# ?

Average Error: 1.7 → 0.3
Time: 8.9s
Precision: binary64
Cost: 6980

# ?

$\left(\left(\left(\left(0 \leq x \land x \leq 1.79 \cdot 10^{+308}\right) \land \left(0 \leq y \land y \leq 1.79 \cdot 10^{+308}\right)\right) \land \left(-1 \leq a \land a \leq 1\right)\right) \land \left(-1.79 \cdot 10^{+308} \leq b \land b \leq 1.79 \cdot 10^{+308}\right)\right) \land \left(-1.79 \cdot 10^{+308} \leq c \land c \leq 1.79 \cdot 10^{+308}\right)$
$\begin{array}{c}[a, b, c] = \mathsf{sort}([a, b, c])\\ \end{array}$
$\left(x + y\right) + \left(a \cdot b\right) \cdot c$
$\begin{array}{l} \mathbf{if}\;b \leq 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot c, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a \cdot c, x\right)\\ \end{array}$
(FPCore (x y a b c) :precision binary64 (+ (+ x y) (* (* a b) c)))
(FPCore (x y a b c)
:precision binary64
(if (<= b 1e-21) (fma a (* b c) (+ x y)) (+ y (fma b (* a c) x))))
double code(double x, double y, double a, double b, double c) {
return (x + y) + ((a * b) * c);
}

double code(double x, double y, double a, double b, double c) {
double tmp;
if (b <= 1e-21) {
tmp = fma(a, (b * c), (x + y));
} else {
tmp = y + fma(b, (a * c), x);
}
return tmp;
}

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

function code(x, y, a, b, c)
tmp = 0.0
if (b <= 1e-21)
tmp = fma(a, Float64(b * c), Float64(x + y));
else
tmp = Float64(y + fma(b, Float64(a * c), x));
end
return tmp
end

code[x_, y_, a_, b_, c_] := N[(N[(x + y), $MachinePrecision] + N[(N[(a * b),$MachinePrecision] * c), $MachinePrecision]),$MachinePrecision]

code[x_, y_, a_, b_, c_] := If[LessEqual[b, 1e-21], N[(a * N[(b * c), $MachinePrecision] + N[(x + y),$MachinePrecision]), $MachinePrecision], N[(y + N[(b * N[(a * c),$MachinePrecision] + x), $MachinePrecision]),$MachinePrecision]]

\left(x + y\right) + \left(a \cdot b\right) \cdot c

\begin{array}{l}
\mathbf{if}\;b \leq 10^{-21}:\\
\;\;\;\;\mathsf{fma}\left(a, b \cdot c, x + y\right)\\

\mathbf{else}:\\
\;\;\;\;y + \mathsf{fma}\left(b, a \cdot c, x\right)\\

\end{array}


# Derivation?

1. Split input into 2 regimes
2. ## if b < 9.99999999999999908e-22

1. Initial program 1.9

$\left(x + y\right) + \left(a \cdot b\right) \cdot c$
2. Simplified0.3

$\leadsto \color{blue}{\mathsf{fma}\left(a, b \cdot c, x + y\right)}$
Proof
[Start]1.9 $\left(x + y\right) + \left(a \cdot b\right) \cdot c$ $\color{blue}{\left(a \cdot b\right) \cdot c + \left(x + y\right)}$ $\color{blue}{a \cdot \left(b \cdot c\right)} + \left(x + y\right)$ $\color{blue}{\mathsf{fma}\left(a, b \cdot c, x + y\right)}$

## if 9.99999999999999908e-22 < b

1. Initial program 0.1

$\left(x + y\right) + \left(a \cdot b\right) \cdot c$
2. Simplified0.3

$\leadsto \color{blue}{y + \mathsf{fma}\left(b, a \cdot c, x\right)}$
Proof
[Start]0.1 $\left(x + y\right) + \left(a \cdot b\right) \cdot c$ $\color{blue}{\left(a \cdot b\right) \cdot c + \left(x + y\right)}$ $\color{blue}{\left(\left(a \cdot b\right) \cdot c + x\right) + y}$ $\color{blue}{y + \left(\left(a \cdot b\right) \cdot c + x\right)}$ $y + \left(\color{blue}{c \cdot \left(a \cdot b\right)} + x\right)$ $y + \left(\color{blue}{\left(c \cdot a\right) \cdot b} + x\right)$ $y + \left(\color{blue}{b \cdot \left(c \cdot a\right)} + x\right)$ $y + \color{blue}{\mathsf{fma}\left(b, c \cdot a, x\right)}$ $y + \mathsf{fma}\left(b, \color{blue}{a \cdot c}, x\right)$
3. Recombined 2 regimes into one program.
4. Final simplification0.3

$\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot c, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a \cdot c, x\right)\\ \end{array}$

# Alternatives

Alternative 1
Error0.3
Cost6980
$\begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\left(x + y\right) + a \cdot \left(b \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(b, a \cdot c, x\right)\\ \end{array}$
Alternative 2
Error0.8
Cost708
$\begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{+48}:\\ \;\;\;\;\left(x + y\right) + a \cdot \left(b \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;y + c \cdot \left(b \cdot a\right)\\ \end{array}$
Alternative 3
Error12.1
Cost452
$\begin{array}{l} \mathbf{if}\;b \leq 1200000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot c\right)\\ \end{array}$
Alternative 4
Error11.3
Cost452
$\begin{array}{l} \mathbf{if}\;b \leq 0.14:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a \cdot c\right)\\ \end{array}$
Alternative 5
Error26.9
Cost448
$y + c \cdot \left(b \cdot a\right)$
Alternative 6
Error11.1
Cost192
$x + y$
Alternative 7
Error37.0
Cost64
$x$
Alternative 8
Error36.2
Cost64
$y$

# Reproduce?

herbie shell --seed 1
(FPCore (x y a b c)
:name "x + y + a * b * c"
:precision binary64
:pre (and (and (and (and (and (<= 0.0 x) (<= x 1.79e+308)) (and (<= 0.0 y) (<= y 1.79e+308))) (and (<= -1.0 a) (<= a 1.0))) (and (<= -1.79e+308 b) (<= b 1.79e+308))) (and (<= -1.79e+308 c) (<= c 1.79e+308)))
(+ (+ x y) (* (* a b) c)))