
(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 (<= x -3.35e-126) (not (<= x 1e-12))) (/ y (/ (- 1.0 (/ y x)) 2.0)) (/ (* x 2.0) (/ (- x y) y))))
double code(double x, double y) {
double tmp;
if ((x <= -3.35e-126) || !(x <= 1e-12)) {
tmp = y / ((1.0 - (y / x)) / 2.0);
} else {
tmp = (x * 2.0) / ((x - y) / y);
}
return tmp;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: tmp
if ((x <= (-3.35d-126)) .or. (.not. (x <= 1d-12))) then
tmp = y / ((1.0d0 - (y / x)) / 2.0d0)
else
tmp = (x * 2.0d0) / ((x - y) / y)
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if ((x <= -3.35e-126) || !(x <= 1e-12)) {
tmp = y / ((1.0 - (y / x)) / 2.0);
} else {
tmp = (x * 2.0) / ((x - y) / y);
}
return tmp;
}
def code(x, y): tmp = 0 if (x <= -3.35e-126) or not (x <= 1e-12): tmp = y / ((1.0 - (y / x)) / 2.0) else: tmp = (x * 2.0) / ((x - y) / y) return tmp
function code(x, y) tmp = 0.0 if ((x <= -3.35e-126) || !(x <= 1e-12)) tmp = Float64(y / Float64(Float64(1.0 - Float64(y / x)) / 2.0)); else tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y)); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if ((x <= -3.35e-126) || ~((x <= 1e-12))) tmp = y / ((1.0 - (y / x)) / 2.0); else tmp = (x * 2.0) / ((x - y) / y); end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[x, -3.35e-126], N[Not[LessEqual[x, 1e-12]], $MachinePrecision]], N[(y / N[(N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;x \leq -3.35 \cdot 10^{-126} \lor \neg \left(x \leq 10^{-12}\right):\\
\;\;\;\;\frac{y}{\frac{1 - \frac{y}{x}}{2}}\\
\mathbf{else}:\\
\;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\
\end{array}
\end{array}
if x < -3.35000000000000003e-126 or 9.9999999999999998e-13 < x Initial program 83.6%
*-commutative83.6%
associate-/l*99.9%
associate-/r*99.9%
div-sub100.0%
*-inverses100.0%
Simplified100.0%
if -3.35000000000000003e-126 < x < 9.9999999999999998e-13Initial program 80.1%
associate-/l*100.0%
Simplified100.0%
Final simplification100.0%
(FPCore (x y) :precision binary64 (if (or (<= x -2.3e-161) (not (<= x 1.25e-100))) (/ y (/ (- 1.0 (/ y x)) 2.0)) (* x -2.0)))
double code(double x, double y) {
double tmp;
if ((x <= -2.3e-161) || !(x <= 1.25e-100)) {
tmp = y / ((1.0 - (y / x)) / 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 <= (-2.3d-161)) .or. (.not. (x <= 1.25d-100))) then
tmp = y / ((1.0d0 - (y / x)) / 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 <= -2.3e-161) || !(x <= 1.25e-100)) {
tmp = y / ((1.0 - (y / x)) / 2.0);
} else {
tmp = x * -2.0;
}
return tmp;
}
def code(x, y): tmp = 0 if (x <= -2.3e-161) or not (x <= 1.25e-100): tmp = y / ((1.0 - (y / x)) / 2.0) else: tmp = x * -2.0 return tmp
function code(x, y) tmp = 0.0 if ((x <= -2.3e-161) || !(x <= 1.25e-100)) tmp = Float64(y / Float64(Float64(1.0 - Float64(y / x)) / 2.0)); else tmp = Float64(x * -2.0); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if ((x <= -2.3e-161) || ~((x <= 1.25e-100))) tmp = y / ((1.0 - (y / x)) / 2.0); else tmp = x * -2.0; end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[x, -2.3e-161], N[Not[LessEqual[x, 1.25e-100]], $MachinePrecision]], N[(y / N[(N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x * -2.0), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;x \leq -2.3 \cdot 10^{-161} \lor \neg \left(x \leq 1.25 \cdot 10^{-100}\right):\\
\;\;\;\;\frac{y}{\frac{1 - \frac{y}{x}}{2}}\\
\mathbf{else}:\\
\;\;\;\;x \cdot -2\\
\end{array}
\end{array}
if x < -2.3e-161 or 1.25e-100 < x Initial program 84.7%
*-commutative84.7%
associate-/l*97.9%
associate-/r*97.9%
div-sub97.9%
*-inverses97.9%
Simplified97.9%
if -2.3e-161 < x < 1.25e-100Initial program 75.7%
*-commutative75.7%
associate-/l*65.3%
associate-/r*65.3%
div-sub65.3%
*-inverses65.3%
Simplified65.3%
Taylor expanded in y around inf 90.8%
*-commutative90.8%
Simplified90.8%
Final simplification96.0%
(FPCore (x y) :precision binary64 (if (or (<= y -3.2e-70) (not (<= y 1.7e-18))) (* x -2.0) (* y 2.0)))
double code(double x, double y) {
double tmp;
if ((y <= -3.2e-70) || !(y <= 1.7e-18)) {
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 <= (-3.2d-70)) .or. (.not. (y <= 1.7d-18))) 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 <= -3.2e-70) || !(y <= 1.7e-18)) {
tmp = x * -2.0;
} else {
tmp = y * 2.0;
}
return tmp;
}
def code(x, y): tmp = 0 if (y <= -3.2e-70) or not (y <= 1.7e-18): tmp = x * -2.0 else: tmp = y * 2.0 return tmp
function code(x, y) tmp = 0.0 if ((y <= -3.2e-70) || !(y <= 1.7e-18)) 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 <= -3.2e-70) || ~((y <= 1.7e-18))) tmp = x * -2.0; else tmp = y * 2.0; end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[y, -3.2e-70], N[Not[LessEqual[y, 1.7e-18]], $MachinePrecision]], N[(x * -2.0), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -3.2 \cdot 10^{-70} \lor \neg \left(y \leq 1.7 \cdot 10^{-18}\right):\\
\;\;\;\;x \cdot -2\\
\mathbf{else}:\\
\;\;\;\;y \cdot 2\\
\end{array}
\end{array}
if y < -3.1999999999999997e-70 or 1.70000000000000001e-18 < y Initial program 81.8%
*-commutative81.8%
associate-/l*79.7%
associate-/r*79.7%
div-sub79.7%
*-inverses79.7%
Simplified79.7%
Taylor expanded in y around inf 76.7%
*-commutative76.7%
Simplified76.7%
if -3.1999999999999997e-70 < y < 1.70000000000000001e-18Initial program 82.8%
*-commutative82.8%
associate-/l*99.9%
associate-/r*99.9%
div-sub100.0%
*-inverses100.0%
Simplified100.0%
Taylor expanded in y around 0 80.7%
Final simplification78.6%
(FPCore (x y) :precision binary64 (* y 2.0))
double code(double x, double y) {
return y * 2.0;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
code = y * 2.0d0
end function
public static double code(double x, double y) {
return y * 2.0;
}
def code(x, y): return y * 2.0
function code(x, y) return Float64(y * 2.0) end
function tmp = code(x, y) tmp = y * 2.0; end
code[x_, y_] := N[(y * 2.0), $MachinePrecision]
\begin{array}{l}
\\
y \cdot 2
\end{array}
Initial program 82.2%
*-commutative82.2%
associate-/l*89.0%
associate-/r*89.0%
div-sub89.0%
*-inverses89.0%
Simplified89.0%
Taylor expanded in y around 0 50.8%
Final simplification50.8%
(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 2023308
(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)))