Average Error: 10.0 → 2.9
Time: 21.2s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}
double f(double x, double y, double z, double t, double a) {
        double r12946560 = x;
        double r12946561 = y;
        double r12946562 = z;
        double r12946563 = r12946561 * r12946562;
        double r12946564 = r12946560 - r12946563;
        double r12946565 = t;
        double r12946566 = a;
        double r12946567 = r12946566 * r12946562;
        double r12946568 = r12946565 - r12946567;
        double r12946569 = r12946564 / r12946568;
        return r12946569;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r12946570 = x;
        double r12946571 = t;
        double r12946572 = a;
        double r12946573 = z;
        double r12946574 = r12946572 * r12946573;
        double r12946575 = r12946571 - r12946574;
        double r12946576 = r12946570 / r12946575;
        double r12946577 = y;
        double r12946578 = r12946571 / r12946573;
        double r12946579 = r12946578 - r12946572;
        double r12946580 = r12946577 / r12946579;
        double r12946581 = r12946576 - r12946580;
        return r12946581;
}

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

Original10.0
Target1.7
Herbie2.9
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344.0:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.0

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

  3. Applied div-sub10.0

    \[\leadsto \color{blue}{\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}}\]
  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. Taylor expanded around 0 2.9

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

  7. Final simplification2.9

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ 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))))