Average Error: 10.7 → 3.1
Time: 13.7s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{1}{a - \frac{t}{z}} \cdot y - \frac{x}{\mathsf{fma}\left(z, a, -t\right)}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{1}{a - \frac{t}{z}} \cdot y - \frac{x}{\mathsf{fma}\left(z, a, -t\right)}
double f(double x, double y, double z, double t, double a) {
        double r427556 = x;
        double r427557 = y;
        double r427558 = z;
        double r427559 = r427557 * r427558;
        double r427560 = r427556 - r427559;
        double r427561 = t;
        double r427562 = a;
        double r427563 = r427562 * r427558;
        double r427564 = r427561 - r427563;
        double r427565 = r427560 / r427564;
        return r427565;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r427566 = 1.0;
        double r427567 = a;
        double r427568 = t;
        double r427569 = z;
        double r427570 = r427568 / r427569;
        double r427571 = r427567 - r427570;
        double r427572 = r427566 / r427571;
        double r427573 = y;
        double r427574 = r427572 * r427573;
        double r427575 = x;
        double r427576 = -r427568;
        double r427577 = fma(r427569, r427567, r427576);
        double r427578 = r427575 / r427577;
        double r427579 = r427574 - r427578;
        return r427579;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.7
Target1.7
Herbie3.1
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.7

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

  2. Simplified10.7

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

  4. Applied frac-2neg10.7

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

  5. Simplified10.7

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

  6. Simplified10.7

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

  8. Applied div-sub10.7

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

  9. Simplified8.3

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

  10. Simplified8.3

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

  11. Taylor expanded around 0 3.0

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

  13. Applied div-inv3.1

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

  14. Final simplification3.1

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

Reproduce

herbie shell --seed 2019196 +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)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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