Average Error: 2.0 → 2.0
Time: 5.4s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{\frac{a}{\sqrt{1 + k \cdot \left(k + 10\right)}} \cdot {k}^{m}}{\sqrt{1 + k \cdot \left(k + 10\right)}}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{\frac{a}{\sqrt{1 + k \cdot \left(k + 10\right)}} \cdot {k}^{m}}{\sqrt{1 + k \cdot \left(k + 10\right)}}
(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))
↓
(FPCore (a k m)
 :precision binary64
 (/
  (* (/ a (sqrt (+ 1.0 (* k (+ k 10.0))))) (pow k m))
  (sqrt (+ 1.0 (* k (+ k 10.0))))))
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 / sqrt(1.0 + (k * (k + 10.0)))) * pow(k, m)) / sqrt(1.0 + (k * (k + 10.0)));
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 2.0

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

  2. Simplified1.9

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

  4. Applied add-sqr-sqrt_binary642.0

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

  5. Applied associate-/r*_binary642.0

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

  6. Simplified2.0

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

  7. Final simplification2.0

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

Reproduce

herbie shell --seed 2020253 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))