Expression, p6

Percentage Accurate: 94.3% → 100.0%

Time: 8.8s

Precision: binary64

Cost: 6976

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

\[\left(\left(\left(-14 \leq a \land a \leq -13\right) \land \left(-3 \leq b \land b \leq -2\right)\right) \land \left(3 \leq c \land c \leq 3.5\right)\right) \land \left(12.5 \leq d \land d \leq 13.5\right)\]

Math

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

↓

\[\mathsf{fma}\left(1, b + c, d + a\right) \cdot 2 \]

FPCore

(FPCore (a b c d) :precision binary64 (* (+ a (+ b (+ c d))) 2.0))

↓

(FPCore (a b c d) :precision binary64 (* (fma 1.0 (+ b c) (+ d a)) 2.0))

C

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

↓

double code(double a, double b, double c, double d) {
	return fma(1.0, (b + c), (d + a)) * 2.0;
}

Julia

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

↓

function code(a, b, c, d)
	return Float64(fma(1.0, Float64(b + c), Float64(d + a)) * 2.0)
end

Wolfram

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

↓

code[a_, b_, c_, d_] := N[(N[(1.0 * N[(b + c), $MachinePrecision] + N[(d + a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Target

Original	94.3%
Target	94.0%
Herbie	100.0%

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

Derivation

1. Initial program 94.2%

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

2. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1, b + c, d + a\right)} \cdot 2 \]
   Step-by-step derivation

   [Start] 94.2%	\[ \left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
   +-commutative [=>] 94.2%	\[ \color{blue}{\left(\left(b + \left(c + d\right)\right) + a\right)} \cdot 2 \]
   associate-+r+ [=>] 95.8%	\[ \left(\color{blue}{\left(\left(b + c\right) + d\right)} + a\right) \cdot 2 \]
   associate-+l+ [=>] 100.0%	\[ \color{blue}{\left(\left(b + c\right) + \left(d + a\right)\right)} \cdot 2 \]
   *-un-lft-identity [=>] 100.0%	\[ \left(\color{blue}{1 \cdot \left(b + c\right)} + \left(d + a\right)\right) \cdot 2 \]
   fma-def [=>] 100.0%	\[ \color{blue}{\mathsf{fma}\left(1, b + c, d + a\right)} \cdot 2 \]

3. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(1, b + c, d + a\right) \cdot 2 \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6976

\[\mathsf{fma}\left(1, b + c, d + a\right) \cdot 2 \]

Alternative 2
Accuracy	14.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;c + d \leq 15.92:\\ \;\;\;\;\left(d + a\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(b + c\right) \cdot 2\\ \end{array} \]

Alternative 3
Accuracy	94.3%
Cost	576

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

Alternative 4
Accuracy	95.7%
Cost	576

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

Alternative 5
Accuracy	100.0%
Cost	576

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

Alternative 6
Accuracy	14.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq -13.8:\\ \;\;\;\;b \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(b + c\right) \cdot 2\\ \end{array} \]

Alternative 7
Accuracy	12.3%
Cost	324

\[\begin{array}{l} \mathbf{if}\;a \leq -13.69:\\ \;\;\;\;b \cdot 2\\ \mathbf{else}:\\ \;\;\;\;c \cdot 2\\ \end{array} \]

Alternative 8
Accuracy	6.3%
Cost	192

\[b \cdot 2 \]

Reproduce

herbie shell --seed 2023272 
(FPCore (a b c d)
  :name "Expression, p6"
  :precision binary64
  :pre (and (and (and (and (<= -14.0 a) (<= a -13.0)) (and (<= -3.0 b) (<= b -2.0))) (and (<= 3.0 c) (<= c 3.5))) (and (<= 12.5 d) (<= d 13.5)))

  :herbie-target
  (+ (* (+ a b) 2.0) (* (+ c d) 2.0))

  (* (+ a (+ b (+ c d))) 2.0))