Average Error: 7.2 → 2.1
Time: 4.7s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 5.304900116922103618735525591865166312009 \cdot 10^{186}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]
\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:\\
\;\;\;\;\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}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 5.304900116922103618735525591865166312009 \cdot 10^{186}:\\
\;\;\;\;\frac{x + \frac{y \cdot z - x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r621679 = x;
        double r621680 = y;
        double r621681 = z;
        double r621682 = r621680 * r621681;
        double r621683 = r621682 - r621679;
        double r621684 = t;
        double r621685 = r621684 * r621681;
        double r621686 = r621685 - r621679;
        double r621687 = r621683 / r621686;
        double r621688 = r621679 + r621687;
        double r621689 = 1.0;
        double r621690 = r621679 + r621689;
        double r621691 = r621688 / r621690;
        return r621691;
}
↓
double f(double x, double y, double z, double t) {
        double r621692 = x;
        double r621693 = y;
        double r621694 = z;
        double r621695 = r621693 * r621694;
        double r621696 = r621695 - r621692;
        double r621697 = t;
        double r621698 = r621697 * r621694;
        double r621699 = r621698 - r621692;
        double r621700 = r621696 / r621699;
        double r621701 = r621692 + r621700;
        double r621702 = 1.0;
        double r621703 = r621692 + r621702;
        double r621704 = r621701 / r621703;
        double r621705 = -inf.0;
        bool r621706 = r621704 <= r621705;
        double r621707 = r621693 / r621699;
        double r621708 = fma(r621707, r621694, r621692);
        double r621709 = 1.0;
        double r621710 = r621703 * r621709;
        double r621711 = r621708 / r621710;
        double r621712 = r621692 / r621699;
        double r621713 = r621712 / r621703;
        double r621714 = r621711 - r621713;
        double r621715 = 5.304900116922104e+186;
        bool r621716 = r621704 <= r621715;
        double r621717 = -r621692;
        double r621718 = fma(r621697, r621694, r621717);
        double r621719 = r621696 / r621718;
        double r621720 = r621692 + r621719;
        double r621721 = r621720 / r621703;
        double r621722 = r621693 / r621697;
        double r621723 = r621692 + r621722;
        double r621724 = r621723 / r621703;
        double r621725 = r621716 ? r621721 : r621724;
        double r621726 = r621706 ? r621714 : r621725;
        return r621726;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.2
Target0.4
Herbie2.1
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ 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. Applied div-sub64.0

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

    6. Simplified6.5

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

    if -inf.0 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 5.304900116922104e+186

    1. Initial program 0.7

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

    3. Applied fma-neg0.7

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

    if 5.304900116922104e+186 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

    1. Initial program 50.9

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

    2. Taylor expanded around inf 14.4

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 5.304900116922103618735525591865166312009 \cdot 10^{186}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))