Average Error: 10.7 → 10.8
Time: 4.5s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\mathsf{fma}\left(z, y, -x\right) \cdot \frac{1}{\mathsf{fma}\left(z, a, -t\right)}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\mathsf{fma}\left(z, y, -x\right) \cdot \frac{1}{\mathsf{fma}\left(z, a, -t\right)}
double f(double x, double y, double z, double t, double a) {
        double r608373 = x;
        double r608374 = y;
        double r608375 = z;
        double r608376 = r608374 * r608375;
        double r608377 = r608373 - r608376;
        double r608378 = t;
        double r608379 = a;
        double r608380 = r608379 * r608375;
        double r608381 = r608378 - r608380;
        double r608382 = r608377 / r608381;
        return r608382;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r608383 = z;
        double r608384 = y;
        double r608385 = x;
        double r608386 = -r608385;
        double r608387 = fma(r608383, r608384, r608386);
        double r608388 = 1.0;
        double r608389 = a;
        double r608390 = t;
        double r608391 = -r608390;
        double r608392 = fma(r608383, r608389, r608391);
        double r608393 = r608388 / r608392;
        double r608394 = r608387 * r608393;
        return r608394;
}

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
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.7

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

  3. Applied frac-2neg10.7

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

  4. Simplified10.7

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

  5. Simplified10.7

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

  7. Applied div-inv10.8

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

  8. Final simplification10.8

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

Reproduce

herbie shell --seed 2020035 +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))))