Average Error: 10.3 → 6.6
Time: 10.1s
Precision: binary64
\[\frac{x - y \cdot z}{t - a \cdot z} \]
↓
\[\begin{array}{l} t_1 := t - z \cdot a\\ \mathbf{if}\;\frac{x - y \cdot z}{t_1} \leq 8.902082254793487 \cdot 10^{+295}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y, z, x\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
t_1 := t - z \cdot a\\
\mathbf{if}\;\frac{x - y \cdot z}{t_1} \leq 8.902082254793487 \cdot 10^{+295}:\\
\;\;\;\;\frac{\mathsf{fma}\left(-y, z, x\right)}{t_1}\\

\mathbf{else}:\\
\;\;\;\;\frac{y}{a}\\


\end{array}

(FPCore (x y z t a) :precision binary64 (/ (- x (* y z)) (- t (* a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))))
   (if (<= (/ (- x (* y z)) t_1) 8.902082254793487e+295)
     (/ (fma (- y) z x) t_1)
     (/ y a))))
double code(double x, double y, double z, double t, double a) {
	return (x - (y * z)) / (t - (a * z));
}
↓
double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double tmp;
	if (((x - (y * z)) / t_1) <= 8.902082254793487e+295) {
		tmp = fma(-y, z, x) / t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.3
Target1.6
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 8.90208225479348708e295

    1. Initial program 6.1

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

    2. Applied egg-rr6.1

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

    if 8.90208225479348708e295 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

    1. Initial program 59.4

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

    2. Taylor expanded in z around inf 13.2

      \[\leadsto \color{blue}{\frac{y}{a}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 8.902082254793487 \cdot 10^{+295}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y, z, x\right)}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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