Average Error: 7.1 → 1.4
Time: 4.3s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 3.6817545759347375 \cdot 10^{290}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\
\;\;\;\;\left(\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\

\mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 3.6817545759347375 \cdot 10^{290}:\\
\;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r706489 = x;
        double r706490 = y;
        double r706491 = z;
        double r706492 = r706490 * r706491;
        double r706493 = r706492 - r706489;
        double r706494 = t;
        double r706495 = r706494 * r706491;
        double r706496 = r706495 - r706489;
        double r706497 = r706493 / r706496;
        double r706498 = r706489 + r706497;
        double r706499 = 1.0;
        double r706500 = r706489 + r706499;
        double r706501 = r706498 / r706500;
        return r706501;
}
↓
double f(double x, double y, double z, double t) {
        double r706502 = x;
        double r706503 = y;
        double r706504 = z;
        double r706505 = r706503 * r706504;
        double r706506 = r706505 - r706502;
        double r706507 = t;
        double r706508 = r706507 * r706504;
        double r706509 = r706508 - r706502;
        double r706510 = r706506 / r706509;
        double r706511 = r706502 + r706510;
        double r706512 = 1.0;
        double r706513 = r706502 + r706512;
        double r706514 = r706511 / r706513;
        double r706515 = -inf.0;
        bool r706516 = r706514 <= r706515;
        double r706517 = r706503 / r706509;
        double r706518 = fma(r706517, r706504, r706502);
        double r706519 = r706502 / r706509;
        double r706520 = r706518 - r706519;
        double r706521 = 1.0;
        double r706522 = r706521 / r706513;
        double r706523 = r706520 * r706522;
        double r706524 = 3.6817545759347375e+290;
        bool r706525 = r706514 <= r706524;
        double r706526 = r706503 / r706507;
        double r706527 = r706502 + r706526;
        double r706528 = r706527 / r706513;
        double r706529 = r706525 ? r706514 : r706528;
        double r706530 = r706516 ? r706523 : r706529;
        return r706530;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.1
Target0.3
Herbie1.4
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < -inf.0

    1. Initial program 64.0

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

    3. Applied div-sub64.0

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

    4. Applied associate-+r-64.0

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

    5. Simplified5.3

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

    7. Applied div-inv5.4

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

    if -inf.0 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 3.6817545759347375e+290

    1. Initial program 0.6

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

    if 3.6817545759347375e+290 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

    1. Initial program 61.7

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

    2. Taylor expanded around inf 9.9

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 3.6817545759347375 \cdot 10^{290}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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)))