Average Error: 23.5 → 20.0
Time: 8.2s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.80427409516903929 \cdot 10^{145} \lor \neg \left(z \le 8822645728086630330000\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.80427409516903929 \cdot 10^{145} \lor \neg \left(z \le 8822645728086630330000\right):\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

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

\end{array}
double code(double x, double y, double z, double t, double a, double b) {
	return (((x * y) + (z * (t - a))) / (y + (z * (b - y))));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if (((z <= -1.8042740951690393e+145) || !(z <= 8.82264572808663e+21))) {
		VAR = ((t / b) - (a / b));
	} else {
		VAR = (fma(x, y, (z * (t - a))) / fma((b - y), z, y));
	}
	return VAR;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original23.5
Target18.2
Herbie20.0
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.8042740951690393e+145 or 8.82264572808663e+21 < z

    1. Initial program 43.7

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

    3. Applied clear-num43.7

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

    4. Simplified43.7

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

    5. Taylor expanded around inf 34.1

      \[\leadsto \color{blue}{\frac{t}{b} - \frac{a}{b}}\]

    if -1.8042740951690393e+145 < z < 8.82264572808663e+21

    1. Initial program 12.0

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

    3. Applied clear-num12.1

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

    4. Simplified12.1

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

    6. Applied clear-num12.1

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

    8. Applied remove-double-div12.0

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

  4. Final simplification20.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.80427409516903929 \cdot 10^{145} \lor \neg \left(z \le 8822645728086630330000\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 +o rules:numerics
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :herbie-target
  (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z))))

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