Average Error: 10.4 → 10.4
Time: 3.5s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{\mathsf{fma}\left(z, y, -x\right)}{\mathsf{fma}\left(z, a, -t\right)}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{\mathsf{fma}\left(z, y, -x\right)}{\mathsf{fma}\left(z, a, -t\right)}
double f(double x, double y, double z, double t, double a) {
        double r643936 = x;
        double r643937 = y;
        double r643938 = z;
        double r643939 = r643937 * r643938;
        double r643940 = r643936 - r643939;
        double r643941 = t;
        double r643942 = a;
        double r643943 = r643942 * r643938;
        double r643944 = r643941 - r643943;
        double r643945 = r643940 / r643944;
        return r643945;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r643946 = z;
        double r643947 = y;
        double r643948 = x;
        double r643949 = -r643948;
        double r643950 = fma(r643946, r643947, r643949);
        double r643951 = a;
        double r643952 = t;
        double r643953 = -r643952;
        double r643954 = fma(r643946, r643951, r643953);
        double r643955 = r643950 / r643954;
        return r643955;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.4
Target1.6
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 10.4

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

  3. Applied frac-2neg10.4

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

  4. Simplified10.4

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

  5. Simplified10.4

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

  6. Final simplification10.4

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

Reproduce

herbie shell --seed 2020100 +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))))