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

↓

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

\frac{x}{y - z \cdot t}

↓

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

double f(double x, double y, double z, double t) {
        double r652578 = x;
        double r652579 = y;
        double r652580 = z;
        double r652581 = t;
        double r652582 = r652580 * r652581;
        double r652583 = r652579 - r652582;
        double r652584 = r652578 / r652583;
        return r652584;
}

↓

double f(double x, double y, double z, double t) {
        double r652585 = x;
        double r652586 = -r652585;
        double r652587 = t;
        double r652588 = z;
        double r652589 = y;
        double r652590 = -r652589;
        double r652591 = fma(r652587, r652588, r652590);
        double r652592 = r652586 / r652591;
        return r652592;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	2.7
Target	1.9
Herbie	2.7

\[\begin{array}{l} \mathbf{if}\;x \lt -1.618195973607049 \cdot 10^{50}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{elif}\;x \lt 2.13783064348764444 \cdot 10^{131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \end{array}\]

Derivation

Initial program 2.7
\[\frac{x}{y - z \cdot t}\]
Using strategy rm
Applied frac-2neg2.7
\[\leadsto \color{blue}{\frac{-x}{-\left(y - z \cdot t\right)}}\]
Simplified2.7
\[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(t, z, -y\right)}}\]
Final simplification2.7
\[\leadsto \frac{-x}{\mathsf{fma}\left(t, z, -y\right)}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1 (- (/ y x) (* (/ z x) t)))))

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