Average Error: 16.5 → 14.2
Time: 4.9s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}
double f(double x, double y, double z, double t, double a, double b) {
        double r663363 = x;
        double r663364 = y;
        double r663365 = z;
        double r663366 = r663364 * r663365;
        double r663367 = t;
        double r663368 = r663366 / r663367;
        double r663369 = r663363 + r663368;
        double r663370 = a;
        double r663371 = 1.0;
        double r663372 = r663370 + r663371;
        double r663373 = b;
        double r663374 = r663364 * r663373;
        double r663375 = r663374 / r663367;
        double r663376 = r663372 + r663375;
        double r663377 = r663369 / r663376;
        return r663377;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r663378 = y;
        double r663379 = t;
        double r663380 = r663378 / r663379;
        double r663381 = z;
        double r663382 = x;
        double r663383 = fma(r663380, r663381, r663382);
        double r663384 = b;
        double r663385 = a;
        double r663386 = 1.0;
        double r663387 = r663385 + r663386;
        double r663388 = fma(r663380, r663384, r663387);
        double r663389 = r663383 / r663388;
        return r663389;
}

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
Target13.2
Herbie14.2
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Initial program 16.5

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

  3. Applied div-inv16.6

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

  4. Simplified15.9

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

  6. Applied *-un-lft-identity15.9

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

  7. Applied associate-*l*15.9

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

  8. Simplified14.2

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

  9. Final simplification14.2

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

Reproduce

herbie shell --seed 2020021 +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))))