Average Error: 16.5 → 13.1
Time: 58.3s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.084229636969371216885683145647115259784 \cdot 10^{78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\\ \mathbf{elif}\;t \le 1092165316512745.5:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -6.084229636969371216885683145647115259784 \cdot 10^{78}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\\

\mathbf{elif}\;t \le 1092165316512745.5:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r26284096 = x;
        double r26284097 = y;
        double r26284098 = z;
        double r26284099 = r26284097 * r26284098;
        double r26284100 = t;
        double r26284101 = r26284099 / r26284100;
        double r26284102 = r26284096 + r26284101;
        double r26284103 = a;
        double r26284104 = 1.0;
        double r26284105 = r26284103 + r26284104;
        double r26284106 = b;
        double r26284107 = r26284097 * r26284106;
        double r26284108 = r26284107 / r26284100;
        double r26284109 = r26284105 + r26284108;
        double r26284110 = r26284102 / r26284109;
        return r26284110;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r26284111 = t;
        double r26284112 = -6.084229636969371e+78;
        bool r26284113 = r26284111 <= r26284112;
        double r26284114 = y;
        double r26284115 = r26284114 / r26284111;
        double r26284116 = z;
        double r26284117 = x;
        double r26284118 = fma(r26284115, r26284116, r26284117);
        double r26284119 = 1.0;
        double r26284120 = b;
        double r26284121 = a;
        double r26284122 = fma(r26284120, r26284115, r26284121);
        double r26284123 = 1.0;
        double r26284124 = r26284122 + r26284123;
        double r26284125 = r26284119 / r26284124;
        double r26284126 = r26284118 * r26284125;
        double r26284127 = 1092165316512745.5;
        bool r26284128 = r26284111 <= r26284127;
        double r26284129 = r26284114 * r26284116;
        double r26284130 = r26284129 / r26284111;
        double r26284131 = r26284117 + r26284130;
        double r26284132 = r26284120 * r26284114;
        double r26284133 = r26284132 / r26284111;
        double r26284134 = r26284123 + r26284121;
        double r26284135 = r26284133 + r26284134;
        double r26284136 = r26284131 / r26284135;
        double r26284137 = r26284128 ? r26284136 : r26284126;
        double r26284138 = r26284113 ? r26284126 : r26284137;
        return r26284138;
}

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

Original16.5
Target13.4
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 2 regimes

  2. if t < -6.084229636969371e+78 or 1092165316512745.5 < t

    1. Initial program 11.3

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

    2. Simplified3.4

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

    4. Applied div-inv3.5

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

    if -6.084229636969371e+78 < t < 1092165316512745.5

    1. Initial program 20.5

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

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.084229636969371216885683145647115259784 \cdot 10^{78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\\ \mathbf{elif}\;t \le 1092165316512745.5:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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))))