
(FPCore (x y) :precision binary64 (/ (* (* x 2.0) y) (- x y)))
double code(double x, double y) {
return ((x * 2.0) * y) / (x - y);
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
code = ((x * 2.0d0) * y) / (x - y)
end function
public static double code(double x, double y) {
return ((x * 2.0) * y) / (x - y);
}
def code(x, y): return ((x * 2.0) * y) / (x - y)
function code(x, y) return Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y)) end
function tmp = code(x, y) tmp = ((x * 2.0) * y) / (x - y); end
code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\frac{\left(x \cdot 2\right) \cdot y}{x - y}
\end{array}
Sampling outcomes in binary64 precision:
Herbie found 6 alternatives:
| Alternative | Accuracy | Speedup |
|---|
(FPCore (x y) :precision binary64 (/ (* (* x 2.0) y) (- x y)))
double code(double x, double y) {
return ((x * 2.0) * y) / (x - y);
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
code = ((x * 2.0d0) * y) / (x - y)
end function
public static double code(double x, double y) {
return ((x * 2.0) * y) / (x - y);
}
def code(x, y): return ((x * 2.0) * y) / (x - y)
function code(x, y) return Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y)) end
function tmp = code(x, y) tmp = ((x * 2.0) * y) / (x - y); end
code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\frac{\left(x \cdot 2\right) \cdot y}{x - y}
\end{array}
(FPCore (x y)
:precision binary64
(let* ((t_0 (/ (* x 2.0) (/ (- x y) y))))
(if (<= y -40000000000000.0)
t_0
(if (<= y 0.001) (* (/ (* x 2.0) (- x y)) y) t_0))))
double code(double x, double y) {
double t_0 = (x * 2.0) / ((x - y) / y);
double tmp;
if (y <= -40000000000000.0) {
tmp = t_0;
} else if (y <= 0.001) {
tmp = ((x * 2.0) / (x - y)) * y;
} else {
tmp = t_0;
}
return tmp;
}
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 * 2.0d0) / ((x - y) / y)
if (y <= (-40000000000000.0d0)) then
tmp = t_0
else if (y <= 0.001d0) then
tmp = ((x * 2.0d0) / (x - y)) * y
else
tmp = t_0
end if
code = tmp
end function
public static double code(double x, double y) {
double t_0 = (x * 2.0) / ((x - y) / y);
double tmp;
if (y <= -40000000000000.0) {
tmp = t_0;
} else if (y <= 0.001) {
tmp = ((x * 2.0) / (x - y)) * y;
} else {
tmp = t_0;
}
return tmp;
}
def code(x, y): t_0 = (x * 2.0) / ((x - y) / y) tmp = 0 if y <= -40000000000000.0: tmp = t_0 elif y <= 0.001: tmp = ((x * 2.0) / (x - y)) * y else: tmp = t_0 return tmp
function code(x, y) t_0 = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y)) tmp = 0.0 if (y <= -40000000000000.0) tmp = t_0; elseif (y <= 0.001) tmp = Float64(Float64(Float64(x * 2.0) / Float64(x - y)) * y); else tmp = t_0; end return tmp end
function tmp_2 = code(x, y) t_0 = (x * 2.0) / ((x - y) / y); tmp = 0.0; if (y <= -40000000000000.0) tmp = t_0; elseif (y <= 0.001) tmp = ((x * 2.0) / (x - y)) * y; else tmp = t_0; end tmp_2 = tmp; end
code[x_, y_] := Block[{t$95$0 = N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -40000000000000.0], t$95$0, If[LessEqual[y, 0.001], N[(N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \frac{x \cdot 2}{\frac{x - y}{y}}\\
\mathbf{if}\;y \leq -40000000000000:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;y \leq 0.001:\\
\;\;\;\;\frac{x \cdot 2}{x - y} \cdot y\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
if y < -4e13 or 1e-3 < y Initial program 77.5%
Applied egg-rr0
if -4e13 < y < 1e-3Initial program 75.0%
Applied egg-rr0
(FPCore (x y)
:precision binary64
(let* ((t_0 (* (* y (/ 2.0 (- x y))) x)))
(if (<= y -3.6e+83)
t_0
(if (<= y 3.3e+30) (* (/ (* x 2.0) (- x y)) y) t_0))))
double code(double x, double y) {
double t_0 = (y * (2.0 / (x - y))) * x;
double tmp;
if (y <= -3.6e+83) {
tmp = t_0;
} else if (y <= 3.3e+30) {
tmp = ((x * 2.0) / (x - y)) * y;
} else {
tmp = t_0;
}
return tmp;
}
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 = (y * (2.0d0 / (x - y))) * x
if (y <= (-3.6d+83)) then
tmp = t_0
else if (y <= 3.3d+30) then
tmp = ((x * 2.0d0) / (x - y)) * y
else
tmp = t_0
end if
code = tmp
end function
public static double code(double x, double y) {
double t_0 = (y * (2.0 / (x - y))) * x;
double tmp;
if (y <= -3.6e+83) {
tmp = t_0;
} else if (y <= 3.3e+30) {
tmp = ((x * 2.0) / (x - y)) * y;
} else {
tmp = t_0;
}
return tmp;
}
def code(x, y): t_0 = (y * (2.0 / (x - y))) * x tmp = 0 if y <= -3.6e+83: tmp = t_0 elif y <= 3.3e+30: tmp = ((x * 2.0) / (x - y)) * y else: tmp = t_0 return tmp
function code(x, y) t_0 = Float64(Float64(y * Float64(2.0 / Float64(x - y))) * x) tmp = 0.0 if (y <= -3.6e+83) tmp = t_0; elseif (y <= 3.3e+30) tmp = Float64(Float64(Float64(x * 2.0) / Float64(x - y)) * y); else tmp = t_0; end return tmp end
function tmp_2 = code(x, y) t_0 = (y * (2.0 / (x - y))) * x; tmp = 0.0; if (y <= -3.6e+83) tmp = t_0; elseif (y <= 3.3e+30) tmp = ((x * 2.0) / (x - y)) * y; else tmp = t_0; end tmp_2 = tmp; end
code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(2.0 / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -3.6e+83], t$95$0, If[LessEqual[y, 3.3e+30], N[(N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \left(y \cdot \frac{2}{x - y}\right) \cdot x\\
\mathbf{if}\;y \leq -3.6 \cdot 10^{+83}:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;y \leq 3.3 \cdot 10^{+30}:\\
\;\;\;\;\frac{x \cdot 2}{x - y} \cdot y\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
if y < -3.5999999999999997e83 or 3.30000000000000026e30 < y Initial program 74.4%
Applied egg-rr0
Applied egg-rr0
if -3.5999999999999997e83 < y < 3.30000000000000026e30Initial program 77.2%
Applied egg-rr0
(FPCore (x y)
:precision binary64
(let* ((t_0 (* (* y (/ 2.0 (- x y))) x)))
(if (<= y -5e-203)
t_0
(if (<= y 4.4e-206) (* y (+ 2.0 (/ (* y 2.0) x))) t_0))))
double code(double x, double y) {
double t_0 = (y * (2.0 / (x - y))) * x;
double tmp;
if (y <= -5e-203) {
tmp = t_0;
} else if (y <= 4.4e-206) {
tmp = y * (2.0 + ((y * 2.0) / x));
} else {
tmp = t_0;
}
return tmp;
}
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 = (y * (2.0d0 / (x - y))) * x
if (y <= (-5d-203)) then
tmp = t_0
else if (y <= 4.4d-206) then
tmp = y * (2.0d0 + ((y * 2.0d0) / x))
else
tmp = t_0
end if
code = tmp
end function
public static double code(double x, double y) {
double t_0 = (y * (2.0 / (x - y))) * x;
double tmp;
if (y <= -5e-203) {
tmp = t_0;
} else if (y <= 4.4e-206) {
tmp = y * (2.0 + ((y * 2.0) / x));
} else {
tmp = t_0;
}
return tmp;
}
def code(x, y): t_0 = (y * (2.0 / (x - y))) * x tmp = 0 if y <= -5e-203: tmp = t_0 elif y <= 4.4e-206: tmp = y * (2.0 + ((y * 2.0) / x)) else: tmp = t_0 return tmp
function code(x, y) t_0 = Float64(Float64(y * Float64(2.0 / Float64(x - y))) * x) tmp = 0.0 if (y <= -5e-203) tmp = t_0; elseif (y <= 4.4e-206) tmp = Float64(y * Float64(2.0 + Float64(Float64(y * 2.0) / x))); else tmp = t_0; end return tmp end
function tmp_2 = code(x, y) t_0 = (y * (2.0 / (x - y))) * x; tmp = 0.0; if (y <= -5e-203) tmp = t_0; elseif (y <= 4.4e-206) tmp = y * (2.0 + ((y * 2.0) / x)); else tmp = t_0; end tmp_2 = tmp; end
code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(2.0 / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -5e-203], t$95$0, If[LessEqual[y, 4.4e-206], N[(y * N[(2.0 + N[(N[(y * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \left(y \cdot \frac{2}{x - y}\right) \cdot x\\
\mathbf{if}\;y \leq -5 \cdot 10^{-203}:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;y \leq 4.4 \cdot 10^{-206}:\\
\;\;\;\;y \cdot \left(2 + \frac{y \cdot 2}{x}\right)\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
if y < -5.0000000000000002e-203 or 4.3999999999999997e-206 < y Initial program 79.0%
Applied egg-rr0
Applied egg-rr0
if -5.0000000000000002e-203 < y < 4.3999999999999997e-206Initial program 63.9%
Taylor expanded in x around inf 0
Simplified0
(FPCore (x y) :precision binary64 (if (<= y -1.05e-37) (* -2.0 x) (if (<= y 6.8e+28) (* y (+ 2.0 (/ (* y 2.0) x))) (* -2.0 x))))
double code(double x, double y) {
double tmp;
if (y <= -1.05e-37) {
tmp = -2.0 * x;
} else if (y <= 6.8e+28) {
tmp = y * (2.0 + ((y * 2.0) / x));
} else {
tmp = -2.0 * x;
}
return tmp;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: tmp
if (y <= (-1.05d-37)) then
tmp = (-2.0d0) * x
else if (y <= 6.8d+28) then
tmp = y * (2.0d0 + ((y * 2.0d0) / x))
else
tmp = (-2.0d0) * x
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if (y <= -1.05e-37) {
tmp = -2.0 * x;
} else if (y <= 6.8e+28) {
tmp = y * (2.0 + ((y * 2.0) / x));
} else {
tmp = -2.0 * x;
}
return tmp;
}
def code(x, y): tmp = 0 if y <= -1.05e-37: tmp = -2.0 * x elif y <= 6.8e+28: tmp = y * (2.0 + ((y * 2.0) / x)) else: tmp = -2.0 * x return tmp
function code(x, y) tmp = 0.0 if (y <= -1.05e-37) tmp = Float64(-2.0 * x); elseif (y <= 6.8e+28) tmp = Float64(y * Float64(2.0 + Float64(Float64(y * 2.0) / x))); else tmp = Float64(-2.0 * x); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if (y <= -1.05e-37) tmp = -2.0 * x; elseif (y <= 6.8e+28) tmp = y * (2.0 + ((y * 2.0) / x)); else tmp = -2.0 * x; end tmp_2 = tmp; end
code[x_, y_] := If[LessEqual[y, -1.05e-37], N[(-2.0 * x), $MachinePrecision], If[LessEqual[y, 6.8e+28], N[(y * N[(2.0 + N[(N[(y * 2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * x), $MachinePrecision]]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -1.05 \cdot 10^{-37}:\\
\;\;\;\;-2 \cdot x\\
\mathbf{elif}\;y \leq 6.8 \cdot 10^{+28}:\\
\;\;\;\;y \cdot \left(2 + \frac{y \cdot 2}{x}\right)\\
\mathbf{else}:\\
\;\;\;\;-2 \cdot x\\
\end{array}
\end{array}
if y < -1.05e-37 or 6.8e28 < y Initial program 77.9%
Taylor expanded in x around 0 0
Simplified0
if -1.05e-37 < y < 6.8e28Initial program 74.5%
Taylor expanded in x around inf 0
Simplified0
(FPCore (x y) :precision binary64 (if (<= y -4.2e-36) (* -2.0 x) (if (<= y 2.9e+29) (* y 2.0) (* -2.0 x))))
double code(double x, double y) {
double tmp;
if (y <= -4.2e-36) {
tmp = -2.0 * x;
} else if (y <= 2.9e+29) {
tmp = y * 2.0;
} else {
tmp = -2.0 * x;
}
return tmp;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: tmp
if (y <= (-4.2d-36)) then
tmp = (-2.0d0) * x
else if (y <= 2.9d+29) then
tmp = y * 2.0d0
else
tmp = (-2.0d0) * x
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if (y <= -4.2e-36) {
tmp = -2.0 * x;
} else if (y <= 2.9e+29) {
tmp = y * 2.0;
} else {
tmp = -2.0 * x;
}
return tmp;
}
def code(x, y): tmp = 0 if y <= -4.2e-36: tmp = -2.0 * x elif y <= 2.9e+29: tmp = y * 2.0 else: tmp = -2.0 * x return tmp
function code(x, y) tmp = 0.0 if (y <= -4.2e-36) tmp = Float64(-2.0 * x); elseif (y <= 2.9e+29) tmp = Float64(y * 2.0); else tmp = Float64(-2.0 * x); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if (y <= -4.2e-36) tmp = -2.0 * x; elseif (y <= 2.9e+29) tmp = y * 2.0; else tmp = -2.0 * x; end tmp_2 = tmp; end
code[x_, y_] := If[LessEqual[y, -4.2e-36], N[(-2.0 * x), $MachinePrecision], If[LessEqual[y, 2.9e+29], N[(y * 2.0), $MachinePrecision], N[(-2.0 * x), $MachinePrecision]]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -4.2 \cdot 10^{-36}:\\
\;\;\;\;-2 \cdot x\\
\mathbf{elif}\;y \leq 2.9 \cdot 10^{+29}:\\
\;\;\;\;y \cdot 2\\
\mathbf{else}:\\
\;\;\;\;-2 \cdot x\\
\end{array}
\end{array}
if y < -4.19999999999999982e-36 or 2.8999999999999999e29 < y Initial program 77.9%
Taylor expanded in x around 0 0
Simplified0
if -4.19999999999999982e-36 < y < 2.8999999999999999e29Initial program 74.5%
Taylor expanded in x around inf 0
Simplified0
(FPCore (x y) :precision binary64 (* -2.0 x))
double code(double x, double y) {
return -2.0 * x;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
code = (-2.0d0) * x
end function
public static double code(double x, double y) {
return -2.0 * x;
}
def code(x, y): return -2.0 * x
function code(x, y) return Float64(-2.0 * x) end
function tmp = code(x, y) tmp = -2.0 * x; end
code[x_, y_] := N[(-2.0 * x), $MachinePrecision]
\begin{array}{l}
\\
-2 \cdot x
\end{array}
Initial program 76.1%
Taylor expanded in x around 0 0
Simplified0
(FPCore (x y)
:precision binary64
(let* ((t_0 (* (/ (* 2.0 x) (- x y)) y)))
(if (< x -1.7210442634149447e+81)
t_0
(if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) t_0))))
double code(double x, double y) {
double t_0 = ((2.0 * x) / (x - y)) * y;
double tmp;
if (x < -1.7210442634149447e+81) {
tmp = t_0;
} else if (x < 83645045635564430.0) {
tmp = (x * 2.0) / ((x - y) / y);
} else {
tmp = t_0;
}
return tmp;
}
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 = ((2.0d0 * x) / (x - y)) * y
if (x < (-1.7210442634149447d+81)) then
tmp = t_0
else if (x < 83645045635564430.0d0) then
tmp = (x * 2.0d0) / ((x - y) / y)
else
tmp = t_0
end if
code = tmp
end function
public static double code(double x, double y) {
double t_0 = ((2.0 * x) / (x - y)) * y;
double tmp;
if (x < -1.7210442634149447e+81) {
tmp = t_0;
} else if (x < 83645045635564430.0) {
tmp = (x * 2.0) / ((x - y) / y);
} else {
tmp = t_0;
}
return tmp;
}
def code(x, y): t_0 = ((2.0 * x) / (x - y)) * y tmp = 0 if x < -1.7210442634149447e+81: tmp = t_0 elif x < 83645045635564430.0: tmp = (x * 2.0) / ((x - y) / y) else: tmp = t_0 return tmp
function code(x, y) t_0 = Float64(Float64(Float64(2.0 * x) / Float64(x - y)) * y) tmp = 0.0 if (x < -1.7210442634149447e+81) tmp = t_0; elseif (x < 83645045635564430.0) tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y)); else tmp = t_0; end return tmp end
function tmp_2 = code(x, y) t_0 = ((2.0 * x) / (x - y)) * y; tmp = 0.0; if (x < -1.7210442634149447e+81) tmp = t_0; elseif (x < 83645045635564430.0) tmp = (x * 2.0) / ((x - y) / y); else tmp = t_0; end tmp_2 = tmp; end
code[x_, y_] := Block[{t$95$0 = N[(N[(N[(2.0 * x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[Less[x, -1.7210442634149447e+81], t$95$0, If[Less[x, 83645045635564430.0], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \frac{2 \cdot x}{x - y} \cdot y\\
\mathbf{if}\;x < -1.7210442634149447 \cdot 10^{+81}:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;x < 83645045635564430:\\
\;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
herbie shell --seed 2024110
(FPCore (x y)
:name "Linear.Projection:perspective from linear-1.19.1.3, B"
:precision binary64
:alt
(if (< x -1.7210442634149447e+81) (* (/ (* 2.0 x) (- x y)) y) (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) (* (/ (* 2.0 x) (- x y)) y)))
(/ (* (* x 2.0) y) (- x y)))