Average Error: 7.5 → 3.7
Time: 21.1s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.440078247488523162933428867079497205001 \cdot 10^{157}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \sqrt[3]{\frac{x}{t \cdot z - x} \cdot \left(\frac{x}{t \cdot z - x} \cdot \frac{x}{t \cdot z - x}\right)}}{x + 1}\\ \mathbf{elif}\;z \le 1.514816533856499451078635282534599335268 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{y \cdot z - x}{\mathsf{fma}\left(t, z, -x\right)} + x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t \cdot z - x} \cdot y, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.440078247488523162933428867079497205001 \cdot 10^{157}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \sqrt[3]{\frac{x}{t \cdot z - x} \cdot \left(\frac{x}{t \cdot z - x} \cdot \frac{x}{t \cdot z - x}\right)}}{x + 1}\\

\mathbf{elif}\;z \le 1.514816533856499451078635282534599335268 \cdot 10^{-166}:\\
\;\;\;\;\frac{\frac{y \cdot z - x}{\mathsf{fma}\left(t, z, -x\right)} + x}{x + 1}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r29477091 = x;
        double r29477092 = y;
        double r29477093 = z;
        double r29477094 = r29477092 * r29477093;
        double r29477095 = r29477094 - r29477091;
        double r29477096 = t;
        double r29477097 = r29477096 * r29477093;
        double r29477098 = r29477097 - r29477091;
        double r29477099 = r29477095 / r29477098;
        double r29477100 = r29477091 + r29477099;
        double r29477101 = 1.0;
        double r29477102 = r29477091 + r29477101;
        double r29477103 = r29477100 / r29477102;
        return r29477103;
}
↓
double f(double x, double y, double z, double t) {
        double r29477104 = z;
        double r29477105 = -5.440078247488523e+157;
        bool r29477106 = r29477104 <= r29477105;
        double r29477107 = y;
        double r29477108 = t;
        double r29477109 = r29477108 * r29477104;
        double r29477110 = x;
        double r29477111 = r29477109 - r29477110;
        double r29477112 = r29477107 / r29477111;
        double r29477113 = fma(r29477112, r29477104, r29477110);
        double r29477114 = r29477110 / r29477111;
        double r29477115 = r29477114 * r29477114;
        double r29477116 = r29477114 * r29477115;
        double r29477117 = cbrt(r29477116);
        double r29477118 = r29477113 - r29477117;
        double r29477119 = 1.0;
        double r29477120 = r29477110 + r29477119;
        double r29477121 = r29477118 / r29477120;
        double r29477122 = 1.5148165338564995e-166;
        bool r29477123 = r29477104 <= r29477122;
        double r29477124 = r29477107 * r29477104;
        double r29477125 = r29477124 - r29477110;
        double r29477126 = -r29477110;
        double r29477127 = fma(r29477108, r29477104, r29477126);
        double r29477128 = r29477125 / r29477127;
        double r29477129 = r29477128 + r29477110;
        double r29477130 = r29477129 / r29477120;
        double r29477131 = 1.0;
        double r29477132 = r29477131 / r29477111;
        double r29477133 = r29477132 * r29477107;
        double r29477134 = fma(r29477133, r29477104, r29477110);
        double r29477135 = r29477134 - r29477114;
        double r29477136 = r29477135 / r29477120;
        double r29477137 = r29477123 ? r29477130 : r29477136;
        double r29477138 = r29477106 ? r29477121 : r29477137;
        return r29477138;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.5
Target0.4
Herbie3.7
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -5.440078247488523e+157

    1. Initial program 23.5

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

    3. Applied div-sub23.5

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

    4. Applied associate-+r-23.5

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

    5. Simplified9.1

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

    7. Applied add-cbrt-cube9.1

      \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \color{blue}{\sqrt[3]{\left(\frac{x}{t \cdot z - x} \cdot \frac{x}{t \cdot z - x}\right) \cdot \frac{x}{t \cdot z - x}}}}{x + 1}\]

    if -5.440078247488523e+157 < z < 1.5148165338564995e-166

    1. Initial program 1.8

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

    3. Applied fma-neg1.8

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

    if 1.5148165338564995e-166 < z

    1. Initial program 9.7

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

    3. Applied div-sub9.7

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

    4. Applied associate-+r-9.7

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

    5. Simplified4.4

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

    7. Applied div-inv4.5

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

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.440078247488523162933428867079497205001 \cdot 10^{157}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \sqrt[3]{\frac{x}{t \cdot z - x} \cdot \left(\frac{x}{t \cdot z - x} \cdot \frac{x}{t \cdot z - x}\right)}}{x + 1}\\ \mathbf{elif}\;z \le 1.514816533856499451078635282534599335268 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{y \cdot z - x}{\mathsf{fma}\left(t, z, -x\right)} + x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t \cdot z - x} \cdot y, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))