Average Error: 7.6 → 3.1
Time: 4.5s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.58103787244507047 \cdot 10^{135}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 6.76130975312649236 \cdot 10^{-45}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.58103787244507047 \cdot 10^{135}:\\
\;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\

\mathbf{elif}\;z \le 6.76130975312649236 \cdot 10^{-45}:\\
\;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r712304 = x;
        double r712305 = y;
        double r712306 = z;
        double r712307 = r712305 * r712306;
        double r712308 = r712307 - r712304;
        double r712309 = t;
        double r712310 = r712309 * r712306;
        double r712311 = r712310 - r712304;
        double r712312 = r712308 / r712311;
        double r712313 = r712304 + r712312;
        double r712314 = 1.0;
        double r712315 = r712304 + r712314;
        double r712316 = r712313 / r712315;
        return r712316;
}
↓
double f(double x, double y, double z, double t) {
        double r712317 = z;
        double r712318 = -4.5810378724450705e+135;
        bool r712319 = r712317 <= r712318;
        double r712320 = x;
        double r712321 = y;
        double r712322 = t;
        double r712323 = r712321 / r712322;
        double r712324 = r712320 + r712323;
        double r712325 = 1.0;
        double r712326 = r712320 + r712325;
        double r712327 = r712324 / r712326;
        double r712328 = 6.761309753126492e-45;
        bool r712329 = r712317 <= r712328;
        double r712330 = r712321 * r712317;
        double r712331 = r712330 - r712320;
        double r712332 = 1.0;
        double r712333 = r712322 * r712317;
        double r712334 = r712333 - r712320;
        double r712335 = r712332 / r712334;
        double r712336 = r712331 * r712335;
        double r712337 = r712320 + r712336;
        double r712338 = r712337 / r712326;
        double r712339 = r712321 / r712334;
        double r712340 = fma(r712339, r712317, r712320);
        double r712341 = r712320 / r712334;
        double r712342 = r712340 - r712341;
        double r712343 = r712326 / r712342;
        double r712344 = r712332 / r712343;
        double r712345 = r712329 ? r712338 : r712344;
        double r712346 = r712319 ? r712327 : r712345;
        return r712346;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

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

Derivation

  1. Split input into 3 regimes

  2. if z < -4.5810378724450705e+135

    1. Initial program 23.1

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

    2. Taylor expanded around inf 6.9

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

    if -4.5810378724450705e+135 < z < 6.761309753126492e-45

    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 6.761309753126492e-45 < z

    1. Initial program 12.6

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

    3. Applied div-sub12.6

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

    4. Applied associate-+r-12.6

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

    5. Simplified4.6

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

    7. Applied clear-num4.7

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.58103787244507047 \cdot 10^{135}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 6.76130975312649236 \cdot 10^{-45}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +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)))