
(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 4 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) y) (- x y))))
(if (or (<= t_0 -4e-67)
(not
(or (<= t_0 -1.8e-296)
(and (not (<= t_0 1e-306)) (<= t_0 2e-49)))))
(* x (* 2.0 (/ y (- x y))))
t_0)))
double code(double x, double y) {
double t_0 = ((x * 2.0) * y) / (x - y);
double tmp;
if ((t_0 <= -4e-67) || !((t_0 <= -1.8e-296) || (!(t_0 <= 1e-306) && (t_0 <= 2e-49)))) {
tmp = x * (2.0 * (y / (x - 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) * y) / (x - y)
if ((t_0 <= (-4d-67)) .or. (.not. (t_0 <= (-1.8d-296)) .or. (.not. (t_0 <= 1d-306)) .and. (t_0 <= 2d-49))) then
tmp = x * (2.0d0 * (y / (x - 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) * y) / (x - y);
double tmp;
if ((t_0 <= -4e-67) || !((t_0 <= -1.8e-296) || (!(t_0 <= 1e-306) && (t_0 <= 2e-49)))) {
tmp = x * (2.0 * (y / (x - y)));
} else {
tmp = t_0;
}
return tmp;
}
def code(x, y): t_0 = ((x * 2.0) * y) / (x - y) tmp = 0 if (t_0 <= -4e-67) or not ((t_0 <= -1.8e-296) or (not (t_0 <= 1e-306) and (t_0 <= 2e-49))): tmp = x * (2.0 * (y / (x - y))) else: tmp = t_0 return tmp
function code(x, y) t_0 = Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y)) tmp = 0.0 if ((t_0 <= -4e-67) || !((t_0 <= -1.8e-296) || (!(t_0 <= 1e-306) && (t_0 <= 2e-49)))) tmp = Float64(x * Float64(2.0 * Float64(y / Float64(x - y)))); else tmp = t_0; end return tmp end
function tmp_2 = code(x, y) t_0 = ((x * 2.0) * y) / (x - y); tmp = 0.0; if ((t_0 <= -4e-67) || ~(((t_0 <= -1.8e-296) || (~((t_0 <= 1e-306)) && (t_0 <= 2e-49))))) tmp = x * (2.0 * (y / (x - y))); else tmp = t_0; end tmp_2 = tmp; end
code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e-67], N[Not[Or[LessEqual[t$95$0, -1.8e-296], And[N[Not[LessEqual[t$95$0, 1e-306]], $MachinePrecision], LessEqual[t$95$0, 2e-49]]]], $MachinePrecision]], N[(x * N[(2.0 * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \frac{\left(x \cdot 2\right) \cdot y}{x - y}\\
\mathbf{if}\;t\_0 \leq -4 \cdot 10^{-67} \lor \neg \left(t\_0 \leq -1.8 \cdot 10^{-296} \lor \neg \left(t\_0 \leq 10^{-306}\right) \land t\_0 \leq 2 \cdot 10^{-49}\right):\\
\;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
if (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -3.99999999999999977e-67 or -1.7999999999999999e-296 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 1.00000000000000003e-306 or 1.99999999999999987e-49 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) Initial program 49.6%
associate-/l*99.5%
associate-*l*99.5%
Simplified99.5%
if -3.99999999999999977e-67 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -1.7999999999999999e-296 or 1.00000000000000003e-306 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 1.99999999999999987e-49Initial program 99.7%
Final simplification99.6%
(FPCore (x y)
:precision binary64
(if (or (<= x -5.8e+25)
(not (or (<= x 3.6) (and (not (<= x 8.8e+52)) (<= x 4.6e+88)))))
(* 2.0 y)
(* x -2.0)))
double code(double x, double y) {
double tmp;
if ((x <= -5.8e+25) || !((x <= 3.6) || (!(x <= 8.8e+52) && (x <= 4.6e+88)))) {
tmp = 2.0 * y;
} else {
tmp = x * -2.0;
}
return tmp;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: tmp
if ((x <= (-5.8d+25)) .or. (.not. (x <= 3.6d0) .or. (.not. (x <= 8.8d+52)) .and. (x <= 4.6d+88))) then
tmp = 2.0d0 * y
else
tmp = x * (-2.0d0)
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if ((x <= -5.8e+25) || !((x <= 3.6) || (!(x <= 8.8e+52) && (x <= 4.6e+88)))) {
tmp = 2.0 * y;
} else {
tmp = x * -2.0;
}
return tmp;
}
def code(x, y): tmp = 0 if (x <= -5.8e+25) or not ((x <= 3.6) or (not (x <= 8.8e+52) and (x <= 4.6e+88))): tmp = 2.0 * y else: tmp = x * -2.0 return tmp
function code(x, y) tmp = 0.0 if ((x <= -5.8e+25) || !((x <= 3.6) || (!(x <= 8.8e+52) && (x <= 4.6e+88)))) tmp = Float64(2.0 * y); else tmp = Float64(x * -2.0); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if ((x <= -5.8e+25) || ~(((x <= 3.6) || (~((x <= 8.8e+52)) && (x <= 4.6e+88))))) tmp = 2.0 * y; else tmp = x * -2.0; end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[x, -5.8e+25], N[Not[Or[LessEqual[x, 3.6], And[N[Not[LessEqual[x, 8.8e+52]], $MachinePrecision], LessEqual[x, 4.6e+88]]]], $MachinePrecision]], N[(2.0 * y), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;x \leq -5.8 \cdot 10^{+25} \lor \neg \left(x \leq 3.6 \lor \neg \left(x \leq 8.8 \cdot 10^{+52}\right) \land x \leq 4.6 \cdot 10^{+88}\right):\\
\;\;\;\;2 \cdot y\\
\mathbf{else}:\\
\;\;\;\;x \cdot -2\\
\end{array}
\end{array}
if x < -5.7999999999999998e25 or 3.60000000000000009 < x < 8.7999999999999999e52 or 4.6000000000000003e88 < x Initial program 70.9%
associate-/l*73.0%
associate-*l*73.0%
Simplified73.0%
Taylor expanded in x around inf 82.7%
*-commutative82.7%
Simplified82.7%
if -5.7999999999999998e25 < x < 3.60000000000000009 or 8.7999999999999999e52 < x < 4.6000000000000003e88Initial program 77.4%
associate-*l/77.3%
associate-/l*77.0%
associate-*l*99.6%
*-commutative99.6%
Simplified99.6%
Taylor expanded in y around inf 75.8%
Final simplification78.9%
(FPCore (x y) :precision binary64 (if (or (<= y -2.7e-128) (not (<= y 2.3e-131))) (* x (* 2.0 (/ y (- x y)))) (* 2.0 y)))
double code(double x, double y) {
double tmp;
if ((y <= -2.7e-128) || !(y <= 2.3e-131)) {
tmp = x * (2.0 * (y / (x - y)));
} else {
tmp = 2.0 * y;
}
return tmp;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: tmp
if ((y <= (-2.7d-128)) .or. (.not. (y <= 2.3d-131))) then
tmp = x * (2.0d0 * (y / (x - y)))
else
tmp = 2.0d0 * y
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if ((y <= -2.7e-128) || !(y <= 2.3e-131)) {
tmp = x * (2.0 * (y / (x - y)));
} else {
tmp = 2.0 * y;
}
return tmp;
}
def code(x, y): tmp = 0 if (y <= -2.7e-128) or not (y <= 2.3e-131): tmp = x * (2.0 * (y / (x - y))) else: tmp = 2.0 * y return tmp
function code(x, y) tmp = 0.0 if ((y <= -2.7e-128) || !(y <= 2.3e-131)) tmp = Float64(x * Float64(2.0 * Float64(y / Float64(x - y)))); else tmp = Float64(2.0 * y); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if ((y <= -2.7e-128) || ~((y <= 2.3e-131))) tmp = x * (2.0 * (y / (x - y))); else tmp = 2.0 * y; end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[y, -2.7e-128], N[Not[LessEqual[y, 2.3e-131]], $MachinePrecision]], N[(x * N[(2.0 * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -2.7 \cdot 10^{-128} \lor \neg \left(y \leq 2.3 \cdot 10^{-131}\right):\\
\;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\
\mathbf{else}:\\
\;\;\;\;2 \cdot y\\
\end{array}
\end{array}
if y < -2.70000000000000006e-128 or 2.30000000000000022e-131 < y Initial program 75.5%
associate-/l*98.1%
associate-*l*98.1%
Simplified98.1%
if -2.70000000000000006e-128 < y < 2.30000000000000022e-131Initial program 71.8%
associate-/l*60.9%
associate-*l*60.9%
Simplified60.9%
Taylor expanded in x around inf 88.4%
*-commutative88.4%
Simplified88.4%
Final simplification95.4%
(FPCore (x y) :precision binary64 (* x -2.0))
double code(double x, double y) {
return x * -2.0;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
code = x * (-2.0d0)
end function
public static double code(double x, double y) {
return x * -2.0;
}
def code(x, y): return x * -2.0
function code(x, y) return Float64(x * -2.0) end
function tmp = code(x, y) tmp = x * -2.0; end
code[x_, y_] := N[(x * -2.0), $MachinePrecision]
\begin{array}{l}
\\
x \cdot -2
\end{array}
Initial program 74.4%
associate-*l/87.7%
associate-/l*87.5%
associate-*l*87.3%
*-commutative87.3%
Simplified87.3%
Taylor expanded in y around inf 49.2%
Final simplification49.2%
(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 2024080
(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)))