Average Error: 2.0 → 2.0
Time: 9.8s
Precision: binary64
Cost: 7168
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\]
\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\frac{a}{\frac{1 + k \cdot \left(k + 10\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 (/ (+ 1.0 (* k (+ k 10.0))) (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 / ((1.0 + (k * (k + 10.0))) / pow(k, m));
}

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

Alternatives

Alternative 1
Error12.0
Cost40448
\[\sqrt[3]{\frac{a \cdot {k}^{m}}{1 + k \cdot 10}} \cdot \left(\sqrt[3]{\frac{a \cdot {k}^{m}}{1 + k \cdot 10}} \cdot \sqrt[3]{\frac{a \cdot {k}^{m}}{1 + k \cdot 10}}\right)\]
Alternative 2
Error13.6
Cost27200
\[\sqrt{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}} \cdot \sqrt{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}}\]
Alternative 3
Error2.1
Cost27136
\[\frac{a \cdot {k}^{m}}{1 + \sqrt[3]{k \cdot \left(k + 10\right)} \cdot \left(\sqrt[3]{k \cdot \left(k + 10\right)} \cdot \sqrt[3]{k \cdot \left(k + 10\right)}\right)}\]
Alternative 4
Error2.1
Cost26624
\[\frac{a \cdot {k}^{m}}{1 + \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right) \cdot \left(\left(k + 10\right) \cdot \sqrt[3]{k}\right)}\]
Alternative 5
Error2.1
Cost20608
\[a \cdot \left(\frac{1}{\sqrt{1 + k \cdot \left(k + 10\right)}} \cdot \frac{{k}^{m}}{\sqrt{1 + k \cdot \left(k + 10\right)}}\right)\]
Alternative 6
Error2.0
Cost20480
\[a \cdot \frac{\frac{{k}^{m}}{\sqrt{1 + k \cdot \left(k + 10\right)}}}{\sqrt{1 + k \cdot \left(k + 10\right)}}\]
Alternative 7
Error2.0
Cost20480
\[\frac{{k}^{m}}{\sqrt{1 + k \cdot \left(k + 10\right)}} \cdot \frac{a}{\sqrt{1 + k \cdot \left(k + 10\right)}}\]
Alternative 8
Error10.1
Cost19968
\[\frac{a \cdot {k}^{m}}{1 + e^{\log \left(k \cdot \left(k + 10\right)\right)}}\]
Alternative 9
Error24.9
Cost14912
\[a \cdot \left(\frac{{k}^{m}}{1 + {\left(k \cdot \left(k + 10\right)\right)}^{3}} \cdot \left(1 + \left(\left(k \cdot \left(k + 10\right)\right) \cdot \left(k \cdot \left(k + 10\right)\right) - k \cdot \left(k + 10\right)\right)\right)\right)\]
Alternative 10
Error18.4
Cost13696
\[a \cdot \frac{\frac{{k}^{m}}{\sqrt{1 + k \cdot \left(k + 10\right)}}}{k}\]
Alternative 11
Error2.1
Cost7296
\[\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a \cdot {k}^{m}}}\]
Alternative 12
Error2.0
Cost7296
\[\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\]
Alternative 13
Error2.0
Cost7168
\[\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\]
Alternative 14
Error2.0
Cost7168
\[a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\]
Alternative 15
Error11.6
Cost7040
\[\frac{a}{\frac{1 + k \cdot 10}{{k}^{m}}}\]
Alternative 16
Error11.6
Cost7040
\[\frac{a \cdot {k}^{m}}{1 + k \cdot 10}\]
Alternative 17
Error2.8
Cost7040
\[\frac{a \cdot {k}^{m}}{1 + k \cdot k}\]
Alternative 18
Error26.0
Cost6912
\[a \cdot \frac{{k}^{m}}{k \cdot k}\]
Alternative 19
Error26.0
Cost6912
\[\frac{a \cdot {k}^{m}}{k \cdot k}\]
Alternative 20
Error15.8
Cost6656
\[a \cdot {k}^{m}\]
Alternative 21
Error23.2
Cost704
\[a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\]
Alternative 22
Error23.2
Cost576
\[\frac{a}{1 + k \cdot \left(k + 10\right)}\]
Alternative 23
Error38.3
Cost448
\[\frac{a}{1 + k \cdot 10}\]
Alternative 24
Error61.9
Cost64
\[1\]
Alternative 25
Error23.7
Cost64
\[0\]
Alternative 26
Error61.9
Cost64
\[-1\]

Error

Derivation

  1. Initial program 2.0

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

  2. Simplified2.0

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

  4. Applied associate-/l*_binary64_23472.0

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

  5. Simplified2.0

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

  6. Final simplification2.0

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

Reproduce

herbie shell --seed 2021042 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))