Average Error: 7.7 → 3.0
Time: 26.9s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.889419895373327148972973112388967775743 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, t, -x\right)}, z, x\right)}} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;z \le 8.620848150685308830177863726582628536017 \cdot 10^{89}:\\ \;\;\;\;\frac{x + \frac{\mathsf{fma}\left(z, y, -x\right) + \left(x - x\right)}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.889419895373327148972973112388967775743 \cdot 10^{-8}:\\
\;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, t, -x\right)}, z, x\right)}} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\

\mathbf{elif}\;z \le 8.620848150685308830177863726582628536017 \cdot 10^{89}:\\
\;\;\;\;\frac{x + \frac{\mathsf{fma}\left(z, y, -x\right) + \left(x - x\right)}{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 r1096087 = x;
        double r1096088 = y;
        double r1096089 = z;
        double r1096090 = r1096088 * r1096089;
        double r1096091 = r1096090 - r1096087;
        double r1096092 = t;
        double r1096093 = r1096092 * r1096089;
        double r1096094 = r1096093 - r1096087;
        double r1096095 = r1096091 / r1096094;
        double r1096096 = r1096087 + r1096095;
        double r1096097 = 1.0;
        double r1096098 = r1096087 + r1096097;
        double r1096099 = r1096096 / r1096098;
        return r1096099;
}
↓
double f(double x, double y, double z, double t) {
        double r1096100 = z;
        double r1096101 = -2.889419895373327e-08;
        bool r1096102 = r1096100 <= r1096101;
        double r1096103 = 1.0;
        double r1096104 = x;
        double r1096105 = 1.0;
        double r1096106 = r1096104 + r1096105;
        double r1096107 = y;
        double r1096108 = t;
        double r1096109 = -r1096104;
        double r1096110 = fma(r1096100, r1096108, r1096109);
        double r1096111 = r1096107 / r1096110;
        double r1096112 = fma(r1096111, r1096100, r1096104);
        double r1096113 = r1096106 / r1096112;
        double r1096114 = r1096103 / r1096113;
        double r1096115 = r1096108 * r1096100;
        double r1096116 = r1096115 - r1096104;
        double r1096117 = r1096104 / r1096116;
        double r1096118 = r1096117 / r1096106;
        double r1096119 = r1096114 - r1096118;
        double r1096120 = 8.620848150685309e+89;
        bool r1096121 = r1096100 <= r1096120;
        double r1096122 = fma(r1096100, r1096107, r1096109);
        double r1096123 = r1096104 - r1096104;
        double r1096124 = r1096122 + r1096123;
        double r1096125 = r1096124 / r1096116;
        double r1096126 = r1096104 + r1096125;
        double r1096127 = r1096126 / r1096106;
        double r1096128 = r1096107 / r1096108;
        double r1096129 = r1096104 + r1096128;
        double r1096130 = r1096129 / r1096106;
        double r1096131 = r1096121 ? r1096127 : r1096130;
        double r1096132 = r1096102 ? r1096119 : r1096131;
        return r1096132;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.7
Target0.4
Herbie3.0
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.889419895373327e-08

    1. Initial program 16.2

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

    3. Applied div-sub16.2

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

    4. Applied associate-+r-16.2

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

    5. Applied div-sub16.2

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

    6. Simplified5.4

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

    8. Applied clear-num5.4

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

    if -2.889419895373327e-08 < z < 8.620848150685309e+89

    1. Initial program 0.5

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

    3. Applied add-cube-cbrt0.8

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

    4. Applied prod-diff0.8

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

    5. Simplified0.5

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

    6. Simplified0.5

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

    if 8.620848150685309e+89 < z

    1. Initial program 19.2

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

    2. Taylor expanded around inf 7.5

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.889419895373327148972973112388967775743 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(z, t, -x\right)}, z, x\right)}} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;z \le 8.620848150685308830177863726582628536017 \cdot 10^{89}:\\ \;\;\;\;\frac{x + \frac{\mathsf{fma}\left(z, y, -x\right) + \left(x - x\right)}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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)))