t}\right) \cdot \frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.4707659967563265 \cdot 10^{138}:\\
\;\;\;\;\left(x + \frac{y}{t}\right) \cdot \frac{1}{x + 1}\\

\mathbf{elif}\;z \le 8.81173598267756785 \cdot 10^{-23}:\\
\;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\

\mathbf{elif}\;z \le 1.9006882053443136 \cdot 10^{254}:\\
\;\;\;\;\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}\\

\mathbf{elif}\;z \le 4.3010845694693669 \cdot 10^{271}:\\
\;\;\;\;\left(x + \frac{y}{t}\right) \cdot \frac{1}{x + 1}\\

\mathbf{else}:\\
\;\;\;\;\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}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r989220 = x;
        double r989221 = y;
        double r989222 = z;
        double r989223 = r989221 * r989222;
        double r989224 = r989223 - r989220;
        double r989225 = t;
        double r989226 = r989225 * r989222;
        double r989227 = r989226 - r989220;
        double r989228 = r989224 / r989227;
        double r989229 = r989220 + r989228;
        double r989230 = 1.0;
        double r989231 = r989220 + r989230;
        double r989232 = r989229 / r989231;
        return r989232;
}
↓
double f(double x, double y, double z, double t) {
        double r989233 = z;
        double r989234 = -4.4707659967563265e+138;
        bool r989235 = r989233 <= r989234;
        double r989236 = x;
        double r989237 = y;
        double r989238 = t;
        double r989239 = r989237 / r989238;
        double r989240 = r989236 + r989239;
        double r989241 = 1.0;
        double r989242 = 1.0;
        double r989243 = r989236 + r989242;
        double r989244 = r989241 / r989243;
        double r989245 = r989240 * r989244;
        double r989246 = 8.811735982677568e-23;
        bool r989247 = r989233 <= r989246;
        double r989248 = r989237 * r989233;
        double r989249 = r989248 - r989236;
        double r989250 = r989238 * r989233;
        double r989251 = r989250 - r989236;
        double r989252 = r989241 / r989251;
        double r989253 = r989249 * r989252;
        double r989254 = r989236 + r989253;
        double r989255 = r989254 / r989243;
        double r989256 = 1.9006882053443136e+254;
        bool r989257 = r989233 <= r989256;
        double r989258 = r989237 / r989251;
        double r989259 = fma(r989258, r989233, r989236);
        double r989260 = r989243 * r989241;
        double r989261 = r989259 / r989260;
        double r989262 = r989236 / r989251;
        double r989263 = r989262 / r989243;
        double r989264 = r989261 - r989263;
        double r989265 = 4.301084569469367e+271;
        bool r989266 = r989233 <= r989265;
        double r989267 = r989266 ? r989245 : r989264;
        double r989268 = r989257 ? r989264 : r989267;
        double r989269 = r989247 ? r989255 : r989268;
        double r989270 = r989235 ? r989245 : r989269;
        return r989270;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.7
Target0.3
Herbie3.2
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -4.4707659967563265e+138 or 1.9006882053443136e+254 < z < 4.301084569469367e+271

    1. Initial program 21.1

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

    3. Applied div-inv21.2

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

    4. Taylor expanded around inf 6.6

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

    if -4.4707659967563265e+138 < z < 8.811735982677568e-23

    1. Initial program 1.5

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

    3. Applied div-inv1.5

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

    if 8.811735982677568e-23 < z < 1.9006882053443136e+254 or 4.301084569469367e+271 < z

    1. Initial program 14.0

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

    3. Applied div-sub14.0

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

    4. Applied associate-+r-14.0

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

    5. Applied div-sub14.0

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

    6. Simplified4.9

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.4707659967563265 \cdot 10^{138}:\\ \;\;\;\;\left(x + \frac{y}{t}\right) \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;z \le 8.81173598267756785 \cdot 10^{-23}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;z \le 1.9006882053443136 \cdot 10^{254}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;z \le 4.3010845694693669 \cdot 10^{271}:\\ \;\;\;\;\left(x + \frac{y}{t}\right) \cdot \frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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)))