
(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))) (t_1 (/ (* x 2.0) (+ (/ x y) -1.0))))
(if (<= t_0 -1e-9)
t_1
(if (<= t_0 -2e-305)
t_0
(if (<= t_0 0.0)
t_1
(if (<= t_0 2e-22) t_0 (* x (* 2.0 (/ y (- x y))))))))))
double code(double x, double y) {
double t_0 = ((x * 2.0) * y) / (x - y);
double t_1 = (x * 2.0) / ((x / y) + -1.0);
double tmp;
if (t_0 <= -1e-9) {
tmp = t_1;
} else if (t_0 <= -2e-305) {
tmp = t_0;
} else if (t_0 <= 0.0) {
tmp = t_1;
} else if (t_0 <= 2e-22) {
tmp = t_0;
} else {
tmp = x * (2.0 * (y / (x - y)));
}
return tmp;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: t_0
real(8) :: t_1
real(8) :: tmp
t_0 = ((x * 2.0d0) * y) / (x - y)
t_1 = (x * 2.0d0) / ((x / y) + (-1.0d0))
if (t_0 <= (-1d-9)) then
tmp = t_1
else if (t_0 <= (-2d-305)) then
tmp = t_0
else if (t_0 <= 0.0d0) then
tmp = t_1
else if (t_0 <= 2d-22) then
tmp = t_0
else
tmp = x * (2.0d0 * (y / (x - y)))
end if
code = tmp
end function
public static double code(double x, double y) {
double t_0 = ((x * 2.0) * y) / (x - y);
double t_1 = (x * 2.0) / ((x / y) + -1.0);
double tmp;
if (t_0 <= -1e-9) {
tmp = t_1;
} else if (t_0 <= -2e-305) {
tmp = t_0;
} else if (t_0 <= 0.0) {
tmp = t_1;
} else if (t_0 <= 2e-22) {
tmp = t_0;
} else {
tmp = x * (2.0 * (y / (x - y)));
}
return tmp;
}
def code(x, y): t_0 = ((x * 2.0) * y) / (x - y) t_1 = (x * 2.0) / ((x / y) + -1.0) tmp = 0 if t_0 <= -1e-9: tmp = t_1 elif t_0 <= -2e-305: tmp = t_0 elif t_0 <= 0.0: tmp = t_1 elif t_0 <= 2e-22: tmp = t_0 else: tmp = x * (2.0 * (y / (x - y))) return tmp
function code(x, y) t_0 = Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y)) t_1 = Float64(Float64(x * 2.0) / Float64(Float64(x / y) + -1.0)) tmp = 0.0 if (t_0 <= -1e-9) tmp = t_1; elseif (t_0 <= -2e-305) tmp = t_0; elseif (t_0 <= 0.0) tmp = t_1; elseif (t_0 <= 2e-22) tmp = t_0; else tmp = Float64(x * Float64(2.0 * Float64(y / Float64(x - y)))); end return tmp end
function tmp_2 = code(x, y) t_0 = ((x * 2.0) * y) / (x - y); t_1 = (x * 2.0) / ((x / y) + -1.0); tmp = 0.0; if (t_0 <= -1e-9) tmp = t_1; elseif (t_0 <= -2e-305) tmp = t_0; elseif (t_0 <= 0.0) tmp = t_1; elseif (t_0 <= 2e-22) tmp = t_0; else tmp = x * (2.0 * (y / (x - y))); 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]}, Block[{t$95$1 = N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-9], t$95$1, If[LessEqual[t$95$0, -2e-305], t$95$0, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e-22], t$95$0, N[(x * N[(2.0 * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \frac{\left(x \cdot 2\right) \cdot y}{x - y}\\
t_1 := \frac{x \cdot 2}{\frac{x}{y} + -1}\\
\mathbf{if}\;t\_0 \leq -1 \cdot 10^{-9}:\\
\;\;\;\;t\_1\\
\mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-305}:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;t\_0 \leq 0:\\
\;\;\;\;t\_1\\
\mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-22}:\\
\;\;\;\;t\_0\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\
\end{array}
\end{array}
if (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -1.00000000000000006e-9 or -1.99999999999999999e-305 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -0.0Initial program 33.9%
associate-/l*99.8%
associate-*l*99.8%
Simplified99.8%
associate-*r*99.8%
clear-num99.7%
un-div-inv99.9%
Applied egg-rr99.9%
div-sub99.9%
sub-neg99.9%
*-inverses99.9%
metadata-eval99.9%
Simplified99.9%
if -1.00000000000000006e-9 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < -1.99999999999999999e-305 or -0.0 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) < 2.0000000000000001e-22Initial program 98.7%
if 2.0000000000000001e-22 < (/.f64 (*.f64 (*.f64 x #s(literal 2 binary64)) y) (-.f64 x y)) Initial program 50.2%
associate-/l*100.0%
associate-*l*100.0%
Simplified100.0%
(FPCore (x y) :precision binary64 (if (or (<= x -9e+163) (not (<= x 1.65e+189))) (* 2.0 y) (* x (* 2.0 (/ y (- x y))))))
double code(double x, double y) {
double tmp;
if ((x <= -9e+163) || !(x <= 1.65e+189)) {
tmp = 2.0 * y;
} else {
tmp = x * (2.0 * (y / (x - y)));
}
return tmp;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: tmp
if ((x <= (-9d+163)) .or. (.not. (x <= 1.65d+189))) then
tmp = 2.0d0 * y
else
tmp = x * (2.0d0 * (y / (x - y)))
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if ((x <= -9e+163) || !(x <= 1.65e+189)) {
tmp = 2.0 * y;
} else {
tmp = x * (2.0 * (y / (x - y)));
}
return tmp;
}
def code(x, y): tmp = 0 if (x <= -9e+163) or not (x <= 1.65e+189): tmp = 2.0 * y else: tmp = x * (2.0 * (y / (x - y))) return tmp
function code(x, y) tmp = 0.0 if ((x <= -9e+163) || !(x <= 1.65e+189)) tmp = Float64(2.0 * y); else tmp = Float64(x * Float64(2.0 * Float64(y / Float64(x - y)))); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if ((x <= -9e+163) || ~((x <= 1.65e+189))) tmp = 2.0 * y; else tmp = x * (2.0 * (y / (x - y))); end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[x, -9e+163], N[Not[LessEqual[x, 1.65e+189]], $MachinePrecision]], N[(2.0 * y), $MachinePrecision], N[(x * N[(2.0 * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;x \leq -9 \cdot 10^{+163} \lor \neg \left(x \leq 1.65 \cdot 10^{+189}\right):\\
\;\;\;\;2 \cdot y\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(2 \cdot \frac{y}{x - y}\right)\\
\end{array}
\end{array}
if x < -8.99999999999999976e163 or 1.6500000000000001e189 < x Initial program 75.7%
associate-/l*58.6%
associate-*l*58.6%
Simplified58.6%
Taylor expanded in x around inf 97.5%
*-commutative97.5%
Simplified97.5%
if -8.99999999999999976e163 < x < 1.6500000000000001e189Initial program 75.8%
associate-/l*94.2%
associate-*l*94.2%
Simplified94.2%
Final simplification94.8%
(FPCore (x y) :precision binary64 (if (or (<= y -4.4e+83) (not (<= y 52.0))) (* x -2.0) (* 2.0 y)))
double code(double x, double y) {
double tmp;
if ((y <= -4.4e+83) || !(y <= 52.0)) {
tmp = x * -2.0;
} 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 <= (-4.4d+83)) .or. (.not. (y <= 52.0d0))) then
tmp = x * (-2.0d0)
else
tmp = 2.0d0 * y
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if ((y <= -4.4e+83) || !(y <= 52.0)) {
tmp = x * -2.0;
} else {
tmp = 2.0 * y;
}
return tmp;
}
def code(x, y): tmp = 0 if (y <= -4.4e+83) or not (y <= 52.0): tmp = x * -2.0 else: tmp = 2.0 * y return tmp
function code(x, y) tmp = 0.0 if ((y <= -4.4e+83) || !(y <= 52.0)) tmp = Float64(x * -2.0); else tmp = Float64(2.0 * y); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if ((y <= -4.4e+83) || ~((y <= 52.0))) tmp = x * -2.0; else tmp = 2.0 * y; end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[y, -4.4e+83], N[Not[LessEqual[y, 52.0]], $MachinePrecision]], N[(x * -2.0), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -4.4 \cdot 10^{+83} \lor \neg \left(y \leq 52\right):\\
\;\;\;\;x \cdot -2\\
\mathbf{else}:\\
\;\;\;\;2 \cdot y\\
\end{array}
\end{array}
if y < -4.39999999999999997e83 or 52 < y Initial program 72.5%
associate-/l*100.0%
associate-*l*100.0%
Simplified100.0%
Taylor expanded in y around inf 87.2%
if -4.39999999999999997e83 < y < 52Initial program 78.1%
associate-/l*78.7%
associate-*l*78.7%
Simplified78.7%
Taylor expanded in x around inf 76.9%
*-commutative76.9%
Simplified76.9%
Final simplification81.3%
(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 75.8%
associate-/l*87.6%
associate-*l*87.6%
Simplified87.6%
Taylor expanded in y around inf 51.4%
(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 2024135
(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)))