
(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%
*-commutative100.0%
associate--l+100.0%
+-commutative100.0%
*-lft-identity100.0%
metadata-eval100.0%
distribute-rgt-out--100.0%
fma-define100.0%
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 -2.8e+96)
x
(if (<= x -8e+81) t_0 (if (<= x -8.5e-152) 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 <= -2.8e+96) {
tmp = x;
} else if (x <= -8e+81) {
tmp = t_0;
} else if (x <= -8.5e-152) {
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 <= (-2.8d+96)) then
tmp = x
else if (x <= (-8d+81)) then
tmp = t_0
else if (x <= (-8.5d-152)) 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 <= -2.8e+96) {
tmp = x;
} else if (x <= -8e+81) {
tmp = t_0;
} else if (x <= -8.5e-152) {
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 <= -2.8e+96: tmp = x elif x <= -8e+81: tmp = t_0 elif x <= -8.5e-152: 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 <= -2.8e+96) tmp = x; elseif (x <= -8e+81) tmp = t_0; elseif (x <= -8.5e-152) 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 <= -2.8e+96)
tmp = x;
elseif (x <= -8e+81)
tmp = t_0;
elseif (x <= -8.5e-152)
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, -2.8e+96], x, If[LessEqual[x, -8e+81], t$95$0, If[LessEqual[x, -8.5e-152], 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 -2.8 \cdot 10^{+96}:\\
\;\;\;\;x\\
\mathbf{elif}\;x \leq -8 \cdot 10^{+81}:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;x \leq -8.5 \cdot 10^{-152}:\\
\;\;\;\;x\\
\mathbf{elif}\;x \leq 1:\\
\;\;\;\;y\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
if x < -2.8e96 or -7.99999999999999937e81 < x < -8.5000000000000007e-152Initial program 100.0%
Taylor expanded in y around 0 50.7%
if -2.8e96 < x < -7.99999999999999937e81 or 1 < x Initial program 100.0%
Taylor expanded in y around inf 49.3%
Taylor expanded in x around inf 47.3%
mul-1-neg47.3%
distribute-lft-neg-out47.3%
*-commutative47.3%
Simplified47.3%
if -8.5000000000000007e-152 < x < 1Initial program 100.0%
Taylor expanded in x around 0 81.2%
Final simplification62.1%
NOTE: x and y should be sorted in increasing order before calling this function. (FPCore (x y) :precision binary64 (if (<= x -4.3e-154) (* x (- 1.0 y)) (if (<= x 1.0) y (* x (- y)))))
assert(x < y);
double code(double x, double y) {
double tmp;
if (x <= -4.3e-154) {
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 <= (-4.3d-154)) 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 <= -4.3e-154) {
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 <= -4.3e-154: 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 <= -4.3e-154) 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 <= -4.3e-154)
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, -4.3e-154], 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 -4.3 \cdot 10^{-154}:\\
\;\;\;\;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 < -4.29999999999999992e-154Initial program 100.0%
Taylor expanded in x around inf 84.3%
if -4.29999999999999992e-154 < x < 1Initial program 100.0%
Taylor expanded in x around 0 81.8%
if 1 < x Initial program 100.0%
Taylor expanded in y around inf 46.9%
Taylor expanded in x around inf 44.9%
mul-1-neg44.9%
distribute-lft-neg-out44.9%
*-commutative44.9%
Simplified44.9%
Final simplification73.2%
NOTE: x and y should be sorted in increasing order before calling this function. (FPCore (x y) :precision binary64 (if (<= x -4.3e-154) (* x (- 1.0 y)) (* y (- 1.0 x))))
assert(x < y);
double code(double x, double y) {
double tmp;
if (x <= -4.3e-154) {
tmp = x * (1.0 - y);
} else {
tmp = y * (1.0 - x);
}
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 <= (-4.3d-154)) then
tmp = x * (1.0d0 - y)
else
tmp = y * (1.0d0 - x)
end if
code = tmp
end function
assert x < y;
public static double code(double x, double y) {
double tmp;
if (x <= -4.3e-154) {
tmp = x * (1.0 - y);
} else {
tmp = y * (1.0 - x);
}
return tmp;
}
[x, y] = sort([x, y]) def code(x, y): tmp = 0 if x <= -4.3e-154: tmp = x * (1.0 - y) else: tmp = y * (1.0 - x) return tmp
x, y = sort([x, y]) function code(x, y) tmp = 0.0 if (x <= -4.3e-154) tmp = Float64(x * Float64(1.0 - y)); else tmp = Float64(y * Float64(1.0 - x)); end return tmp end
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
tmp = 0.0;
if (x <= -4.3e-154)
tmp = x * (1.0 - y);
else
tmp = y * (1.0 - x);
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, -4.3e-154], N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\begin{array}{l}
\mathbf{if}\;x \leq -4.3 \cdot 10^{-154}:\\
\;\;\;\;x \cdot \left(1 - y\right)\\
\mathbf{else}:\\
\;\;\;\;y \cdot \left(1 - x\right)\\
\end{array}
\end{array}
if x < -4.29999999999999992e-154Initial program 100.0%
Taylor expanded in x around inf 84.3%
if -4.29999999999999992e-154 < x Initial program 100.0%
Taylor expanded in y around inf 68.6%
Final simplification74.1%
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 (<= x -2.5e-151) x y))
assert(x < y);
double code(double x, double y) {
double tmp;
if (x <= -2.5e-151) {
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 (x <= (-2.5d-151)) 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 (x <= -2.5e-151) {
tmp = x;
} else {
tmp = y;
}
return tmp;
}
[x, y] = sort([x, y]) def code(x, y): tmp = 0 if x <= -2.5e-151: tmp = x else: tmp = y return tmp
x, y = sort([x, y]) function code(x, y) tmp = 0.0 if (x <= -2.5e-151) 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 (x <= -2.5e-151)
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[x, -2.5e-151], x, y]
\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\begin{array}{l}
\mathbf{if}\;x \leq -2.5 \cdot 10^{-151}:\\
\;\;\;\;x\\
\mathbf{else}:\\
\;\;\;\;y\\
\end{array}
\end{array}
if x < -2.50000000000000002e-151Initial program 100.0%
Taylor expanded in y around 0 49.0%
if -2.50000000000000002e-151 < x Initial program 100.0%
Taylor expanded in x around 0 50.1%
Final simplification49.7%
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 38.1%
Final simplification38.1%
herbie shell --seed 2024053
(FPCore (x y)
:name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, A"
:precision binary64
(- (+ x y) (* x y)))