Average Error: 10.3 → 2.9
Time: 3.5s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{x}{t - a \cdot z} - \frac{1}{\frac{\frac{t}{z} - a}{y}}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{x}{t - a \cdot z} - \frac{1}{\frac{\frac{t}{z} - a}{y}}
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) {
	return ((x / (t - (a * z))) - (1.0 / (((t / z) - a) / y)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.3
Target1.5
Herbie2.9
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.3

    \[\frac{x - y \cdot z}{t - a \cdot z}\]
  2. Using strategy rm

  3. Applied div-sub10.3

    \[\leadsto \color{blue}{\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}}\]
  4. Using strategy rm

  5. Applied associate-/l*7.7

    \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{\frac{y}{\frac{t - a \cdot z}{z}}}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity7.7

    \[\leadsto \frac{x}{t - a \cdot z} - \frac{y}{\frac{t - a \cdot z}{\color{blue}{1 \cdot z}}}\]

  8. Applied *-un-lft-identity7.7

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

  9. Applied times-frac7.7

    \[\leadsto \frac{x}{t - a \cdot z} - \frac{y}{\color{blue}{\frac{1}{1} \cdot \frac{t - a \cdot z}{z}}}\]

  10. Simplified7.7

    \[\leadsto \frac{x}{t - a \cdot z} - \frac{y}{\color{blue}{1} \cdot \frac{t - a \cdot z}{z}}\]

  11. Simplified2.7

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

  13. Applied clear-num2.9

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

  14. Simplified2.9

    \[\leadsto \frac{x}{t - a \cdot z} - \frac{1}{\color{blue}{\frac{\frac{t}{z} - a}{y}}}\]

  15. Final simplification2.9

    \[\leadsto \frac{x}{t - a \cdot z} - \frac{1}{\frac{\frac{t}{z} - a}{y}}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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