Average Error: 0.0 → 0.2
Time: 2.3s
Precision: binary64
\[\left(\left(\left(56789 \leq a \land a \leq 98765\right) \land \left(0 \leq b \land b \leq 1\right)\right) \land \left(0 \leq c \land c \leq 0.0016773\right)\right) \land \left(0 \leq d \land d \leq 0.0016773\right)\]
\[ \begin{array}{c}[b, c, d] = \mathsf{sort}([b, c, d])\\ \end{array} \]
\[a \cdot \left(\left(b + c\right) + d\right) \]
↓
\[\mathsf{fma}\left(a, d, a \cdot c\right) \]
(FPCore (a b c d) :precision binary64 (* a (+ (+ b c) d)))
↓
(FPCore (a b c d) :precision binary64 (fma a d (* a 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(a, d, (a * c));
}

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

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

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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original0.0
Target0.0
Herbie0.2
\[a \cdot b + a \cdot \left(c + d\right) \]

Derivation

  1. Initial program 0.0

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

  2. Applied egg-rr0.0

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

  3. Taylor expanded in b around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022159 
(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)))