Average Error: 10.7 → 10.8
Time: 3.4s
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 r788746 = x;
        double r788747 = y;
        double r788748 = z;
        double r788749 = r788747 * r788748;
        double r788750 = r788746 - r788749;
        double r788751 = t;
        double r788752 = a;
        double r788753 = r788752 * r788748;
        double r788754 = r788751 - r788753;
        double r788755 = r788750 / r788754;
        return r788755;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r788756 = z;
        double r788757 = y;
        double r788758 = x;
        double r788759 = -r788758;
        double r788760 = fma(r788756, r788757, r788759);
        double r788761 = 1.0;
        double r788762 = a;
        double r788763 = t;
        double r788764 = -r788763;
        double r788765 = fma(r788756, r788762, r788764);
        double r788766 = r788761 / r788765;
        double r788767 = r788760 * r788766;
        return r788767;
}

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 2020060 +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))))