Average Error: 0.0 → 0.0

Time: 4.1s

Precision: binary64

Cost: 576

\[56789 \leq a \land a \leq 98765 \land 0 \leq b \land b \leq 1 \land 0 \leq c \land c \leq 0.0016773 \land 0 \leq d \land d \leq 0.0016773\]

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

↓

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

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

↓

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

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

↓

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

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

↓

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

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

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

Alternatives

Alternative 1
Error	0.8
Cost	26560

\[\left(\sqrt{a} \cdot \sqrt{d + \left(b + c\right)}\right) \cdot \left(\sqrt{a} \cdot \sqrt{d + \left(b + c\right)}\right)\]

Alternative 2
Error	1.5
Cost	20672

\[\sqrt[3]{a \cdot \left(d + \left(b + c\right)\right)} \cdot \left(\sqrt[3]{a \cdot \left(d + \left(b + c\right)\right)} \cdot \sqrt[3]{a \cdot \left(d + \left(b + c\right)\right)}\right)\]

Alternative 3
Error	1.3
Cost	19904

\[\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(\left(d + \left(b + c\right)\right) \cdot \sqrt[3]{a}\right)\]

Alternative 4
Error	0.5
Cost	13760

\[\sqrt{a \cdot \left(d + \left(b + c\right)\right)} \cdot \sqrt{a \cdot \left(d + \left(b + c\right)\right)}\]

Alternative 5
Error	0.5
Cost	13632

\[\sqrt{d + \left(b + c\right)} \cdot \left(a \cdot \sqrt{d + \left(b + c\right)}\right)\]

Alternative 6
Error	22.0
Cost	13504

\[\sqrt{d + \left(b + c\right)} \cdot \left(a \cdot \sqrt{d + c}\right)\]

Alternative 7
Error	0.6
Cost	13376

\[\sqrt{a} \cdot \left(\sqrt{a} \cdot \left(d + \left(b + c\right)\right)\right)\]

Alternative 8
Error	16.8
Cost	13312

\[\sqrt[3]{{\left(a \cdot \left(d + \left(b + c\right)\right)\right)}^{3}}\]

Alternative 9
Error	4.5
Cost	13248

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

Alternative 10
Error	40.7
Cost	13120

\[\sqrt{a} \cdot \left(b \cdot \sqrt{a}\right)\]

Alternative 11
Error	7.4
Cost	1216

\[\frac{a \cdot \left(\left(b + c\right) \cdot \left(b + c\right) - d \cdot d\right)}{\left(b + c\right) - d}\]

Alternative 12
Error	5.2
Cost	1088

\[a \cdot d + \frac{a \cdot \left(b \cdot b - c \cdot c\right)}{b - c}\]

Alternative 13
Error	0.0
Cost	704

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

Alternative 14
Error	0.0
Cost	576

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

Alternative 15
Error	20.7
Cost	448

\[a \cdot b + a \cdot d\]

Alternative 16
Error	22.1
Cost	448

\[a \cdot d + a \cdot c\]

Alternative 17
Error	0.0
Cost	448

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

Alternative 18
Error	20.7
Cost	320

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

Alternative 19
Error	22.1
Cost	320

\[a \cdot \left(d + c\right)\]

Alternative 20
Error	20.8
Cost	320

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

Alternative 21
Error	40.6
Cost	192

\[a \cdot b\]

Alternative 22
Error	41.8
Cost	192

\[a \cdot d\]

Alternative 23
Error	41.9
Cost	192

\[a \cdot c\]

Alternative 24
Error	59.2
Cost	64

\[1\]

Alternative 25
Error	61.5
Cost	64

\[0\]

Alternative 26
Error	62.8
Cost	64

\[-1\]

Error

Derivation

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

Reproduce

herbie shell --seed 2021042 
(FPCore (a b c d)
  :name "Expression, p14"
  :precision binary64
  :pre (and (<= 56789.0 a 98765.0) (<= 0.0 b 1.0) (<= 0.0 c 0.0016773) (<= 0.0 d 0.0016773))

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

  (* a (+ (+ b c) d)))