Average Error: 17.0 → 12.9
Time: 23.2s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.858089223190981120235222646202607437687 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\ \mathbf{elif}\;t \le 2014489387.42927074432373046875:\\ \;\;\;\;\frac{\frac{y \cdot z}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)} \cdot \mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.858089223190981120235222646202607437687 \cdot 10^{-48}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\

\mathbf{elif}\;t \le 2014489387.42927074432373046875:\\
\;\;\;\;\frac{\frac{y \cdot z}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1\right)}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r26771146 = x;
        double r26771147 = y;
        double r26771148 = z;
        double r26771149 = r26771147 * r26771148;
        double r26771150 = t;
        double r26771151 = r26771149 / r26771150;
        double r26771152 = r26771146 + r26771151;
        double r26771153 = a;
        double r26771154 = 1.0;
        double r26771155 = r26771153 + r26771154;
        double r26771156 = b;
        double r26771157 = r26771147 * r26771156;
        double r26771158 = r26771157 / r26771150;
        double r26771159 = r26771155 + r26771158;
        double r26771160 = r26771152 / r26771159;
        return r26771160;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r26771161 = t;
        double r26771162 = -1.858089223190981e-48;
        bool r26771163 = r26771161 <= r26771162;
        double r26771164 = z;
        double r26771165 = r26771164 / r26771161;
        double r26771166 = y;
        double r26771167 = x;
        double r26771168 = fma(r26771165, r26771166, r26771167);
        double r26771169 = r26771166 / r26771161;
        double r26771170 = b;
        double r26771171 = a;
        double r26771172 = 1.0;
        double r26771173 = r26771171 + r26771172;
        double r26771174 = fma(r26771169, r26771170, r26771173);
        double r26771175 = r26771168 / r26771174;
        double r26771176 = 2014489387.4292707;
        bool r26771177 = r26771161 <= r26771176;
        double r26771178 = r26771166 * r26771164;
        double r26771179 = r26771178 / r26771161;
        double r26771180 = r26771179 + r26771167;
        double r26771181 = r26771170 * r26771166;
        double r26771182 = r26771181 / r26771161;
        double r26771183 = r26771182 + r26771173;
        double r26771184 = r26771180 / r26771183;
        double r26771185 = 1.0;
        double r26771186 = fma(r26771169, r26771164, r26771167);
        double r26771187 = r26771185 / r26771186;
        double r26771188 = r26771187 * r26771174;
        double r26771189 = r26771185 / r26771188;
        double r26771190 = r26771177 ? r26771184 : r26771189;
        double r26771191 = r26771163 ? r26771175 : r26771190;
        return r26771191;
}

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

Original17.0
Target13.5
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -1.858089223190981e-48

    1. Initial program 11.7

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

    2. Simplified5.4

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

    4. Applied clear-num5.8

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

    6. Applied div-inv5.8

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

    8. Applied div-inv5.8

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

    9. Simplified4.9

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

    if -1.858089223190981e-48 < t < 2014489387.4292707

    1. Initial program 22.5

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

    if 2014489387.4292707 < t

    1. Initial program 13.6

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

    2. Simplified4.5

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

    4. Applied clear-num5.0

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

    6. Applied div-inv5.0

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

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.858089223190981120235222646202607437687 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\ \mathbf{elif}\;t \le 2014489387.42927074432373046875:\\ \;\;\;\;\frac{\frac{y \cdot z}{t} + x}{\frac{b \cdot y}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)} \cdot \mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +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))))