Average Error: 16.8 → 5.7
Time: 8.5s
Precision: binary64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
↓
\[\begin{array}{l} t_1 := \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{if}\;y \leq -1.5249508773597182 \cdot 10^{-19} \lor \neg \left(y \leq 6884034804144.34\right):\\ \;\;\;\;\frac{1}{\frac{t}{y \cdot z} + \mathsf{fma}\left(\frac{t}{z}, \frac{a}{y}, \frac{b}{z}\right)} + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \end{array} \]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
t_1 := \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\
\mathbf{if}\;y \leq -1.5249508773597182 \cdot 10^{-19} \lor \neg \left(y \leq 6884034804144.34\right):\\
\;\;\;\;\frac{1}{\frac{t}{y \cdot z} + \mathsf{fma}\left(\frac{t}{z}, \frac{a}{y}, \frac{b}{z}\right)} + t_1\\

\mathbf{else}:\\
\;\;\;\;t_1 + \frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\


\end{array}

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
↓
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ 1.0 (+ a (/ (* y b) t))))))
   (if (or (<= y -1.5249508773597182e-19) (not (<= y 6884034804144.34)))
     (+ (/ 1.0 (+ (/ t (* y z)) (fma (/ t z) (/ a y) (/ b z)))) t_1)
     (+ t_1 (/ (* y z) (fma y b (fma a t t)))))))
double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 + (a + ((y * b) / t)));
	double tmp;
	if ((y <= -1.5249508773597182e-19) || !(y <= 6884034804144.34)) {
		tmp = (1.0 / ((t / (y * z)) + fma((t / z), (a / y), (b / z)))) + t_1;
	} else {
		tmp = t_1 + ((y * z) / fma(y, b, fma(a, t, t)));
	}
	return tmp;
}

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

Original16.8
Target13.1
Herbie5.7
\[\begin{array}{l} \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.52495087735971815e-19 or 6884034804144.3398 < y

    1. Initial program 30.1

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

    2. Applied clear-num_binary6430.1

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

    3. Taylor expanded in x around 0 28.1

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

    4. Applied clear-num_binary6428.1

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

    5. Taylor expanded in a around 0 12.1

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

    6. Simplified9.8

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

    if -1.52495087735971815e-19 < y < 6884034804144.3398

    1. Initial program 3.8

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

    2. Applied clear-num_binary643.9

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

    3. Taylor expanded in x around 0 3.1

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

    4. Taylor expanded in z around inf 1.7

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

    5. Simplified1.7

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

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5249508773597182 \cdot 10^{-19} \lor \neg \left(y \leq 6884034804144.34\right):\\ \;\;\;\;\frac{1}{\frac{t}{y \cdot z} + \mathsf{fma}\left(\frac{t}{z}, \frac{a}{y}, \frac{b}{z}\right)} + \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)} + \frac{y \cdot z}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021274 
(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) (* 1.0 (* (+ 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))))