
(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 (if (or (<= y -3.25e-72) (not (<= y 4.5e+135))) (* (/ x (+ (/ x y) -1.0)) 2.0) (* y (/ (* x 2.0) (- x y)))))
double code(double x, double y) {
double tmp;
if ((y <= -3.25e-72) || !(y <= 4.5e+135)) {
tmp = (x / ((x / y) + -1.0)) * 2.0;
} else {
tmp = y * ((x * 2.0) / (x - y));
}
return tmp;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: tmp
if ((y <= (-3.25d-72)) .or. (.not. (y <= 4.5d+135))) then
tmp = (x / ((x / y) + (-1.0d0))) * 2.0d0
else
tmp = y * ((x * 2.0d0) / (x - y))
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if ((y <= -3.25e-72) || !(y <= 4.5e+135)) {
tmp = (x / ((x / y) + -1.0)) * 2.0;
} else {
tmp = y * ((x * 2.0) / (x - y));
}
return tmp;
}
def code(x, y): tmp = 0 if (y <= -3.25e-72) or not (y <= 4.5e+135): tmp = (x / ((x / y) + -1.0)) * 2.0 else: tmp = y * ((x * 2.0) / (x - y)) return tmp
function code(x, y) tmp = 0.0 if ((y <= -3.25e-72) || !(y <= 4.5e+135)) tmp = Float64(Float64(x / Float64(Float64(x / y) + -1.0)) * 2.0); else tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y))); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if ((y <= -3.25e-72) || ~((y <= 4.5e+135))) tmp = (x / ((x / y) + -1.0)) * 2.0; else tmp = y * ((x * 2.0) / (x - y)); end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[y, -3.25e-72], N[Not[LessEqual[y, 4.5e+135]], $MachinePrecision]], N[(N[(x / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -3.25 \cdot 10^{-72} \lor \neg \left(y \leq 4.5 \cdot 10^{+135}\right):\\
\;\;\;\;\frac{x}{\frac{x}{y} + -1} \cdot 2\\
\mathbf{else}:\\
\;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\
\end{array}
\end{array}
if y < -3.2499999999999998e-72 or 4.50000000000000007e135 < y Initial program 75.5%
associate-/l*99.9%
associate-*l/99.9%
div-sub99.9%
*-inverses99.9%
metadata-eval99.9%
sub-neg99.9%
metadata-eval99.9%
metadata-eval99.9%
Simplified99.9%
if -3.2499999999999998e-72 < y < 4.50000000000000007e135Initial program 72.8%
associate-*l/99.9%
Simplified99.9%
Final simplification99.9%
(FPCore (x y) :precision binary64 (if (or (<= y -1.55e-225) (not (<= y 1.75e-164))) (* (/ x (+ (/ x y) -1.0)) 2.0) (* y 2.0)))
double code(double x, double y) {
double tmp;
if ((y <= -1.55e-225) || !(y <= 1.75e-164)) {
tmp = (x / ((x / y) + -1.0)) * 2.0;
} else {
tmp = y * 2.0;
}
return tmp;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: tmp
if ((y <= (-1.55d-225)) .or. (.not. (y <= 1.75d-164))) then
tmp = (x / ((x / y) + (-1.0d0))) * 2.0d0
else
tmp = y * 2.0d0
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if ((y <= -1.55e-225) || !(y <= 1.75e-164)) {
tmp = (x / ((x / y) + -1.0)) * 2.0;
} else {
tmp = y * 2.0;
}
return tmp;
}
def code(x, y): tmp = 0 if (y <= -1.55e-225) or not (y <= 1.75e-164): tmp = (x / ((x / y) + -1.0)) * 2.0 else: tmp = y * 2.0 return tmp
function code(x, y) tmp = 0.0 if ((y <= -1.55e-225) || !(y <= 1.75e-164)) tmp = Float64(Float64(x / Float64(Float64(x / y) + -1.0)) * 2.0); else tmp = Float64(y * 2.0); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if ((y <= -1.55e-225) || ~((y <= 1.75e-164))) tmp = (x / ((x / y) + -1.0)) * 2.0; else tmp = y * 2.0; end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[y, -1.55e-225], N[Not[LessEqual[y, 1.75e-164]], $MachinePrecision]], N[(N[(x / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -1.55 \cdot 10^{-225} \lor \neg \left(y \leq 1.75 \cdot 10^{-164}\right):\\
\;\;\;\;\frac{x}{\frac{x}{y} + -1} \cdot 2\\
\mathbf{else}:\\
\;\;\;\;y \cdot 2\\
\end{array}
\end{array}
if y < -1.54999999999999998e-225 or 1.75e-164 < y Initial program 76.0%
associate-/l*98.0%
associate-*l/98.0%
div-sub97.9%
*-inverses97.9%
metadata-eval97.9%
sub-neg97.9%
metadata-eval97.9%
metadata-eval97.9%
Simplified97.9%
if -1.54999999999999998e-225 < y < 1.75e-164Initial program 67.2%
associate-/l*59.5%
associate-*l/59.5%
div-sub59.5%
*-inverses59.5%
metadata-eval59.5%
sub-neg59.5%
metadata-eval59.5%
metadata-eval59.5%
Simplified59.5%
Taylor expanded in x around inf 91.3%
Final simplification96.5%
(FPCore (x y) :precision binary64 (if (or (<= y -4.8e+15) (not (<= y 1.22e+64))) (* x -2.0) (* y 2.0)))
double code(double x, double y) {
double tmp;
if ((y <= -4.8e+15) || !(y <= 1.22e+64)) {
tmp = x * -2.0;
} else {
tmp = y * 2.0;
}
return tmp;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: tmp
if ((y <= (-4.8d+15)) .or. (.not. (y <= 1.22d+64))) then
tmp = x * (-2.0d0)
else
tmp = y * 2.0d0
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if ((y <= -4.8e+15) || !(y <= 1.22e+64)) {
tmp = x * -2.0;
} else {
tmp = y * 2.0;
}
return tmp;
}
def code(x, y): tmp = 0 if (y <= -4.8e+15) or not (y <= 1.22e+64): tmp = x * -2.0 else: tmp = y * 2.0 return tmp
function code(x, y) tmp = 0.0 if ((y <= -4.8e+15) || !(y <= 1.22e+64)) tmp = Float64(x * -2.0); else tmp = Float64(y * 2.0); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if ((y <= -4.8e+15) || ~((y <= 1.22e+64))) tmp = x * -2.0; else tmp = y * 2.0; end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[y, -4.8e+15], N[Not[LessEqual[y, 1.22e+64]], $MachinePrecision]], N[(x * -2.0), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -4.8 \cdot 10^{+15} \lor \neg \left(y \leq 1.22 \cdot 10^{+64}\right):\\
\;\;\;\;x \cdot -2\\
\mathbf{else}:\\
\;\;\;\;y \cdot 2\\
\end{array}
\end{array}
if y < -4.8e15 or 1.21999999999999994e64 < y Initial program 72.3%
associate-*l/77.9%
Simplified77.9%
Taylor expanded in x around 0 76.4%
*-commutative76.4%
Simplified76.4%
if -4.8e15 < y < 1.21999999999999994e64Initial program 75.2%
associate-/l*82.3%
associate-*l/82.3%
div-sub82.3%
*-inverses82.3%
metadata-eval82.3%
sub-neg82.3%
metadata-eval82.3%
metadata-eval82.3%
Simplified82.3%
Taylor expanded in x around inf 76.6%
Final simplification76.5%
(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 74.0%
associate-*l/91.1%
Simplified91.1%
Taylor expanded in x around 0 44.9%
*-commutative44.9%
Simplified44.9%
Final simplification44.9%
(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 2023322
(FPCore (x y)
:name "Linear.Projection:perspective from linear-1.19.1.3, B"
:precision binary64
:herbie-target
(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)))