
(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 (* (* 2.0 x) (/ y (- x y)))))
(if (<= y -2.6e-62)
t_0
(if (<= y 4.9e-60) (* (/ x (- x y)) (* 2.0 y)) t_0))))
double code(double x, double y) {
double t_0 = (2.0 * x) * (y / (x - y));
double tmp;
if (y <= -2.6e-62) {
tmp = t_0;
} else if (y <= 4.9e-60) {
tmp = (x / (x - y)) * (2.0 * 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) * (y / (x - y))
if (y <= (-2.6d-62)) then
tmp = t_0
else if (y <= 4.9d-60) then
tmp = (x / (x - y)) * (2.0d0 * 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) * (y / (x - y));
double tmp;
if (y <= -2.6e-62) {
tmp = t_0;
} else if (y <= 4.9e-60) {
tmp = (x / (x - y)) * (2.0 * y);
} else {
tmp = t_0;
}
return tmp;
}
def code(x, y): t_0 = (2.0 * x) * (y / (x - y)) tmp = 0 if y <= -2.6e-62: tmp = t_0 elif y <= 4.9e-60: tmp = (x / (x - y)) * (2.0 * y) else: tmp = t_0 return tmp
function code(x, y) t_0 = Float64(Float64(2.0 * x) * Float64(y / Float64(x - y))) tmp = 0.0 if (y <= -2.6e-62) tmp = t_0; elseif (y <= 4.9e-60) tmp = Float64(Float64(x / Float64(x - y)) * Float64(2.0 * y)); else tmp = t_0; end return tmp end
function tmp_2 = code(x, y) t_0 = (2.0 * x) * (y / (x - y)); tmp = 0.0; if (y <= -2.6e-62) tmp = t_0; elseif (y <= 4.9e-60) tmp = (x / (x - y)) * (2.0 * y); else tmp = t_0; end tmp_2 = tmp; end
code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 * x), $MachinePrecision] * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.6e-62], t$95$0, If[LessEqual[y, 4.9e-60], N[(N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision] * N[(2.0 * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \left(2 \cdot x\right) \cdot \frac{y}{x - y}\\
\mathbf{if}\;y \leq -2.6 \cdot 10^{-62}:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;y \leq 4.9 \cdot 10^{-60}:\\
\;\;\;\;\frac{x}{x - y} \cdot \left(2 \cdot y\right)\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
if y < -2.5999999999999999e-62 or 4.89999999999999988e-60 < y Initial program 79.9%
lift-/.f64N/A
lift-*.f64N/A
associate-/l*N/A
*-commutativeN/A
lower-*.f64N/A
lower-/.f6499.9
lift-*.f64N/A
*-commutativeN/A
lower-*.f6499.9
Applied rewrites99.9%
if -2.5999999999999999e-62 < y < 4.89999999999999988e-60Initial program 76.3%
lift-/.f64N/A
lift-*.f64N/A
lift-*.f64N/A
associate-*l*N/A
*-commutativeN/A
associate-/l*N/A
lower-*.f64N/A
*-commutativeN/A
lower-*.f64N/A
lower-/.f64100.0
Applied rewrites100.0%
Final simplification100.0%
(FPCore (x y)
:precision binary64
(let* ((t_0 (* (/ x (- x y)) (* 2.0 y))))
(if (<= x -2.2e-122)
t_0
(if (<= x 6.5e-114) (* -2.0 (fma (/ x y) x x)) t_0))))
double code(double x, double y) {
double t_0 = (x / (x - y)) * (2.0 * y);
double tmp;
if (x <= -2.2e-122) {
tmp = t_0;
} else if (x <= 6.5e-114) {
tmp = -2.0 * fma((x / y), x, x);
} else {
tmp = t_0;
}
return tmp;
}
function code(x, y) t_0 = Float64(Float64(x / Float64(x - y)) * Float64(2.0 * y)) tmp = 0.0 if (x <= -2.2e-122) tmp = t_0; elseif (x <= 6.5e-114) tmp = Float64(-2.0 * fma(Float64(x / y), x, x)); else tmp = t_0; end return tmp end
code[x_, y_] := Block[{t$95$0 = N[(N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision] * N[(2.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.2e-122], t$95$0, If[LessEqual[x, 6.5e-114], N[(-2.0 * N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \frac{x}{x - y} \cdot \left(2 \cdot y\right)\\
\mathbf{if}\;x \leq -2.2 \cdot 10^{-122}:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;x \leq 6.5 \cdot 10^{-114}:\\
\;\;\;\;-2 \cdot \mathsf{fma}\left(\frac{x}{y}, x, x\right)\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
if x < -2.2e-122 or 6.4999999999999998e-114 < x Initial program 82.4%
lift-/.f64N/A
lift-*.f64N/A
lift-*.f64N/A
associate-*l*N/A
*-commutativeN/A
associate-/l*N/A
lower-*.f64N/A
*-commutativeN/A
lower-*.f64N/A
lower-/.f6498.8
Applied rewrites98.8%
if -2.2e-122 < x < 6.4999999999999998e-114Initial program 69.7%
Taylor expanded in y around inf
distribute-lft-outN/A
*-commutativeN/A
lower-*.f64N/A
+-commutativeN/A
unpow2N/A
associate-/l*N/A
*-commutativeN/A
lower-fma.f64N/A
lower-/.f6489.6
Applied rewrites89.6%
Final simplification96.0%
(FPCore (x y) :precision binary64 (if (<= x -4.6e+111) (* 2.0 y) (if (<= x 2.4e+29) (* -2.0 x) (* 2.0 y))))
double code(double x, double y) {
double tmp;
if (x <= -4.6e+111) {
tmp = 2.0 * y;
} else if (x <= 2.4e+29) {
tmp = -2.0 * x;
} 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 (x <= (-4.6d+111)) then
tmp = 2.0d0 * y
else if (x <= 2.4d+29) then
tmp = (-2.0d0) * x
else
tmp = 2.0d0 * y
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if (x <= -4.6e+111) {
tmp = 2.0 * y;
} else if (x <= 2.4e+29) {
tmp = -2.0 * x;
} else {
tmp = 2.0 * y;
}
return tmp;
}
def code(x, y): tmp = 0 if x <= -4.6e+111: tmp = 2.0 * y elif x <= 2.4e+29: tmp = -2.0 * x else: tmp = 2.0 * y return tmp
function code(x, y) tmp = 0.0 if (x <= -4.6e+111) tmp = Float64(2.0 * y); elseif (x <= 2.4e+29) tmp = Float64(-2.0 * x); else tmp = Float64(2.0 * y); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if (x <= -4.6e+111) tmp = 2.0 * y; elseif (x <= 2.4e+29) tmp = -2.0 * x; else tmp = 2.0 * y; end tmp_2 = tmp; end
code[x_, y_] := If[LessEqual[x, -4.6e+111], N[(2.0 * y), $MachinePrecision], If[LessEqual[x, 2.4e+29], N[(-2.0 * x), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;x \leq -4.6 \cdot 10^{+111}:\\
\;\;\;\;2 \cdot y\\
\mathbf{elif}\;x \leq 2.4 \cdot 10^{+29}:\\
\;\;\;\;-2 \cdot x\\
\mathbf{else}:\\
\;\;\;\;2 \cdot y\\
\end{array}
\end{array}
if x < -4.60000000000000004e111 or 2.4000000000000001e29 < x Initial program 72.9%
Taylor expanded in y around 0
*-commutativeN/A
lower-*.f6481.3
Applied rewrites81.3%
if -4.60000000000000004e111 < x < 2.4000000000000001e29Initial program 82.2%
Taylor expanded in y around inf
lower-*.f6474.4
Applied rewrites74.4%
Final simplification77.2%
(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 78.5%
Taylor expanded in y around inf
lower-*.f6451.8
Applied rewrites51.8%
(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 2024244
(FPCore (x y)
:name "Linear.Projection:perspective from linear-1.19.1.3, B"
:precision binary64
:alt
(! :herbie-platform default (if (< x -1721044263414944700000000000000000000000000000000000000000000000000000000000000000) (* (/ (* 2 x) (- x y)) y) (if (< x 83645045635564430) (/ (* x 2) (/ (- x y) y)) (* (/ (* 2 x) (- x y)) y))))
(/ (* (* x 2.0) y) (- x y)))