Average Error: 0.2 → 0.0
Time: 3.0s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\left(2 \cdot {\left(a \cdot b\right)}^{2} + \left({b}^{4} + {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(\left(2 \cdot {\left(a \cdot b\right)}^{2} + \left({b}^{4} + {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1
(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))
↓
(FPCore (a b)
 :precision binary64
 (-
  (+ (+ (* 2.0 (pow (* a b) 2.0)) (+ (pow b 4.0) (pow a 4.0))) (* 4.0 (* b b)))
  1.0))
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 (((2.0 * pow((a * b), 2.0)) + (pow(b, 4.0) + pow(a, 4.0))) + (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. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  5. Applied add-exp-log_binary64_21620.0

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

  6. Applied add-exp-log_binary64_216231.5

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

  7. Applied prod-exp_binary64_217331.5

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

  8. Applied add-exp-log_binary64_216231.5

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

  9. Applied prod-exp_binary64_217331.5

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

  10. Applied add-exp-log_binary64_216248.0

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

  11. Applied add-exp-log_binary64_216248.0

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

  12. Applied prod-exp_binary64_217348.0

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

  13. Applied prod-exp_binary64_217348.0

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

  14. Simplified22.8

    \[\leadsto \left(\left(e^{\color{blue}{\log 2 + 2 \cdot \log \left(a \cdot b\right)}} + \left({b}^{4} + {a}^{4}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\]
  15. Using strategy rm

  16. Applied exp-sum_binary64_217022.8

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

  17. Simplified22.8

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

  18. Simplified0.0

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

  19. Final simplification0.0

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

Reproduce

herbie shell --seed 2020343 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))