Average Error: 2.0 → 2.0
Time: 7.8s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\frac{a}{\frac{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}{{k}^{m}}} \]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{a}{\frac{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}{{k}^{m}}}

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))
↓
(FPCore (a k m)
 :precision binary64
 (/ a (/ (fma k 10.0 (fma k k 1.0)) (pow k m))))
double code(double a, double k, double m) {
	return (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}
↓
double code(double a, double k, double m) {
	return a / (fma(k, 10.0, fma(k, k, 1.0)) / pow(k, m));
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Initial program 2.0

    \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

  2. Applied associate-/l*_binary642.0

    \[\leadsto \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}} \]

  3. Simplified2.0

    \[\leadsto \frac{a}{\color{blue}{\frac{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}{{k}^{m}}}} \]

  4. Final simplification2.0

    \[\leadsto \frac{a}{\frac{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}{{k}^{m}}} \]

Reproduce

herbie shell --seed 2022088 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))