Expression, p14

Average Accuracy: 99.9% → 100.0%

Time: 6.9s

Precision: binary64

Cost: 6848

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

Math

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

↓

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

FPCore

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

code[a_, b_, c_, d_] := N[(d * a + N[(a * N[(c + b), $MachinePrecision]), $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 99.9%

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

2. Taylor expanded in d around 0 99.9%

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

3. Applied egg-rr 100.0%

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

   [Start] 99.9	\[ a \cdot \left(c + b\right) + a \cdot d \]
   +-commutative [=>] 99.9	\[ \color{blue}{a \cdot d + a \cdot \left(c + b\right)} \]
   *-commutative [=>] 99.9	\[ \color{blue}{d \cdot a} + a \cdot \left(c + b\right) \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(d, a, a \cdot \left(c + b\right)\right)} \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	51.7%
Cost	1380

\[\begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{-262}:\\ \;\;\;\;a \cdot c\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-251}:\\ \;\;\;\;d \cdot a\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-244}:\\ \;\;\;\;a \cdot c\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-199}:\\ \;\;\;\;d \cdot a\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-184}:\\ \;\;\;\;a \cdot c\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-151}:\\ \;\;\;\;d \cdot a\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-130}:\\ \;\;\;\;a \cdot c\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-104}:\\ \;\;\;\;d \cdot a\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-84}:\\ \;\;\;\;a \cdot c\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 2
Accuracy	78.4%
Cost	982

\[\begin{array}{l} \mathbf{if}\;d \leq 7 \cdot 10^{-147} \lor \neg \left(d \leq 8.5 \cdot 10^{-142}\right) \land \left(d \leq 1.5 \cdot 10^{-130} \lor \neg \left(d \leq 1.65 \cdot 10^{-119}\right) \land d \leq 2.4 \cdot 10^{-108}\right):\\ \;\;\;\;a \cdot \left(c + b\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot a\\ \end{array} \]

Alternative 3
Accuracy	51.6%
Cost	589

\[\begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-130} \lor \neg \left(b \leq 6.5 \cdot 10^{-104}\right) \land b \leq 2.4 \cdot 10^{-84}:\\ \;\;\;\;a \cdot c\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 4
Accuracy	86.0%
Cost	584

\[\begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-130}:\\ \;\;\;\;a \cdot \left(d + c\right)\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-106}:\\ \;\;\;\;a \cdot \left(d + b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(c + b\right)\\ \end{array} \]

Alternative 5
Accuracy	99.9%
Cost	576

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

Alternative 6
Accuracy	85.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{-108}:\\ \;\;\;\;a \cdot \left(d + c\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(c + b\right)\\ \end{array} \]

Alternative 7
Accuracy	99.9%
Cost	448

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

Alternative 8
Accuracy	34.9%
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023143 
(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)))