
(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 5 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 -2e+21) (not (<= y 1.85e-51))) (/ (* x 2.0) (/ (- x y) y)) (* y (/ (* x 2.0) (- x y)))))
double code(double x, double y) {
double tmp;
if ((y <= -2e+21) || !(y <= 1.85e-51)) {
tmp = (x * 2.0) / ((x - y) / y);
} 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 <= (-2d+21)) .or. (.not. (y <= 1.85d-51))) then
tmp = (x * 2.0d0) / ((x - y) / y)
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 <= -2e+21) || !(y <= 1.85e-51)) {
tmp = (x * 2.0) / ((x - y) / y);
} else {
tmp = y * ((x * 2.0) / (x - y));
}
return tmp;
}
def code(x, y): tmp = 0 if (y <= -2e+21) or not (y <= 1.85e-51): tmp = (x * 2.0) / ((x - y) / y) else: tmp = y * ((x * 2.0) / (x - y)) return tmp
function code(x, y) tmp = 0.0 if ((y <= -2e+21) || !(y <= 1.85e-51)) tmp = Float64(Float64(x * 2.0) / Float64(Float64(x - y) / y)); 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 <= -2e+21) || ~((y <= 1.85e-51))) tmp = (x * 2.0) / ((x - y) / y); else tmp = y * ((x * 2.0) / (x - y)); end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[y, -2e+21], N[Not[LessEqual[y, 1.85e-51]], $MachinePrecision]], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $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 -2 \cdot 10^{+21} \lor \neg \left(y \leq 1.85 \cdot 10^{-51}\right):\\
\;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\
\mathbf{else}:\\
\;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\
\end{array}
\end{array}
if y < -2e21 or 1.84999999999999987e-51 < y Initial program 81.6%
associate-/l*100.0%
Simplified100.0%
if -2e21 < y < 1.84999999999999987e-51Initial program 71.5%
associate-*l/99.9%
Simplified99.9%
Final simplification100.0%
(FPCore (x y) :precision binary64 (if (or (<= y -3.8e+59) (not (<= y 1.02e+203))) (* x -2.0) (* y (/ (* x 2.0) (- x y)))))
double code(double x, double y) {
double tmp;
if ((y <= -3.8e+59) || !(y <= 1.02e+203)) {
tmp = x * -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 <= (-3.8d+59)) .or. (.not. (y <= 1.02d+203))) then
tmp = x * (-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 <= -3.8e+59) || !(y <= 1.02e+203)) {
tmp = x * -2.0;
} else {
tmp = y * ((x * 2.0) / (x - y));
}
return tmp;
}
def code(x, y): tmp = 0 if (y <= -3.8e+59) or not (y <= 1.02e+203): tmp = x * -2.0 else: tmp = y * ((x * 2.0) / (x - y)) return tmp
function code(x, y) tmp = 0.0 if ((y <= -3.8e+59) || !(y <= 1.02e+203)) tmp = Float64(x * -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 <= -3.8e+59) || ~((y <= 1.02e+203))) tmp = x * -2.0; else tmp = y * ((x * 2.0) / (x - y)); end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[y, -3.8e+59], N[Not[LessEqual[y, 1.02e+203]], $MachinePrecision]], N[(x * -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 -3.8 \cdot 10^{+59} \lor \neg \left(y \leq 1.02 \cdot 10^{+203}\right):\\
\;\;\;\;x \cdot -2\\
\mathbf{else}:\\
\;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\
\end{array}
\end{array}
if y < -3.8000000000000001e59 or 1.02e203 < y Initial program 76.2%
associate-*l/59.6%
Simplified59.6%
Taylor expanded in x around 0 90.8%
if -3.8000000000000001e59 < y < 1.02e203Initial program 77.0%
associate-*l/98.0%
Simplified98.0%
Final simplification96.0%
(FPCore (x y) :precision binary64 (if (or (<= y -2e+21) (not (<= y 6e-45))) (/ 2.0 (/ (+ (/ x y) -1.0) x)) (* y (/ (* x 2.0) (- x y)))))
double code(double x, double y) {
double tmp;
if ((y <= -2e+21) || !(y <= 6e-45)) {
tmp = 2.0 / (((x / y) + -1.0) / x);
} 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 <= (-2d+21)) .or. (.not. (y <= 6d-45))) then
tmp = 2.0d0 / (((x / y) + (-1.0d0)) / x)
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 <= -2e+21) || !(y <= 6e-45)) {
tmp = 2.0 / (((x / y) + -1.0) / x);
} else {
tmp = y * ((x * 2.0) / (x - y));
}
return tmp;
}
def code(x, y): tmp = 0 if (y <= -2e+21) or not (y <= 6e-45): tmp = 2.0 / (((x / y) + -1.0) / x) else: tmp = y * ((x * 2.0) / (x - y)) return tmp
function code(x, y) tmp = 0.0 if ((y <= -2e+21) || !(y <= 6e-45)) tmp = Float64(2.0 / Float64(Float64(Float64(x / y) + -1.0) / x)); 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 <= -2e+21) || ~((y <= 6e-45))) tmp = 2.0 / (((x / y) + -1.0) / x); else tmp = y * ((x * 2.0) / (x - y)); end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[y, -2e+21], N[Not[LessEqual[y, 6e-45]], $MachinePrecision]], N[(2.0 / N[(N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $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 -2 \cdot 10^{+21} \lor \neg \left(y \leq 6 \cdot 10^{-45}\right):\\
\;\;\;\;\frac{2}{\frac{\frac{x}{y} + -1}{x}}\\
\mathbf{else}:\\
\;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\
\end{array}
\end{array}
if y < -2e21 or 6.00000000000000022e-45 < y Initial program 80.9%
associate-*l/74.9%
Simplified74.9%
associate-/r/100.0%
*-commutative100.0%
associate-/l*99.8%
div-sub99.8%
*-inverses99.8%
sub-neg99.8%
metadata-eval99.8%
Applied egg-rr99.8%
if -2e21 < y < 6.00000000000000022e-45Initial program 72.6%
associate-*l/100.0%
Simplified100.0%
Final simplification99.9%
(FPCore (x y) :precision binary64 (if (or (<= y -1.45e-58) (not (<= y 5.5e-45))) (* x -2.0) (* y 2.0)))
double code(double x, double y) {
double tmp;
if ((y <= -1.45e-58) || !(y <= 5.5e-45)) {
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 <= (-1.45d-58)) .or. (.not. (y <= 5.5d-45))) 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 <= -1.45e-58) || !(y <= 5.5e-45)) {
tmp = x * -2.0;
} else {
tmp = y * 2.0;
}
return tmp;
}
def code(x, y): tmp = 0 if (y <= -1.45e-58) or not (y <= 5.5e-45): tmp = x * -2.0 else: tmp = y * 2.0 return tmp
function code(x, y) tmp = 0.0 if ((y <= -1.45e-58) || !(y <= 5.5e-45)) 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 <= -1.45e-58) || ~((y <= 5.5e-45))) tmp = x * -2.0; else tmp = y * 2.0; end tmp_2 = tmp; end
code[x_, y_] := If[Or[LessEqual[y, -1.45e-58], N[Not[LessEqual[y, 5.5e-45]], $MachinePrecision]], N[(x * -2.0), $MachinePrecision], N[(y * 2.0), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -1.45 \cdot 10^{-58} \lor \neg \left(y \leq 5.5 \cdot 10^{-45}\right):\\
\;\;\;\;x \cdot -2\\
\mathbf{else}:\\
\;\;\;\;y \cdot 2\\
\end{array}
\end{array}
if y < -1.44999999999999995e-58 or 5.5000000000000003e-45 < y Initial program 82.3%
associate-*l/77.8%
Simplified77.8%
Taylor expanded in x around 0 79.0%
if -1.44999999999999995e-58 < y < 5.5000000000000003e-45Initial program 69.4%
associate-*l/100.0%
Simplified100.0%
Taylor expanded in x around inf 80.5%
Final simplification79.6%
(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 76.8%
associate-*l/87.2%
Simplified87.2%
Taylor expanded in x around 0 53.7%
Final simplification53.7%
(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 2024031
(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)))