Expression 4, p15

Average Accuracy: 100.0% → 100.0%

Time: 3.1s

Precision: binary64

Cost: 6976

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

\[ \begin{array}{c}[a, b] = \mathsf{sort}([a, b])\\ \end{array} \]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	100.0%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 100.0%

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

2. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \left(a + b\right) \cdot \left(a + b\right) \]
   distribute-rgt-in [=>] 100.0	\[ \color{blue}{a \cdot \left(a + b\right) + b \cdot \left(a + b\right)} \]
   +-commutative [=>] 100.0	\[ \color{blue}{b \cdot \left(a + b\right) + a \cdot \left(a + b\right)} \]

3. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ b \cdot \left(a + b\right) + a \cdot \left(a + b\right) \]
   *-commutative [=>] 100.0	\[ \color{blue}{\left(a + b\right) \cdot b} + a \cdot \left(a + b\right) \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(a + b, b, a \cdot \left(a + b\right)\right)} \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	704

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

Alternative 2
Accuracy	99.7%
Cost	448

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

Alternative 3
Accuracy	100.0%
Cost	448

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

Alternative 4
Accuracy	98.6%
Cost	192

\[b \cdot b \]

Reproduce

herbie shell --seed 2023138 
(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)))