Average Error: 0.0 → 0.0

Time: 3.1s

Precision: binary64

Cost: 704

\[5 \leq a \land a \leq 10 \land 0 \leq b \land b \leq 0.001\]

\[\left(a + b\right) \cdot \left(a + b\right)\]

↓

\[a \cdot \left(a + b\right) + b \cdot \left(a + b\right)\]

\left(a + b\right) \cdot \left(a + b\right)

↓

a \cdot \left(a + b\right) + b \cdot \left(a + b\right)

(FPCore (a b) :precision binary64 (* (+ a b) (+ a b)))

↓

(FPCore (a b) :precision binary64 (+ (* a (+ a b)) (* b (+ a b))))

double code(double a, double b) {
	return (a + b) * (a + b);
}

↓

double code(double a, double b) {
	return (a * (a + b)) + (b * (a + b));
}

Error

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

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	0.0
Target	0.0
Herbie	0.0

\[\left(\left(b \cdot a + b \cdot b\right) + b \cdot a\right) + a \cdot a\]

Alternatives

Alternative 1
Error	0.8
Cost	27584

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

Alternative 2
Error	0.3
Cost	14528

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

Alternative 3
Error	1.0
Cost	14080

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

Alternative 4
Error	0.9
Cost	13120

\[e^{2 \cdot \log \left(a + b\right)}\]

Alternative 5
Error	1.0
Cost	13056

\[{\left(\sqrt{a + b}\right)}^{4}\]

Alternative 6
Error	0.0
Cost	704

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

Alternative 7
Error	0.2
Cost	448

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

Alternative 8
Error	0.0
Cost	448

\[\left(a + b\right) \cdot \left(a + b\right)\]

Alternative 9
Error	0.8
Cost	192

\[a \cdot a\]

Alternative 10
Error	54.5
Cost	64

\[1\]

Alternative 11
Error	62.0
Cost	64

\[0\]

Alternative 12
Error	63.0
Cost	64

\[-1\]

Error

Derivation

Initial program 0.0
\[\left(a + b\right) \cdot \left(a + b\right)\]
Using strategy rm
Applied distribute-rgt-in_binary64_13530.0
\[\leadsto \color{blue}{a \cdot \left(a + b\right) + b \cdot \left(a + b\right)}\]
Simplified0.0
\[\leadsto \color{blue}{a \cdot \left(a + b\right) + b \cdot \left(a + b\right)}\]
Final simplification0.0
\[\leadsto a \cdot \left(a + b\right) + b \cdot \left(a + b\right)\]

Reproduce

herbie shell --seed 2021042 
(FPCore (a b)
  :name "Expression 4, p15"
  :precision binary64
  :pre (and (<= 5.0 a 10.0) (<= 0.0 b 0.001))

  :herbie-target
  (+ (+ (+ (* b a) (* b b)) (* b a)) (* a a))

  (* (+ a b) (+ a b)))