Average Error: 0.2 → 0.0

Time: 5.5s

Precision: binary64

Cost: 14272

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

↓

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

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

↓

\left(\left(b \cdot b\right) \cdot 4 + \left(\left(a \cdot a\right) \cdot \left(b \cdot \left(b \cdot 2\right)\right) + \left({b}^{4} + {a}^{4}\right)\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
 (-
  (+
   (* (* b b) 4.0)
   (+ (* (* a a) (* b (* b 2.0))) (+ (pow b 4.0) (pow a 4.0))))
  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 (((b * b) * 4.0) + (((a * a) * (b * (b * 2.0))) + (pow(b, 4.0) + pow(a, 4.0)))) - 1.0;
}

Error

The X axis uses an exponential scale — Bits error versus `a`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	0.5
Cost	40576

\[\left(\left(b \cdot b\right) \cdot 4 + \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \left(\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}\right)\right) - 1\]

Alternative 2
Error	42.2
Cost	34816

\[\frac{{\left(a \cdot a + b \cdot b\right)}^{6} + 64 \cdot {b}^{6}}{{\left(a \cdot a + b \cdot b\right)}^{4} + \left({b}^{4} \cdot 16 - \left(\left(b \cdot b\right) \cdot 4\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{2}\right)} - 1\]

Alternative 3
Error	0.2
Cost	27584

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

Alternative 4
Error	0.7
Cost	27200

\[\left(\left(b \cdot b\right) \cdot 4 + {\left(\sqrt[3]{a \cdot a + b \cdot b}\right)}^{4} \cdot {\left(\sqrt[3]{a \cdot a + b \cdot b}\right)}^{2}\right) - 1\]

Alternative 5
Error	26.7
Cost	21120

\[\frac{{\left(a \cdot a + b \cdot b\right)}^{4} - {b}^{4} \cdot 16}{{\left(a \cdot a + b \cdot b\right)}^{2} - \left(b \cdot b\right) \cdot 4} - 1\]

Alternative 6
Error	16.1
Cost	20288

\[\sqrt[3]{{\left(\left(b \cdot b\right) \cdot 4 + {\left(a \cdot a + b \cdot b\right)}^{2}\right)}^{3}} - 1\]

Alternative 7
Error	16.1
Cost	13824

\[\left(\left(b \cdot b\right) \cdot 4 + \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{6}}\right) - 1\]

Alternative 8
Error	12.0
Cost	7424

\[b \cdot \left({b}^{3} + b \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\right) - 1\]

Alternative 9
Error	0.2
Cost	7424

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

Alternative 10
Error	10.9
Cost	7040

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

Alternative 11
Error	12.2
Cost	7040

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

Alternative 12
Error	13.1
Cost	6656

\[{b}^{4} - 1\]

Alternative 13
Error	11.7
Cost	6656

\[{a}^{4} - 1\]

Alternative 14
Error	0.2
Cost	1472

\[\left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + \left(b \cdot b\right) \cdot 4\right) - 1\]

Alternative 15
Error	12.1
Cost	1088

\[b \cdot \left(b \cdot \left(b \cdot b + \left(4 + \left(a \cdot a\right) \cdot 2\right)\right)\right) - 1\]

Alternative 16
Error	12.1
Cost	1088

\[\left(b \cdot b\right) \cdot \left(4 + \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\right) - 1\]

Alternative 17
Error	22.6
Cost	832

\[b \cdot \left(b \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\right) - 1\]

Alternative 18
Error	23.4
Cost	704

\[b \cdot \left(b \cdot \left(\left(a \cdot a\right) \cdot 2\right)\right) - 1\]

Alternative 19
Error	62.1
Cost	64

\[1\]

Alternative 20
Error	62.2
Cost	64

\[0\]

Alternative 21
Error	23.7
Cost	64

\[-1\]

Error

Derivation

Initial program 0.2
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
Taylor expanded around 0 0.0
\[\leadsto \left(\color{blue}{\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({b}^{4} + {a}^{4}\right)\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
Simplified0.0
\[\leadsto \left(\color{blue}{\left(\left(a \cdot a\right) \cdot \left(b \cdot \left(b \cdot 2\right)\right) + \left({b}^{4} + {a}^{4}\right)\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
Simplified0.0
\[\leadsto \color{blue}{\left(\left(b \cdot b\right) \cdot 4 + \left(\left(a \cdot a\right) \cdot \left(b \cdot \left(b \cdot 2\right)\right) + \left({b}^{4} + {a}^{4}\right)\right)\right) - 1}\]
Final simplification0.0
\[\leadsto \left(\left(b \cdot b\right) \cdot 4 + \left(\left(a \cdot a\right) \cdot \left(b \cdot \left(b \cdot 2\right)\right) + \left({b}^{4} + {a}^{4}\right)\right)\right) - 1\]

Reproduce

herbie shell --seed 2021042 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))