Average Error: 3.1 → 2.8
Time: 8.4s
Precision: binary64
\[[z, t] = \mathsf{sort}([z, t]) \\]
\[\frac{x}{y - z \cdot t} \]
↓
\[\begin{array}{l} t_1 := t \cdot \frac{z}{x}\\ \mathbf{if}\;x \leq 1.105352214638888 \cdot 10^{+179}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, -z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(y, \frac{1}{x}, -t_1\right) + \mathsf{fma}\left(\frac{-z}{x}, t, t_1\right)\right)}^{-1}\\ \end{array} \]
\frac{x}{y - z \cdot t}
↓
\begin{array}{l}
t_1 := t \cdot \frac{z}{x}\\
\mathbf{if}\;x \leq 1.105352214638888 \cdot 10^{+179}:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(t, -z, y\right)}\\

\mathbf{else}:\\
\;\;\;\;{\left(\mathsf{fma}\left(y, \frac{1}{x}, -t_1\right) + \mathsf{fma}\left(\frac{-z}{x}, t, t_1\right)\right)}^{-1}\\


\end{array}

(FPCore (x y z t) :precision binary64 (/ x (- y (* z t))))
↓
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (/ z x))))
   (if (<= x 1.105352214638888e+179)
     (/ x (fma t (- z) y))
     (pow (+ (fma y (/ 1.0 x) (- t_1)) (fma (/ (- z) x) t t_1)) -1.0))))
double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}
↓
double code(double x, double y, double z, double t) {
	double t_1 = t * (z / x);
	double tmp;
	if (x <= 1.105352214638888e+179) {
		tmp = x / fma(t, -z, y);
	} else {
		tmp = pow((fma(y, (1.0 / x), -t_1) + fma((-z / x), t, t_1)), -1.0);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original3.1
Target1.9
Herbie2.8
\[\begin{array}{l} \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if x < 1.10535221463888801e179

    1. Initial program 2.4

      \[\frac{x}{y - z \cdot t} \]

    2. Applied egg-rr2.4

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, -z, y\right)}} \]

    if 1.10535221463888801e179 < x

    1. Initial program 9.4

      \[\frac{x}{y - z \cdot t} \]

    2. Applied egg-rr9.5

      \[\leadsto \color{blue}{{\left(\frac{y - z \cdot t}{x}\right)}^{-1}} \]

    3. Applied egg-rr6.4

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

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.105352214638888 \cdot 10^{+179}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, -z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(y, \frac{1}{x}, -t \cdot \frac{z}{x}\right) + \mathsf{fma}\left(\frac{-z}{x}, t, t \cdot \frac{z}{x}\right)\right)}^{-1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1.0 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1.0 (- (/ y x) (* (/ z x) t)))))

  (/ x (- y (* z t))))