Average Error: 10.6 → 3.1
Time: 3.2s
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 r755342 = x;
        double r755343 = y;
        double r755344 = z;
        double r755345 = r755343 * r755344;
        double r755346 = r755342 - r755345;
        double r755347 = t;
        double r755348 = a;
        double r755349 = r755348 * r755344;
        double r755350 = r755347 - r755349;
        double r755351 = r755346 / r755350;
        return r755351;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r755352 = x;
        double r755353 = 1.0;
        double r755354 = t;
        double r755355 = a;
        double r755356 = z;
        double r755357 = r755355 * r755356;
        double r755358 = r755354 - r755357;
        double r755359 = r755353 / r755358;
        double r755360 = r755352 * r755359;
        double r755361 = y;
        double r755362 = r755354 / r755356;
        double r755363 = r755362 - r755355;
        double r755364 = r755361 / r755363;
        double r755365 = r755360 - r755364;
        return r755365;
}

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 
(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))))