Average Error: 7.0 → 2.5
Time: 10.1s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.20761997366181834 \cdot 10^{-298} \lor \neg \left(x \le 1.374279956057484 \cdot 10^{-254}\right):\\ \;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(y, \frac{z}{t \cdot z - x}, x\right)}} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le 2.20761997366181834 \cdot 10^{-298} \lor \neg \left(x \le 1.374279956057484 \cdot 10^{-254}\right):\\
\;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(y, \frac{z}{t \cdot z - x}, x\right)}} - \frac{\frac{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 r663645 = x;
        double r663646 = y;
        double r663647 = z;
        double r663648 = r663646 * r663647;
        double r663649 = r663648 - r663645;
        double r663650 = t;
        double r663651 = r663650 * r663647;
        double r663652 = r663651 - r663645;
        double r663653 = r663649 / r663652;
        double r663654 = r663645 + r663653;
        double r663655 = 1.0;
        double r663656 = r663645 + r663655;
        double r663657 = r663654 / r663656;
        return r663657;
}
↓
double f(double x, double y, double z, double t) {
        double r663658 = x;
        double r663659 = 2.2076199736618183e-298;
        bool r663660 = r663658 <= r663659;
        double r663661 = 1.3742799560574841e-254;
        bool r663662 = r663658 <= r663661;
        double r663663 = !r663662;
        bool r663664 = r663660 || r663663;
        double r663665 = 1.0;
        double r663666 = 1.0;
        double r663667 = r663658 + r663666;
        double r663668 = y;
        double r663669 = z;
        double r663670 = t;
        double r663671 = r663670 * r663669;
        double r663672 = r663671 - r663658;
        double r663673 = r663669 / r663672;
        double r663674 = fma(r663668, r663673, r663658);
        double r663675 = r663667 / r663674;
        double r663676 = r663665 / r663675;
        double r663677 = r663658 / r663672;
        double r663678 = r663677 / r663667;
        double r663679 = r663676 - r663678;
        double r663680 = r663668 / r663670;
        double r663681 = r663658 + r663680;
        double r663682 = r663681 / r663667;
        double r663683 = r663664 ? r663679 : r663682;
        return r663683;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.0
Target0.4
Herbie2.5
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 2.2076199736618183e-298 or 1.3742799560574841e-254 < x

    1. Initial program 6.9

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

    3. Applied div-sub6.9

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

    4. Applied associate-+r-6.9

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

    5. Applied div-sub6.9

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t \cdot z - x}}{x + 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity6.9

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

    8. Applied times-frac2.1

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

    9. Simplified2.1

      \[\leadsto \frac{x + \color{blue}{y} \cdot \frac{z}{t \cdot z - x}}{x + 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\]
    10. Using strategy rm

    11. Applied clear-num2.1

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

    12. Simplified2.1

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

    if 2.2076199736618183e-298 < x < 1.3742799560574841e-254

    1. Initial program 10.8

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

    2. Taylor expanded around inf 14.9

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.20761997366181834 \cdot 10^{-298} \lor \neg \left(x \le 1.374279956057484 \cdot 10^{-254}\right):\\ \;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(y, \frac{z}{t \cdot z - x}, x\right)}} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))