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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.04768639156161125 \cdot 10^{39}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}{\frac{y}{\frac{t}{z}} + x}}\\ \mathbf{elif}\;t \le 1.66916686530934162 \cdot 10^{47}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;t \le -5.04768639156161125 \cdot 10^{39}:\\
\;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}{\frac{y}{\frac{t}{z}} + x}}\\

\mathbf{elif}\;t \le 1.66916686530934162 \cdot 10^{47}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\

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

\end{array}

double f(double x, double y, double z, double t, double a, double b) {
        double r798373 = x;
        double r798374 = y;
        double r798375 = z;
        double r798376 = r798374 * r798375;
        double r798377 = t;
        double r798378 = r798376 / r798377;
        double r798379 = r798373 + r798378;
        double r798380 = a;
        double r798381 = 1.0;
        double r798382 = r798380 + r798381;
        double r798383 = b;
        double r798384 = r798374 * r798383;
        double r798385 = r798384 / r798377;
        double r798386 = r798382 + r798385;
        double r798387 = r798379 / r798386;
        return r798387;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r798388 = t;
        double r798389 = -5.047686391561611e+39;
        bool r798390 = r798388 <= r798389;
        double r798391 = 1.0;
        double r798392 = y;
        double r798393 = r798392 / r798388;
        double r798394 = b;
        double r798395 = a;
        double r798396 = fma(r798393, r798394, r798395);
        double r798397 = 1.0;
        double r798398 = r798396 + r798397;
        double r798399 = z;
        double r798400 = r798388 / r798399;
        double r798401 = r798392 / r798400;
        double r798402 = x;
        double r798403 = r798401 + r798402;
        double r798404 = r798398 / r798403;
        double r798405 = r798391 / r798404;
        double r798406 = 1.6691668653093416e+47;
        bool r798407 = r798388 <= r798406;
        double r798408 = r798392 * r798399;
        double r798409 = r798408 / r798388;
        double r798410 = r798402 + r798409;
        double r798411 = r798395 + r798397;
        double r798412 = r798392 * r798394;
        double r798413 = r798412 / r798388;
        double r798414 = r798411 + r798413;
        double r798415 = r798410 / r798414;
        double r798416 = fma(r798393, r798399, r798402);
        double r798417 = r798391 / r798398;
        double r798418 = r798416 * r798417;
        double r798419 = r798407 ? r798415 : r798418;
        double r798420 = r798390 ? r798405 : r798419;
        return r798420;
}

Original	16.6
Target	13.1
Herbie	12.9

Derivation

Split input into 3 regimes
if t < -5.047686391561611e+39
1. Initial program 12.4
  \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
2. Simplified3.6
  \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}}\]
3. Using strategy rm
4. Applied clear-num4.0
  \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}}\]
5. Using strategy rm
6. Applied fma-udef4.0
  \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}{\color{blue}{\frac{y}{t} \cdot z + x}}}\]
7. Simplified3.9
  \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}{\color{blue}{\frac{y}{\frac{t}{z}}} + x}}\]
if -5.047686391561611e+39 < t < 1.6691668653093416e+47
1. Initial program 20.6
  \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
if 1.6691668653093416e+47 < t
1. Initial program 11.0
  \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
2. Simplified2.9
  \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}}\]
3. Using strategy rm
4. Applied div-inv3.0
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}}\]
Recombined 3 regimes into one program.
Final simplification12.9
\[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.04768639156161125 \cdot 10^{39}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}{\frac{y}{\frac{t}{z}} + x}}\\ \mathbf{elif}\;t \le 1.66916686530934162 \cdot 10^{47}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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))))

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Target

Derivation

`if t < -5.047686391561611e+39`

`if -5.047686391561611e+39 < t < 1.6691668653093416e+47`

`if 1.6691668653093416e+47 < t`

Reproduce