Average Error: 0.2 → 0.2
Time: 5.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{4}}{2}\right)} \cdot {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{4}}{2}\right)}\right) \cdot {\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}^{3} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{4}}{2}\right)} \cdot {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{4}}{2}\right)}\right) \cdot {\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}^{3} + 4 \cdot \left(b \cdot b\right)\right) - 1
double code(double a, double b) {
	return ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0);
}
↓
double code(double a, double b) {
	return ((((pow(fma(a, a, (b * b)), ((2.0 / 4.0) / 2.0)) * pow(fma(a, a, (b * b)), ((2.0 / 4.0) / 2.0))) * pow(pow(fma(a, a, (b * b)), ((2.0 / 2.0) / 2.0)), 3.0)) + (4.0 * (b * b))) - 1.0);
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
  2. Using strategy rm

  3. Applied sqr-pow0.2

    \[\leadsto \left(\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)}} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  4. Simplified0.2

    \[\leadsto \left(\color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{2}{2}\right)}} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{2}{2}\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  5. Simplified0.2

    \[\leadsto \left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{2}{2}\right)}} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
  6. Using strategy rm

  7. Applied sqr-pow0.2

    \[\leadsto \left(\color{blue}{\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)} \cdot {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{2}{2}\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  8. Applied associate-*l*0.1

    \[\leadsto \left(\color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot \left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{2}{2}\right)}\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  9. Simplified0.1

    \[\leadsto \left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}^{3}} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
  10. Using strategy rm

  11. Applied sqr-pow0.2

    \[\leadsto \left(\color{blue}{\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{\frac{2}{2}}{2}}{2}\right)} \cdot {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{\frac{2}{2}}{2}}{2}\right)}\right)} \cdot {\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}^{3} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  12. Simplified0.2

    \[\leadsto \left(\left(\color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{4}}{2}\right)}} \cdot {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{\frac{2}{2}}{2}}{2}\right)}\right) \cdot {\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}^{3} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  13. Simplified0.2

    \[\leadsto \left(\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{4}}{2}\right)} \cdot \color{blue}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{4}}{2}\right)}}\right) \cdot {\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}^{3} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  14. Final simplification0.2

    \[\leadsto \left(\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{4}}{2}\right)} \cdot {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{4}}{2}\right)}\right) \cdot {\left({\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}^{3} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))