
(FPCore (x y) :precision binary64 (- (+ x y) (* x y)))
double code(double x, double y) {
return (x + y) - (x * y);
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
code = (x + y) - (x * y)
end function
public static double code(double x, double y) {
return (x + y) - (x * y);
}
def code(x, y): return (x + y) - (x * y)
function code(x, y) return Float64(Float64(x + y) - Float64(x * y)) end
function tmp = code(x, y) tmp = (x + y) - (x * y); end
code[x_, y_] := N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\left(x + y\right) - x \cdot y
\end{array}
Sampling outcomes in binary64 precision:
Herbie found 7 alternatives:
| Alternative | Accuracy | Speedup |
|---|
(FPCore (x y) :precision binary64 (- (+ x y) (* x y)))
double code(double x, double y) {
return (x + y) - (x * y);
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
code = (x + y) - (x * y)
end function
public static double code(double x, double y) {
return (x + y) - (x * y);
}
def code(x, y): return (x + y) - (x * y)
function code(x, y) return Float64(Float64(x + y) - Float64(x * y)) end
function tmp = code(x, y) tmp = (x + y) - (x * y); end
code[x_, y_] := N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\left(x + y\right) - x \cdot y
\end{array}
NOTE: x and y should be sorted in increasing order before calling this function. (FPCore (x y) :precision binary64 (fma x (- 1.0 y) y))
assert(x < y);
double code(double x, double y) {
return fma(x, (1.0 - y), y);
}
x, y = sort([x, y]) function code(x, y) return fma(x, Float64(1.0 - y), y) end
NOTE: x and y should be sorted in increasing order before calling this function. code[x_, y_] := N[(x * N[(1.0 - y), $MachinePrecision] + y), $MachinePrecision]
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\mathsf{fma}\left(x, 1 - y, y\right)
\end{array}
Initial program 100.0%
+-commutative100.0%
associate-+r-100.0%
*-commutative100.0%
+-commutative100.0%
*-lft-identity100.0%
metadata-eval100.0%
distribute-rgt-out--100.0%
fma-def100.0%
metadata-eval100.0%
Simplified100.0%
Final simplification100.0%
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
:precision binary64
(let* ((t_0 (* x (- y))))
(if (<= x -1.3e+201)
t_0
(if (<= x -8e+98)
x
(if (<= x -3.2e+79)
t_0
(if (<= x -2.5e-95) x (if (<= x 1.0) y t_0)))))))assert(x < y);
double code(double x, double y) {
double t_0 = x * -y;
double tmp;
if (x <= -1.3e+201) {
tmp = t_0;
} else if (x <= -8e+98) {
tmp = x;
} else if (x <= -3.2e+79) {
tmp = t_0;
} else if (x <= -2.5e-95) {
tmp = x;
} else if (x <= 1.0) {
tmp = y;
} else {
tmp = t_0;
}
return tmp;
}
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: t_0
real(8) :: tmp
t_0 = x * -y
if (x <= (-1.3d+201)) then
tmp = t_0
else if (x <= (-8d+98)) then
tmp = x
else if (x <= (-3.2d+79)) then
tmp = t_0
else if (x <= (-2.5d-95)) then
tmp = x
else if (x <= 1.0d0) then
tmp = y
else
tmp = t_0
end if
code = tmp
end function
assert x < y;
public static double code(double x, double y) {
double t_0 = x * -y;
double tmp;
if (x <= -1.3e+201) {
tmp = t_0;
} else if (x <= -8e+98) {
tmp = x;
} else if (x <= -3.2e+79) {
tmp = t_0;
} else if (x <= -2.5e-95) {
tmp = x;
} else if (x <= 1.0) {
tmp = y;
} else {
tmp = t_0;
}
return tmp;
}
[x, y] = sort([x, y]) def code(x, y): t_0 = x * -y tmp = 0 if x <= -1.3e+201: tmp = t_0 elif x <= -8e+98: tmp = x elif x <= -3.2e+79: tmp = t_0 elif x <= -2.5e-95: tmp = x elif x <= 1.0: tmp = y else: tmp = t_0 return tmp
x, y = sort([x, y]) function code(x, y) t_0 = Float64(x * Float64(-y)) tmp = 0.0 if (x <= -1.3e+201) tmp = t_0; elseif (x <= -8e+98) tmp = x; elseif (x <= -3.2e+79) tmp = t_0; elseif (x <= -2.5e-95) tmp = x; elseif (x <= 1.0) tmp = y; else tmp = t_0; end return tmp end
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
t_0 = x * -y;
tmp = 0.0;
if (x <= -1.3e+201)
tmp = t_0;
elseif (x <= -8e+98)
tmp = x;
elseif (x <= -3.2e+79)
tmp = t_0;
elseif (x <= -2.5e-95)
tmp = x;
elseif (x <= 1.0)
tmp = y;
else
tmp = t_0;
end
tmp_2 = tmp;
end
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x * (-y)), $MachinePrecision]}, If[LessEqual[x, -1.3e+201], t$95$0, If[LessEqual[x, -8e+98], x, If[LessEqual[x, -3.2e+79], t$95$0, If[LessEqual[x, -2.5e-95], x, If[LessEqual[x, 1.0], y, t$95$0]]]]]]
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\begin{array}{l}
t_0 := x \cdot \left(-y\right)\\
\mathbf{if}\;x \leq -1.3 \cdot 10^{+201}:\\
\;\;\;\;t_0\\
\mathbf{elif}\;x \leq -8 \cdot 10^{+98}:\\
\;\;\;\;x\\
\mathbf{elif}\;x \leq -3.2 \cdot 10^{+79}:\\
\;\;\;\;t_0\\
\mathbf{elif}\;x \leq -2.5 \cdot 10^{-95}:\\
\;\;\;\;x\\
\mathbf{elif}\;x \leq 1:\\
\;\;\;\;y\\
\mathbf{else}:\\
\;\;\;\;t_0\\
\end{array}
\end{array}
if x < -1.29999999999999993e201 or -7.99999999999999998e98 < x < -3.20000000000000003e79 or 1 < x Initial program 100.0%
Taylor expanded in x around inf 98.8%
Taylor expanded in y around inf 59.9%
mul-1-neg59.9%
distribute-rgt-neg-out59.9%
Simplified59.9%
if -1.29999999999999993e201 < x < -7.99999999999999998e98 or -3.20000000000000003e79 < x < -2.4999999999999999e-95Initial program 99.9%
Taylor expanded in y around 0 61.6%
if -2.4999999999999999e-95 < x < 1Initial program 100.0%
Taylor expanded in x around 0 79.9%
Final simplification68.7%
NOTE: x and y should be sorted in increasing order before calling this function. (FPCore (x y) :precision binary64 (if (<= x -2.2e-95) (* x (- 1.0 y)) (if (<= x 1.0) y (* x (- y)))))
assert(x < y);
double code(double x, double y) {
double tmp;
if (x <= -2.2e-95) {
tmp = x * (1.0 - y);
} else if (x <= 1.0) {
tmp = y;
} else {
tmp = x * -y;
}
return tmp;
}
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: tmp
if (x <= (-2.2d-95)) then
tmp = x * (1.0d0 - y)
else if (x <= 1.0d0) then
tmp = y
else
tmp = x * -y
end if
code = tmp
end function
assert x < y;
public static double code(double x, double y) {
double tmp;
if (x <= -2.2e-95) {
tmp = x * (1.0 - y);
} else if (x <= 1.0) {
tmp = y;
} else {
tmp = x * -y;
}
return tmp;
}
[x, y] = sort([x, y]) def code(x, y): tmp = 0 if x <= -2.2e-95: tmp = x * (1.0 - y) elif x <= 1.0: tmp = y else: tmp = x * -y return tmp
x, y = sort([x, y]) function code(x, y) tmp = 0.0 if (x <= -2.2e-95) tmp = Float64(x * Float64(1.0 - y)); elseif (x <= 1.0) tmp = y; else tmp = Float64(x * Float64(-y)); end return tmp end
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
tmp = 0.0;
if (x <= -2.2e-95)
tmp = x * (1.0 - y);
elseif (x <= 1.0)
tmp = y;
else
tmp = x * -y;
end
tmp_2 = tmp;
end
NOTE: x and y should be sorted in increasing order before calling this function. code[x_, y_] := If[LessEqual[x, -2.2e-95], N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], y, N[(x * (-y)), $MachinePrecision]]]
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\begin{array}{l}
\mathbf{if}\;x \leq -2.2 \cdot 10^{-95}:\\
\;\;\;\;x \cdot \left(1 - y\right)\\
\mathbf{elif}\;x \leq 1:\\
\;\;\;\;y\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(-y\right)\\
\end{array}
\end{array}
if x < -2.1999999999999999e-95Initial program 99.9%
Taylor expanded in x around inf 89.8%
if -2.1999999999999999e-95 < x < 1Initial program 100.0%
Taylor expanded in x around 0 79.9%
if 1 < x Initial program 100.0%
Taylor expanded in x around inf 98.3%
Taylor expanded in y around inf 55.6%
mul-1-neg55.6%
distribute-rgt-neg-out55.6%
Simplified55.6%
Final simplification77.5%
NOTE: x and y should be sorted in increasing order before calling this function. (FPCore (x y) :precision binary64 (if (<= y 2.65e-65) (* x (- 1.0 y)) (- y (* x y))))
assert(x < y);
double code(double x, double y) {
double tmp;
if (y <= 2.65e-65) {
tmp = x * (1.0 - y);
} else {
tmp = y - (x * y);
}
return tmp;
}
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: tmp
if (y <= 2.65d-65) then
tmp = x * (1.0d0 - y)
else
tmp = y - (x * y)
end if
code = tmp
end function
assert x < y;
public static double code(double x, double y) {
double tmp;
if (y <= 2.65e-65) {
tmp = x * (1.0 - y);
} else {
tmp = y - (x * y);
}
return tmp;
}
[x, y] = sort([x, y]) def code(x, y): tmp = 0 if y <= 2.65e-65: tmp = x * (1.0 - y) else: tmp = y - (x * y) return tmp
x, y = sort([x, y]) function code(x, y) tmp = 0.0 if (y <= 2.65e-65) tmp = Float64(x * Float64(1.0 - y)); else tmp = Float64(y - Float64(x * y)); end return tmp end
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
tmp = 0.0;
if (y <= 2.65e-65)
tmp = x * (1.0 - y);
else
tmp = y - (x * y);
end
tmp_2 = tmp;
end
NOTE: x and y should be sorted in increasing order before calling this function. code[x_, y_] := If[LessEqual[y, 2.65e-65], N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], N[(y - N[(x * y), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\begin{array}{l}
\mathbf{if}\;y \leq 2.65 \cdot 10^{-65}:\\
\;\;\;\;x \cdot \left(1 - y\right)\\
\mathbf{else}:\\
\;\;\;\;y - x \cdot y\\
\end{array}
\end{array}
if y < 2.65000000000000019e-65Initial program 100.0%
Taylor expanded in x around inf 68.5%
if 2.65000000000000019e-65 < y Initial program 100.0%
Taylor expanded in y around inf 92.4%
distribute-lft-out--92.4%
*-rgt-identity92.4%
Simplified92.4%
Final simplification76.1%
NOTE: x and y should be sorted in increasing order before calling this function. (FPCore (x y) :precision binary64 (- (+ x y) (* x y)))
assert(x < y);
double code(double x, double y) {
return (x + y) - (x * y);
}
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
code = (x + y) - (x * y)
end function
assert x < y;
public static double code(double x, double y) {
return (x + y) - (x * y);
}
[x, y] = sort([x, y]) def code(x, y): return (x + y) - (x * y)
x, y = sort([x, y]) function code(x, y) return Float64(Float64(x + y) - Float64(x * y)) end
x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
tmp = (x + y) - (x * y);
end
NOTE: x and y should be sorted in increasing order before calling this function. code[x_, y_] := N[(N[(x + y), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\left(x + y\right) - x \cdot y
\end{array}
Initial program 100.0%
Final simplification100.0%
NOTE: x and y should be sorted in increasing order before calling this function. (FPCore (x y) :precision binary64 (if (<= y 5.6e-64) x y))
assert(x < y);
double code(double x, double y) {
double tmp;
if (y <= 5.6e-64) {
tmp = x;
} else {
tmp = y;
}
return tmp;
}
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: tmp
if (y <= 5.6d-64) then
tmp = x
else
tmp = y
end if
code = tmp
end function
assert x < y;
public static double code(double x, double y) {
double tmp;
if (y <= 5.6e-64) {
tmp = x;
} else {
tmp = y;
}
return tmp;
}
[x, y] = sort([x, y]) def code(x, y): tmp = 0 if y <= 5.6e-64: tmp = x else: tmp = y return tmp
x, y = sort([x, y]) function code(x, y) tmp = 0.0 if (y <= 5.6e-64) tmp = x; else tmp = y; end return tmp end
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
tmp = 0.0;
if (y <= 5.6e-64)
tmp = x;
else
tmp = y;
end
tmp_2 = tmp;
end
NOTE: x and y should be sorted in increasing order before calling this function. code[x_, y_] := If[LessEqual[y, 5.6e-64], x, y]
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\begin{array}{l}
\mathbf{if}\;y \leq 5.6 \cdot 10^{-64}:\\
\;\;\;\;x\\
\mathbf{else}:\\
\;\;\;\;y\\
\end{array}
\end{array}
if y < 5.60000000000000008e-64Initial program 100.0%
Taylor expanded in y around 0 51.7%
if 5.60000000000000008e-64 < y Initial program 100.0%
Taylor expanded in x around 0 49.7%
Final simplification51.1%
NOTE: x and y should be sorted in increasing order before calling this function. (FPCore (x y) :precision binary64 x)
assert(x < y);
double code(double x, double y) {
return x;
}
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
code = x
end function
assert x < y;
public static double code(double x, double y) {
return x;
}
[x, y] = sort([x, y]) def code(x, y): return x
x, y = sort([x, y]) function code(x, y) return x end
x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
tmp = x;
end
NOTE: x and y should be sorted in increasing order before calling this function. code[x_, y_] := x
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
x
\end{array}
Initial program 100.0%
Taylor expanded in y around 0 37.4%
Final simplification37.4%
herbie shell --seed 2023271
(FPCore (x y)
:name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
:precision binary64
(- (+ x y) (* x y)))