Average Error: 16.0 → 12.4
Time: 11.3s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.93323669588324535 \cdot 10^{36} \lor \neg \left(t \le 1.6432470003784239 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}{\frac{z}{t} \cdot y + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -3.93323669588324535 \cdot 10^{36} \lor \neg \left(t \le 1.6432470003784239 \cdot 10^{-15}\right):\\
\;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}{\frac{z}{t} \cdot y + x}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r808012 = x;
        double r808013 = y;
        double r808014 = z;
        double r808015 = r808013 * r808014;
        double r808016 = t;
        double r808017 = r808015 / r808016;
        double r808018 = r808012 + r808017;
        double r808019 = a;
        double r808020 = 1.0;
        double r808021 = r808019 + r808020;
        double r808022 = b;
        double r808023 = r808013 * r808022;
        double r808024 = r808023 / r808016;
        double r808025 = r808021 + r808024;
        double r808026 = r808018 / r808025;
        return r808026;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r808027 = t;
        double r808028 = -3.9332366958832453e+36;
        bool r808029 = r808027 <= r808028;
        double r808030 = 1.6432470003784239e-15;
        bool r808031 = r808027 <= r808030;
        double r808032 = !r808031;
        bool r808033 = r808029 || r808032;
        double r808034 = 1.0;
        double r808035 = y;
        double r808036 = r808035 / r808027;
        double r808037 = b;
        double r808038 = a;
        double r808039 = fma(r808036, r808037, r808038);
        double r808040 = 1.0;
        double r808041 = r808039 + r808040;
        double r808042 = z;
        double r808043 = r808042 / r808027;
        double r808044 = r808043 * r808035;
        double r808045 = x;
        double r808046 = r808044 + r808045;
        double r808047 = r808041 / r808046;
        double r808048 = r808034 / r808047;
        double r808049 = r808035 * r808042;
        double r808050 = r808049 / r808027;
        double r808051 = r808045 + r808050;
        double r808052 = r808038 + r808040;
        double r808053 = r808035 * r808037;
        double r808054 = r808053 / r808027;
        double r808055 = r808052 + r808054;
        double r808056 = r808051 / r808055;
        double r808057 = r808033 ? r808048 : r808056;
        return r808057;
}

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

Original16.0
Target12.9
Herbie12.4
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -3.9332366958832453e+36 or 1.6432470003784239e-15 < t

    1. Initial program 11.5

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

    2. Simplified3.8

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

    4. Applied clear-num4.2

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

    6. Applied fma-udef4.2

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

    7. Simplified4.2

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

    if -3.9332366958832453e+36 < t < 1.6432470003784239e-15

    1. Initial program 20.4

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

  4. Final simplification12.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.93323669588324535 \cdot 10^{36} \lor \neg \left(t \le 1.6432470003784239 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}{\frac{z}{t} \cdot y + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))