
(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 -5e+32) (not (<= y 1e-33))) (* (/ x (+ (/ x y) -1.0)) 2.0) (* y (/ (* x 2.0) (- x y)))))
double code(double x, double y) {
double tmp;
if ((y <= -5e+32) || !(y <= 1e-33)) {
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 <= (-5d+32)) .or. (.not. (y <= 1d-33))) 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 <= -5e+32) || !(y <= 1e-33)) {
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 <= -5e+32) or not (y <= 1e-33): 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 <= -5e+32) || !(y <= 1e-33)) 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 <= -5e+32) || ~((y <= 1e-33))) 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, -5e+32], N[Not[LessEqual[y, 1e-33]], $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 -5 \cdot 10^{+32} \lor \neg \left(y \leq 10^{-33}\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 < -4.9999999999999997e32 or 1.0000000000000001e-33 < y Initial program 76.9%
associate-/l*99.3%
associate-*l/99.9%
div-sub100.0%
*-inverses100.0%
metadata-eval100.0%
sub-neg100.0%
metadata-eval100.0%
metadata-eval100.0%
Simplified100.0%
if -4.9999999999999997e32 < y < 1.0000000000000001e-33Initial program 79.7%
associate-*l/100.0%
Simplified100.0%
Final simplification100.0%
(FPCore (x y)
:precision binary64
(if (or (<= x -4.5e+15)
(and (not (<= x 2.45e-91)) (or (<= x 2e-55) (not (<= x 2e+74)))))
(* y 2.0)
(* x -2.0)))
double code(double x, double y) {
double tmp;
if ((x <= -4.5e+15) || (!(x <= 2.45e-91) && ((x <= 2e-55) || !(x <= 2e+74)))) {
tmp = y * 2.0;
} 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 <= (-4.5d+15)) .or. (.not. (x <= 2.45d-91)) .and. (x <= 2d-55) .or. (.not. (x <= 2d+74))) then
tmp = y * 2.0d0
else
tmp = x * (-2.0d0)
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if ((x <= -4.5e+15) || (!(x <= 2.45e-91) && ((x <= 2e-55) || !(x <= 2e+74)))) {
tmp = y * 2.0;
} else {
tmp = x * -2.0;
}
return tmp;
}
def code(x, y): tmp = 0 if (x <= -4.5e+15) or (not (x <= 2.45e-91) and ((x <= 2e-55) or not (x <= 2e+74))): tmp = y * 2.0 else: tmp = x * -2.0 return tmp
function code(x, y) tmp = 0.0 if ((x <= -4.5e+15) || (!(x <= 2.45e-91) && ((x <= 2e-55) || !(x <= 2e+74)))) tmp = Float64(y * 2.0); else tmp = Float64(x * -2.0); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if ((x <= -4.5e+15) || (~((x <= 2.45e-91)) && ((x <= 2e-55) || ~((x <= 2e+74))))) tmp = y * 2.0; else tmp = x * -2.0; end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[x, -4.5e+15], And[N[Not[LessEqual[x, 2.45e-91]], $MachinePrecision], Or[LessEqual[x, 2e-55], N[Not[LessEqual[x, 2e+74]], $MachinePrecision]]]], N[(y * 2.0), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;x \leq -4.5 \cdot 10^{+15} \lor \neg \left(x \leq 2.45 \cdot 10^{-91}\right) \land \left(x \leq 2 \cdot 10^{-55} \lor \neg \left(x \leq 2 \cdot 10^{+74}\right)\right):\\
\;\;\;\;y \cdot 2\\
\mathbf{else}:\\
\;\;\;\;x \cdot -2\\
\end{array}
\end{array}
if x < -4.5e15 or 2.4499999999999999e-91 < x < 1.99999999999999999e-55 or 1.9999999999999999e74 < x Initial program 74.2%
associate-/l*73.6%
associate-*l/74.4%
div-sub74.4%
*-inverses74.4%
metadata-eval74.4%
sub-neg74.4%
metadata-eval74.4%
metadata-eval74.4%
Simplified74.4%
Taylor expanded in x around inf 84.7%
if -4.5e15 < x < 2.4499999999999999e-91 or 1.99999999999999999e-55 < x < 1.9999999999999999e74Initial program 81.7%
associate-*l/78.8%
Simplified78.8%
Taylor expanded in x around 0 82.7%
*-commutative82.7%
Simplified82.7%
Final simplification83.6%
(FPCore (x y) :precision binary64 (if (or (<= y -8.5e-81) (not (<= y 2.9e-181))) (* (/ x (+ (/ x y) -1.0)) 2.0) (* y 2.0)))
double code(double x, double y) {
double tmp;
if ((y <= -8.5e-81) || !(y <= 2.9e-181)) {
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 <= (-8.5d-81)) .or. (.not. (y <= 2.9d-181))) 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 <= -8.5e-81) || !(y <= 2.9e-181)) {
tmp = (x / ((x / y) + -1.0)) * 2.0;
} else {
tmp = y * 2.0;
}
return tmp;
}
def code(x, y): tmp = 0 if (y <= -8.5e-81) or not (y <= 2.9e-181): tmp = (x / ((x / y) + -1.0)) * 2.0 else: tmp = y * 2.0 return tmp
function code(x, y) tmp = 0.0 if ((y <= -8.5e-81) || !(y <= 2.9e-181)) 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 <= -8.5e-81) || ~((y <= 2.9e-181))) 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, -8.5e-81], N[Not[LessEqual[y, 2.9e-181]], $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 -8.5 \cdot 10^{-81} \lor \neg \left(y \leq 2.9 \cdot 10^{-181}\right):\\
\;\;\;\;\frac{x}{\frac{x}{y} + -1} \cdot 2\\
\mathbf{else}:\\
\;\;\;\;y \cdot 2\\
\end{array}
\end{array}
if y < -8.5000000000000001e-81 or 2.8999999999999998e-181 < y Initial program 78.2%
associate-/l*98.9%
associate-*l/99.4%
div-sub99.4%
*-inverses99.4%
metadata-eval99.4%
sub-neg99.4%
metadata-eval99.4%
metadata-eval99.4%
Simplified99.4%
if -8.5000000000000001e-81 < y < 2.8999999999999998e-181Initial program 78.2%
associate-/l*55.8%
associate-*l/55.8%
div-sub55.8%
*-inverses55.8%
metadata-eval55.8%
sub-neg55.8%
metadata-eval55.8%
metadata-eval55.8%
Simplified55.8%
Taylor expanded in x around inf 91.5%
Final simplification97.3%
(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 78.2%
associate-*l/88.4%
Simplified88.4%
Taylor expanded in x around 0 51.9%
*-commutative51.9%
Simplified51.9%
Final simplification51.9%
(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 2024026
(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)))