
(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 -4.7e+98) (not (<= y 1e-64))) (/ x (- (* 0.5 (/ x y)) 0.5)) (* (/ y (- 1.0 (/ y x))) 2.0)))
double code(double x, double y) {
double tmp;
if ((y <= -4.7e+98) || !(y <= 1e-64)) {
tmp = x / ((0.5 * (x / y)) - 0.5);
} else {
tmp = (y / (1.0 - (y / 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 ((y <= (-4.7d+98)) .or. (.not. (y <= 1d-64))) then
tmp = x / ((0.5d0 * (x / y)) - 0.5d0)
else
tmp = (y / (1.0d0 - (y / x))) * 2.0d0
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if ((y <= -4.7e+98) || !(y <= 1e-64)) {
tmp = x / ((0.5 * (x / y)) - 0.5);
} else {
tmp = (y / (1.0 - (y / x))) * 2.0;
}
return tmp;
}
def code(x, y): tmp = 0 if (y <= -4.7e+98) or not (y <= 1e-64): tmp = x / ((0.5 * (x / y)) - 0.5) else: tmp = (y / (1.0 - (y / x))) * 2.0 return tmp
function code(x, y) tmp = 0.0 if ((y <= -4.7e+98) || !(y <= 1e-64)) tmp = Float64(x / Float64(Float64(0.5 * Float64(x / y)) - 0.5)); else tmp = Float64(Float64(y / Float64(1.0 - Float64(y / x))) * 2.0); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if ((y <= -4.7e+98) || ~((y <= 1e-64))) tmp = x / ((0.5 * (x / y)) - 0.5); else tmp = (y / (1.0 - (y / x))) * 2.0; end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[y, -4.7e+98], N[Not[LessEqual[y, 1e-64]], $MachinePrecision]], N[(x / N[(N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -4.7 \cdot 10^{+98} \lor \neg \left(y \leq 10^{-64}\right):\\
\;\;\;\;\frac{x}{0.5 \cdot \frac{x}{y} - 0.5}\\
\mathbf{else}:\\
\;\;\;\;\frac{y}{1 - \frac{y}{x}} \cdot 2\\
\end{array}
\end{array}
if y < -4.6999999999999997e98 or 9.99999999999999965e-65 < y Initial program 77.3%
associate-*l/79.2%
associate-/l*79.2%
associate-/r/99.9%
Simplified99.9%
Taylor expanded in x around 0 99.9%
if -4.6999999999999997e98 < y < 9.99999999999999965e-65Initial program 75.9%
*-commutative75.9%
associate-/l*99.3%
associate-/r*100.0%
associate-/r/100.0%
div-sub100.0%
*-inverses100.0%
Simplified100.0%
Final simplification100.0%
(FPCore (x y) :precision binary64 (if (or (<= x -2.65e-126) (not (<= x 1.4e-236))) (* (/ x (- x y)) (* y 2.0)) (/ x -0.5)))
double code(double x, double y) {
double tmp;
if ((x <= -2.65e-126) || !(x <= 1.4e-236)) {
tmp = (x / (x - y)) * (y * 2.0);
} else {
tmp = x / -0.5;
}
return tmp;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: tmp
if ((x <= (-2.65d-126)) .or. (.not. (x <= 1.4d-236))) then
tmp = (x / (x - y)) * (y * 2.0d0)
else
tmp = x / (-0.5d0)
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if ((x <= -2.65e-126) || !(x <= 1.4e-236)) {
tmp = (x / (x - y)) * (y * 2.0);
} else {
tmp = x / -0.5;
}
return tmp;
}
def code(x, y): tmp = 0 if (x <= -2.65e-126) or not (x <= 1.4e-236): tmp = (x / (x - y)) * (y * 2.0) else: tmp = x / -0.5 return tmp
function code(x, y) tmp = 0.0 if ((x <= -2.65e-126) || !(x <= 1.4e-236)) tmp = Float64(Float64(x / Float64(x - y)) * Float64(y * 2.0)); else tmp = Float64(x / -0.5); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if ((x <= -2.65e-126) || ~((x <= 1.4e-236))) tmp = (x / (x - y)) * (y * 2.0); else tmp = x / -0.5; end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[x, -2.65e-126], N[Not[LessEqual[x, 1.4e-236]], $MachinePrecision]], N[(N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision] * N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(x / -0.5), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;x \leq -2.65 \cdot 10^{-126} \lor \neg \left(x \leq 1.4 \cdot 10^{-236}\right):\\
\;\;\;\;\frac{x}{x - y} \cdot \left(y \cdot 2\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{x}{-0.5}\\
\end{array}
\end{array}
if x < -2.64999999999999997e-126 or 1.39999999999999993e-236 < x Initial program 76.7%
associate-*l/97.2%
associate-/l*97.7%
associate-/r/88.0%
Simplified88.0%
associate-/l/88.0%
associate-/r/97.7%
Applied egg-rr97.7%
if -2.64999999999999997e-126 < x < 1.39999999999999993e-236Initial program 76.0%
associate-*l/54.2%
associate-/l*54.2%
associate-/r/100.0%
Simplified100.0%
Taylor expanded in x around 0 91.5%
Final simplification96.6%
(FPCore (x y)
:precision binary64
(if (<= y -1.35e-23)
(/ x -0.5)
(if (or (<= y 4e-15) (and (not (<= y 3e+18)) (<= y 3.2e+50)))
(* y 2.0)
(/ x -0.5))))
double code(double x, double y) {
double tmp;
if (y <= -1.35e-23) {
tmp = x / -0.5;
} else if ((y <= 4e-15) || (!(y <= 3e+18) && (y <= 3.2e+50))) {
tmp = y * 2.0;
} else {
tmp = x / -0.5;
}
return tmp;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: tmp
if (y <= (-1.35d-23)) then
tmp = x / (-0.5d0)
else if ((y <= 4d-15) .or. (.not. (y <= 3d+18)) .and. (y <= 3.2d+50)) then
tmp = y * 2.0d0
else
tmp = x / (-0.5d0)
end if
code = tmp
end function
public static double code(double x, double y) {
double tmp;
if (y <= -1.35e-23) {
tmp = x / -0.5;
} else if ((y <= 4e-15) || (!(y <= 3e+18) && (y <= 3.2e+50))) {
tmp = y * 2.0;
} else {
tmp = x / -0.5;
}
return tmp;
}
def code(x, y): tmp = 0 if y <= -1.35e-23: tmp = x / -0.5 elif (y <= 4e-15) or (not (y <= 3e+18) and (y <= 3.2e+50)): tmp = y * 2.0 else: tmp = x / -0.5 return tmp
function code(x, y) tmp = 0.0 if (y <= -1.35e-23) tmp = Float64(x / -0.5); elseif ((y <= 4e-15) || (!(y <= 3e+18) && (y <= 3.2e+50))) tmp = Float64(y * 2.0); else tmp = Float64(x / -0.5); end return tmp end
function tmp_2 = code(x, y) tmp = 0.0; if (y <= -1.35e-23) tmp = x / -0.5; elseif ((y <= 4e-15) || (~((y <= 3e+18)) && (y <= 3.2e+50))) tmp = y * 2.0; else tmp = x / -0.5; end tmp_2 = tmp; end
code[x_, y_] := If[LessEqual[y, -1.35e-23], N[(x / -0.5), $MachinePrecision], If[Or[LessEqual[y, 4e-15], And[N[Not[LessEqual[y, 3e+18]], $MachinePrecision], LessEqual[y, 3.2e+50]]], N[(y * 2.0), $MachinePrecision], N[(x / -0.5), $MachinePrecision]]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -1.35 \cdot 10^{-23}:\\
\;\;\;\;\frac{x}{-0.5}\\
\mathbf{elif}\;y \leq 4 \cdot 10^{-15} \lor \neg \left(y \leq 3 \cdot 10^{+18}\right) \land y \leq 3.2 \cdot 10^{+50}:\\
\;\;\;\;y \cdot 2\\
\mathbf{else}:\\
\;\;\;\;\frac{x}{-0.5}\\
\end{array}
\end{array}
if y < -1.34999999999999992e-23 or 4.0000000000000003e-15 < y < 3e18 or 3.19999999999999983e50 < y Initial program 77.0%
associate-*l/79.5%
associate-/l*80.3%
associate-/r/99.2%
Simplified99.2%
Taylor expanded in x around 0 79.1%
if -1.34999999999999992e-23 < y < 4.0000000000000003e-15 or 3e18 < y < 3.19999999999999983e50Initial program 76.1%
*-commutative76.1%
associate-/l*100.0%
associate-/r*100.0%
associate-/r/100.0%
div-sub100.0%
*-inverses100.0%
Simplified100.0%
Taylor expanded in y around 0 81.4%
Final simplification80.3%
(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 76.5%
*-commutative76.5%
associate-/l*89.3%
associate-/r*89.7%
associate-/r/89.7%
div-sub89.7%
*-inverses89.7%
Simplified89.7%
Taylor expanded in y around 0 51.5%
Final simplification51.5%
(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 2023271
(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)))