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

↓

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

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

↓

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

double f(double x, double y, double z, double t, double a, double b) {
        double r788018 = x;
        double r788019 = y;
        double r788020 = z;
        double r788021 = r788019 * r788020;
        double r788022 = t;
        double r788023 = r788021 / r788022;
        double r788024 = r788018 + r788023;
        double r788025 = a;
        double r788026 = 1.0;
        double r788027 = r788025 + r788026;
        double r788028 = b;
        double r788029 = r788019 * r788028;
        double r788030 = r788029 / r788022;
        double r788031 = r788027 + r788030;
        double r788032 = r788024 / r788031;
        return r788032;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r788033 = 1.0;
        double r788034 = y;
        double r788035 = t;
        double r788036 = r788034 / r788035;
        double r788037 = z;
        double r788038 = x;
        double r788039 = fma(r788036, r788037, r788038);
        double r788040 = r788033 * r788039;
        double r788041 = a;
        double r788042 = 1.0;
        double r788043 = r788041 + r788042;
        double r788044 = r788035 / r788034;
        double r788045 = b;
        double r788046 = r788044 / r788045;
        double r788047 = r788033 / r788046;
        double r788048 = r788043 + r788047;
        double r788049 = r788040 / r788048;
        return r788049;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	17.0
Target	13.6
Herbie	15.0

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

Derivation

Initial program 17.0
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
Using strategy rm
Applied *-un-lft-identity17.0
\[\leadsto \frac{x + \color{blue}{1 \cdot \frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
Applied *-un-lft-identity17.0
\[\leadsto \frac{\color{blue}{1 \cdot x} + 1 \cdot \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
Applied distribute-lft-out17.0
\[\leadsto \frac{\color{blue}{1 \cdot \left(x + \frac{y \cdot z}{t}\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
Simplified17.2
\[\leadsto \frac{1 \cdot \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
Using strategy rm
Applied clear-num17.2
\[\leadsto \frac{1 \cdot \mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + \color{blue}{\frac{1}{\frac{t}{y \cdot b}}}}\]
Using strategy rm
Applied associate-/r*15.0
\[\leadsto \frac{1 \cdot \mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + \frac{1}{\color{blue}{\frac{\frac{t}{y}}{b}}}}\]
Final simplification15.0
\[\leadsto \frac{1 \cdot \mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + \frac{1}{\frac{\frac{t}{y}}{b}}}\]

Reproduce

herbie shell --seed 2020018 +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))))