Expression 4, p15

Average Error: 0.0 → 0.0

Time: 3.3s

Precision: binary64

Cost: 7104

\[\left(5 \leq a \land a \leq 10\right) \land \left(0 \leq b \land b \leq 0.001\right)\]

Math

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

↓

\[\mathsf{fma}\left(2, b \cdot a, b \cdot b\right) + a \cdot a \]

FPCore

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

↓

(FPCore (a b) :precision binary64 (+ (fma 2.0 (* b a) (* b b)) (* a a)))

C

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

↓

double code(double a, double b) {
	return fma(2.0, (b * a), (b * b)) + (a * a);
}

Julia

function code(a, b)
	return Float64(Float64(a + b) * Float64(a + b))
end

↓

function code(a, b)
	return Float64(fma(2.0, Float64(b * a), Float64(b * b)) + Float64(a * a))
end

Wolfram

code[a_, b_] := N[(N[(a + b), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_] := N[(N[(2.0 * N[(b * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

2. Taylor expanded in a around 0 0.0

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

3. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, b \cdot a, b \cdot b\right) + a \cdot a} \]
   Proof

   [Start] 0.0	\[ 2 \cdot \left(a \cdot b\right) + \left({b}^{2} + {a}^{2}\right) \]
   associate-+r+ [=>] 0.0	\[ \color{blue}{\left(2 \cdot \left(a \cdot b\right) + {b}^{2}\right) + {a}^{2}} \]
   unpow2 [=>] 0.0	\[ \left(2 \cdot \left(a \cdot b\right) + \color{blue}{b \cdot b}\right) + {a}^{2} \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(2, a \cdot b, b \cdot b\right)} + {a}^{2} \]
   *-commutative [=>] 0.0	\[ \mathsf{fma}\left(2, \color{blue}{b \cdot a}, b \cdot b\right) + {a}^{2} \]
   unpow2 [=>] 0.0	\[ \mathsf{fma}\left(2, b \cdot a, b \cdot b\right) + \color{blue}{a \cdot a} \]

4. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(2, b \cdot a, b \cdot b\right) + a \cdot a \]

Alternatives

Alternative 1
Error	0.0
Cost	704

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

Alternative 2
Error	0.0
Cost	704

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

Alternative 3
Error	60.6
Cost	448

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

Alternative 4
Error	0.0
Cost	448

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

Alternative 5
Error	61.4
Cost	192

\[b \cdot b \]

Reproduce

herbie shell --seed 2022354 
(FPCore (a b)
  :name "Expression 4, p15"
  :precision binary64
  :pre (and (and (<= 5.0 a) (<= a 10.0)) (and (<= 0.0 b) (<= b 0.001)))

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

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