Average Error: 7.7 → 3.0
Time: 25.6s
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 r515183 = x;
        double r515184 = y;
        double r515185 = z;
        double r515186 = r515184 * r515185;
        double r515187 = r515186 - r515183;
        double r515188 = t;
        double r515189 = r515188 * r515185;
        double r515190 = r515189 - r515183;
        double r515191 = r515187 / r515190;
        double r515192 = r515183 + r515191;
        double r515193 = 1.0;
        double r515194 = r515183 + r515193;
        double r515195 = r515192 / r515194;
        return r515195;
}
↓
double f(double x, double y, double z, double t) {
        double r515196 = z;
        double r515197 = -2.889419895373327e-08;
        bool r515198 = r515196 <= r515197;
        double r515199 = 1.0;
        double r515200 = x;
        double r515201 = 1.0;
        double r515202 = r515200 + r515201;
        double r515203 = y;
        double r515204 = t;
        double r515205 = -r515200;
        double r515206 = fma(r515196, r515204, r515205);
        double r515207 = r515203 / r515206;
        double r515208 = fma(r515207, r515196, r515200);
        double r515209 = r515202 / r515208;
        double r515210 = r515199 / r515209;
        double r515211 = r515204 * r515196;
        double r515212 = r515211 - r515200;
        double r515213 = r515200 / r515212;
        double r515214 = r515213 / r515202;
        double r515215 = r515210 - r515214;
        double r515216 = 8.620848150685309e+89;
        bool r515217 = r515196 <= r515216;
        double r515218 = fma(r515196, r515203, r515205);
        double r515219 = r515200 - r515200;
        double r515220 = r515218 + r515219;
        double r515221 = r515220 / r515212;
        double r515222 = r515200 + r515221;
        double r515223 = r515222 / r515202;
        double r515224 = r515203 / r515204;
        double r515225 = r515200 + r515224;
        double r515226 = r515225 / r515202;
        double r515227 = r515217 ? r515223 : r515226;
        double r515228 = r515198 ? r515215 : r515227;
        return r515228;
}

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)))