Average Error: 16.9 → 13.3
Time: 46.9s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.925140021497838792533109596521415441427 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \mathbf{elif}\;t \le 2.894819849023154659261238020079688553345 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{z \cdot y}{t} + x} \cdot \sqrt[3]{\frac{z \cdot y}{t} + x}}{\frac{\frac{b \cdot y}{t} + \left(a + 1\right)}{\sqrt[3]{\frac{z \cdot y}{t} + x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.925140021497838792533109596521415441427 \cdot 10^{-52}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\

\mathbf{elif}\;t \le 2.894819849023154659261238020079688553345 \cdot 10^{-23}:\\
\;\;\;\;\frac{\sqrt[3]{\frac{z \cdot y}{t} + x} \cdot \sqrt[3]{\frac{z \cdot y}{t} + x}}{\frac{\frac{b \cdot y}{t} + \left(a + 1\right)}{\sqrt[3]{\frac{z \cdot y}{t} + x}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r33098151 = x;
        double r33098152 = y;
        double r33098153 = z;
        double r33098154 = r33098152 * r33098153;
        double r33098155 = t;
        double r33098156 = r33098154 / r33098155;
        double r33098157 = r33098151 + r33098156;
        double r33098158 = a;
        double r33098159 = 1.0;
        double r33098160 = r33098158 + r33098159;
        double r33098161 = b;
        double r33098162 = r33098152 * r33098161;
        double r33098163 = r33098162 / r33098155;
        double r33098164 = r33098160 + r33098163;
        double r33098165 = r33098157 / r33098164;
        return r33098165;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r33098166 = t;
        double r33098167 = -1.9251400214978388e-52;
        bool r33098168 = r33098166 <= r33098167;
        double r33098169 = z;
        double r33098170 = r33098169 / r33098166;
        double r33098171 = y;
        double r33098172 = x;
        double r33098173 = fma(r33098170, r33098171, r33098172);
        double r33098174 = a;
        double r33098175 = 1.0;
        double r33098176 = r33098174 + r33098175;
        double r33098177 = cbrt(r33098166);
        double r33098178 = r33098177 * r33098177;
        double r33098179 = r33098171 / r33098178;
        double r33098180 = b;
        double r33098181 = r33098180 / r33098177;
        double r33098182 = r33098179 * r33098181;
        double r33098183 = r33098176 + r33098182;
        double r33098184 = r33098173 / r33098183;
        double r33098185 = 2.8948198490231547e-23;
        bool r33098186 = r33098166 <= r33098185;
        double r33098187 = r33098169 * r33098171;
        double r33098188 = r33098187 / r33098166;
        double r33098189 = r33098188 + r33098172;
        double r33098190 = cbrt(r33098189);
        double r33098191 = r33098190 * r33098190;
        double r33098192 = r33098180 * r33098171;
        double r33098193 = r33098192 / r33098166;
        double r33098194 = r33098193 + r33098176;
        double r33098195 = r33098194 / r33098190;
        double r33098196 = r33098191 / r33098195;
        double r33098197 = r33098186 ? r33098196 : r33098184;
        double r33098198 = r33098168 ? r33098184 : r33098197;
        return r33098198;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original16.9
Target13.6
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.9251400214978388e-52 or 2.8948198490231547e-23 < t

    1. Initial program 12.0

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]

    2. Taylor expanded around 0 12.0

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

    3. Simplified9.2

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

    5. Applied add-cube-cbrt9.3

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

    6. Applied times-frac5.4

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

    if -1.9251400214978388e-52 < t < 2.8948198490231547e-23

    1. Initial program 23.4

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt23.9

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

    4. Applied associate-/l*23.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x + \frac{y \cdot z}{t}} \cdot \sqrt[3]{x + \frac{y \cdot z}{t}}}{\frac{\left(a + 1\right) + \frac{y \cdot b}{t}}{\sqrt[3]{x + \frac{y \cdot z}{t}}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.925140021497838792533109596521415441427 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \mathbf{elif}\;t \le 2.894819849023154659261238020079688553345 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{z \cdot y}{t} + x} \cdot \sqrt[3]{\frac{z \cdot y}{t} + x}}{\frac{\frac{b \cdot y}{t} + \left(a + 1\right)}{\sqrt[3]{\frac{z \cdot y}{t} + x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\left(a + 1\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))