Expression, p14

Percentage Accurate: 99.9% → 100.0%

Time: 6.3s

Precision: binary64

Cost: 6848

• Unsound rule application detected in e-graph. Results may not be sound.

\[\left(\left(\left(56789 \leq a \land a \leq 98765\right) \land \left(0 \leq b \land b \leq 1\right)\right) \land \left(0 \leq c \land c \leq 0.0016773\right)\right) \land \left(0 \leq d \land d \leq 0.0016773\right)\]

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

Math

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

↓

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

FPCore

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

↓

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

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 fma((b + d), a, (a * c));
}

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	99.9%
Target	99.9%
Herbie	100.0%

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

Derivation

1. Initial program 100.0%

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

2. Simplified 99.9%

   \[\leadsto \color{blue}{a \cdot \left(c + \left(b + d\right)\right)} \]
   Step-by-step derivation

   [Start] 100.0	\[ a \cdot \left(\left(b + c\right) + d\right) \]
   +-commutative [=>] 100.0	\[ a \cdot \left(\color{blue}{\left(c + b\right)} + d\right) \]
   associate-+l+ [=>] 99.9	\[ a \cdot \color{blue}{\left(c + \left(b + d\right)\right)} \]

3. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b + d, a, a \cdot c\right)} \]
   Step-by-step derivation

   [Start] 99.9	\[ a \cdot \left(c + \left(b + d\right)\right) \]
   +-commutative [=>] 99.9	\[ a \cdot \color{blue}{\left(\left(b + d\right) + c\right)} \]
   distribute-rgt-in [=>] 99.9	\[ \color{blue}{\left(b + d\right) \cdot a + c \cdot a} \]
   fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(b + d, a, c \cdot a\right)} \]
   *-commutative [=>] 99.9	\[ \mathsf{fma}\left(b + d, a, \color{blue}{a \cdot c}\right) \]

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	448

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

Alternative 2
Accuracy	99.7%
Cost	320

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

Alternative 3
Accuracy	93.6%
Cost	192

\[d \cdot a \]

Reproduce

herbie shell --seed 2023161 
(FPCore (a b c d)
  :name "Expression, p14"
  :precision binary64
  :pre (and (and (and (and (<= 56789.0 a) (<= a 98765.0)) (and (<= 0.0 b) (<= b 1.0))) (and (<= 0.0 c) (<= c 0.0016773))) (and (<= 0.0 d) (<= d 0.0016773)))

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

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