Average Error: 10.6 → 3.1
Time: 3.7s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}
double f(double x, double y, double z, double t, double a) {
        double r720248 = x;
        double r720249 = y;
        double r720250 = z;
        double r720251 = r720249 * r720250;
        double r720252 = r720248 - r720251;
        double r720253 = t;
        double r720254 = a;
        double r720255 = r720254 * r720250;
        double r720256 = r720253 - r720255;
        double r720257 = r720252 / r720256;
        return r720257;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r720258 = x;
        double r720259 = 1.0;
        double r720260 = t;
        double r720261 = a;
        double r720262 = z;
        double r720263 = r720261 * r720262;
        double r720264 = r720260 - r720263;
        double r720265 = r720259 / r720264;
        double r720266 = r720258 * r720265;
        double r720267 = y;
        double r720268 = r720260 / r720262;
        double r720269 = r720268 - r720261;
        double r720270 = r720267 / r720269;
        double r720271 = r720266 - r720270;
        return r720271;
}

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

Original10.6
Target1.7
Herbie3.1
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.6

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

  3. Applied div-sub10.6

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

  5. Applied associate-/l*8.2

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

  7. Applied div-sub8.2

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

  8. Simplified3.1

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

  10. Applied div-inv3.1

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

  11. Final simplification3.1

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(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))))