Average Error: 16.1 → 13.1
Time: 6.1s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.065454724711100881523316892095087502286 \cdot 10^{-112}:\\ \;\;\;\;\frac{x + y \cdot \frac{\frac{1}{t}}{\frac{1}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{elif}\;y \le 5.680955193031466208337968211151925843386 \cdot 10^{-149}:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{\frac{y \cdot z}{t}} \cdot \sqrt[3]{\frac{y \cdot z}{t}}\right) \cdot \sqrt[3]{\frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;y \le -7.065454724711100881523316892095087502286 \cdot 10^{-112}:\\
\;\;\;\;\frac{x + y \cdot \frac{\frac{1}{t}}{\frac{1}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\

\mathbf{elif}\;y \le 5.680955193031466208337968211151925843386 \cdot 10^{-149}:\\
\;\;\;\;\frac{x + \left(\sqrt[3]{\frac{y \cdot z}{t}} \cdot \sqrt[3]{\frac{y \cdot z}{t}}\right) \cdot \sqrt[3]{\frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r680213 = x;
        double r680214 = y;
        double r680215 = z;
        double r680216 = r680214 * r680215;
        double r680217 = t;
        double r680218 = r680216 / r680217;
        double r680219 = r680213 + r680218;
        double r680220 = a;
        double r680221 = 1.0;
        double r680222 = r680220 + r680221;
        double r680223 = b;
        double r680224 = r680214 * r680223;
        double r680225 = r680224 / r680217;
        double r680226 = r680222 + r680225;
        double r680227 = r680219 / r680226;
        return r680227;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r680228 = y;
        double r680229 = -7.065454724711101e-112;
        bool r680230 = r680228 <= r680229;
        double r680231 = x;
        double r680232 = 1.0;
        double r680233 = t;
        double r680234 = r680232 / r680233;
        double r680235 = z;
        double r680236 = r680232 / r680235;
        double r680237 = r680234 / r680236;
        double r680238 = r680228 * r680237;
        double r680239 = r680231 + r680238;
        double r680240 = a;
        double r680241 = 1.0;
        double r680242 = r680240 + r680241;
        double r680243 = b;
        double r680244 = r680233 / r680243;
        double r680245 = r680228 / r680244;
        double r680246 = r680242 + r680245;
        double r680247 = r680239 / r680246;
        double r680248 = 5.680955193031466e-149;
        bool r680249 = r680228 <= r680248;
        double r680250 = r680228 * r680235;
        double r680251 = r680250 / r680233;
        double r680252 = cbrt(r680251);
        double r680253 = r680252 * r680252;
        double r680254 = r680253 * r680252;
        double r680255 = r680231 + r680254;
        double r680256 = r680228 * r680243;
        double r680257 = r680256 / r680233;
        double r680258 = r680242 + r680257;
        double r680259 = r680255 / r680258;
        double r680260 = r680233 / r680235;
        double r680261 = r680228 / r680260;
        double r680262 = r680231 + r680261;
        double r680263 = r680262 / r680246;
        double r680264 = r680249 ? r680259 : r680263;
        double r680265 = r680230 ? r680247 : r680264;
        return r680265;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.1
Target12.9
Herbie13.1
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -7.065454724711101e-112

    1. Initial program 22.8

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

    3. Applied associate-/l*20.6

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

    5. Applied associate-/l*18.2

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

    7. Applied div-inv18.3

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

    9. Applied div-inv18.3

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

    10. Applied associate-/r*18.3

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

    if -7.065454724711101e-112 < y < 5.680955193031466e-149

    1. Initial program 1.7

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

    3. Applied add-cube-cbrt1.9

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

    if 5.680955193031466e-149 < y

    1. Initial program 21.5

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

    3. Applied associate-/l*19.5

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

    5. Applied associate-/l*17.4

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

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.065454724711100881523316892095087502286 \cdot 10^{-112}:\\ \;\;\;\;\frac{x + y \cdot \frac{\frac{1}{t}}{\frac{1}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{elif}\;y \le 5.680955193031466208337968211151925843386 \cdot 10^{-149}:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{\frac{y \cdot z}{t}} \cdot \sqrt[3]{\frac{y \cdot z}{t}}\right) \cdot \sqrt[3]{\frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1) (/ (* y b) t))))