
(FPCore (x y) :precision binary64 (/ (* x y) (+ y 1.0)))
double code(double x, double y) {
return (x * y) / (y + 1.0);
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
code = (x * y) / (y + 1.0d0)
end function
public static double code(double x, double y) {
return (x * y) / (y + 1.0);
}
def code(x, y): return (x * y) / (y + 1.0)
function code(x, y) return Float64(Float64(x * y) / Float64(y + 1.0)) end
function tmp = code(x, y) tmp = (x * y) / (y + 1.0); end
code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\frac{x \cdot y}{y + 1}
\end{array}
Sampling outcomes in binary64 precision:
Herbie found 5 alternatives:
| Alternative | Accuracy | Speedup |
|---|
(FPCore (x y) :precision binary64 (/ (* x y) (+ y 1.0)))
double code(double x, double y) {
return (x * y) / (y + 1.0);
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
code = (x * y) / (y + 1.0d0)
end function
public static double code(double x, double y) {
return (x * y) / (y + 1.0);
}
def code(x, y): return (x * y) / (y + 1.0)
function code(x, y) return Float64(Float64(x * y) / Float64(y + 1.0)) end
function tmp = code(x, y) tmp = (x * y) / (y + 1.0); end
code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\frac{x \cdot y}{y + 1}
\end{array}
(FPCore (x y) :precision binary64 (let* ((t_0 (- x (/ x y)))) (if (<= y -1.1e+72) t_0 (if (<= y 70000000.0) (* (/ x (+ 1.0 y)) y) t_0))))
double code(double x, double y) {
double t_0 = x - (x / y);
double tmp;
if (y <= -1.1e+72) {
tmp = t_0;
} else if (y <= 70000000.0) {
tmp = (x / (1.0 + 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 = x - (x / y)
if (y <= (-1.1d+72)) then
tmp = t_0
else if (y <= 70000000.0d0) then
tmp = (x / (1.0d0 + y)) * y
else
tmp = t_0
end if
code = tmp
end function
public static double code(double x, double y) {
double t_0 = x - (x / y);
double tmp;
if (y <= -1.1e+72) {
tmp = t_0;
} else if (y <= 70000000.0) {
tmp = (x / (1.0 + y)) * y;
} else {
tmp = t_0;
}
return tmp;
}
def code(x, y): t_0 = x - (x / y) tmp = 0 if y <= -1.1e+72: tmp = t_0 elif y <= 70000000.0: tmp = (x / (1.0 + y)) * y else: tmp = t_0 return tmp
function code(x, y) t_0 = Float64(x - Float64(x / y)) tmp = 0.0 if (y <= -1.1e+72) tmp = t_0; elseif (y <= 70000000.0) tmp = Float64(Float64(x / Float64(1.0 + y)) * y); else tmp = t_0; end return tmp end
function tmp_2 = code(x, y) t_0 = x - (x / y); tmp = 0.0; if (y <= -1.1e+72) tmp = t_0; elseif (y <= 70000000.0) tmp = (x / (1.0 + y)) * y; else tmp = t_0; end tmp_2 = tmp; end
code[x_, y_] := Block[{t$95$0 = N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.1e+72], t$95$0, If[LessEqual[y, 70000000.0], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := x - \frac{x}{y}\\
\mathbf{if}\;y \leq -1.1 \cdot 10^{+72}:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;y \leq 70000000:\\
\;\;\;\;\frac{x}{1 + y} \cdot y\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
if y < -1.1e72 or 7e7 < y Initial program 75.1%
Taylor expanded in y around inf
mul-1-negN/A
unsub-negN/A
lower--.f64N/A
lower-/.f64100.0
Applied rewrites100.0%
if -1.1e72 < y < 7e7Initial program 99.4%
lift-/.f64N/A
lift-*.f64N/A
*-commutativeN/A
associate-/l*N/A
*-commutativeN/A
lower-*.f64N/A
lower-/.f64100.0
lift-+.f64N/A
+-commutativeN/A
lower-+.f64100.0
Applied rewrites100.0%
(FPCore (x y) :precision binary64 (let* ((t_0 (/ (* y x) (+ 1.0 y)))) (if (<= t_0 1e+166) t_0 (* (/ x (+ 1.0 y)) y))))
double code(double x, double y) {
double t_0 = (y * x) / (1.0 + y);
double tmp;
if (t_0 <= 1e+166) {
tmp = t_0;
} else {
tmp = (x / (1.0 + y)) * 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) :: tmp
t_0 = (y * x) / (1.0d0 + y)
if (t_0 <= 1d+166) then
tmp = t_0
else
tmp = (x / (1.0d0 + y)) * y
end if
code = tmp
end function
public static double code(double x, double y) {
double t_0 = (y * x) / (1.0 + y);
double tmp;
if (t_0 <= 1e+166) {
tmp = t_0;
} else {
tmp = (x / (1.0 + y)) * y;
}
return tmp;
}
def code(x, y): t_0 = (y * x) / (1.0 + y) tmp = 0 if t_0 <= 1e+166: tmp = t_0 else: tmp = (x / (1.0 + y)) * y return tmp
function code(x, y) t_0 = Float64(Float64(y * x) / Float64(1.0 + y)) tmp = 0.0 if (t_0 <= 1e+166) tmp = t_0; else tmp = Float64(Float64(x / Float64(1.0 + y)) * y); end return tmp end
function tmp_2 = code(x, y) t_0 = (y * x) / (1.0 + y); tmp = 0.0; if (t_0 <= 1e+166) tmp = t_0; else tmp = (x / (1.0 + y)) * y; end tmp_2 = tmp; end
code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+166], t$95$0, N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \frac{y \cdot x}{1 + y}\\
\mathbf{if}\;t\_0 \leq 10^{+166}:\\
\;\;\;\;t\_0\\
\mathbf{else}:\\
\;\;\;\;\frac{x}{1 + y} \cdot y\\
\end{array}
\end{array}
if (/.f64 (*.f64 x y) (+.f64 y #s(literal 1 binary64))) < 9.9999999999999994e165Initial program 94.1%
if 9.9999999999999994e165 < (/.f64 (*.f64 x y) (+.f64 y #s(literal 1 binary64))) Initial program 23.2%
lift-/.f64N/A
lift-*.f64N/A
*-commutativeN/A
associate-/l*N/A
*-commutativeN/A
lower-*.f64N/A
lower-/.f6499.9
lift-+.f64N/A
+-commutativeN/A
lower-+.f6499.9
Applied rewrites99.9%
Final simplification94.5%
(FPCore (x y) :precision binary64 (let* ((t_0 (- x (/ x y)))) (if (<= y -1.0) t_0 (if (<= y 1.0) (fma (- x) (* y y) (* y x)) t_0))))
double code(double x, double y) {
double t_0 = x - (x / y);
double tmp;
if (y <= -1.0) {
tmp = t_0;
} else if (y <= 1.0) {
tmp = fma(-x, (y * y), (y * x));
} else {
tmp = t_0;
}
return tmp;
}
function code(x, y) t_0 = Float64(x - Float64(x / y)) tmp = 0.0 if (y <= -1.0) tmp = t_0; elseif (y <= 1.0) tmp = fma(Float64(-x), Float64(y * y), Float64(y * x)); else tmp = t_0; end return tmp end
code[x_, y_] := Block[{t$95$0 = N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.0], N[((-x) * N[(y * y), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := x - \frac{x}{y}\\
\mathbf{if}\;y \leq -1:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;y \leq 1:\\
\;\;\;\;\mathsf{fma}\left(-x, y \cdot y, y \cdot x\right)\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
if y < -1 or 1 < y Initial program 77.4%
Taylor expanded in y around inf
mul-1-negN/A
unsub-negN/A
lower--.f64N/A
lower-/.f6499.9
Applied rewrites99.9%
if -1 < y < 1Initial program 100.0%
Taylor expanded in y around 0
*-commutativeN/A
lower-*.f64N/A
mul-1-negN/A
unsub-negN/A
lower--.f64N/A
*-commutativeN/A
lower-*.f6499.3
Applied rewrites99.3%
Applied rewrites99.3%
(FPCore (x y) :precision binary64 (let* ((t_0 (- x (/ x y)))) (if (<= y -1.0) t_0 (if (<= y 1.0) (* (- x (* y x)) y) t_0))))
double code(double x, double y) {
double t_0 = x - (x / y);
double tmp;
if (y <= -1.0) {
tmp = t_0;
} else if (y <= 1.0) {
tmp = (x - (y * x)) * 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 = x - (x / y)
if (y <= (-1.0d0)) then
tmp = t_0
else if (y <= 1.0d0) then
tmp = (x - (y * x)) * y
else
tmp = t_0
end if
code = tmp
end function
public static double code(double x, double y) {
double t_0 = x - (x / y);
double tmp;
if (y <= -1.0) {
tmp = t_0;
} else if (y <= 1.0) {
tmp = (x - (y * x)) * y;
} else {
tmp = t_0;
}
return tmp;
}
def code(x, y): t_0 = x - (x / y) tmp = 0 if y <= -1.0: tmp = t_0 elif y <= 1.0: tmp = (x - (y * x)) * y else: tmp = t_0 return tmp
function code(x, y) t_0 = Float64(x - Float64(x / y)) tmp = 0.0 if (y <= -1.0) tmp = t_0; elseif (y <= 1.0) tmp = Float64(Float64(x - Float64(y * x)) * y); else tmp = t_0; end return tmp end
function tmp_2 = code(x, y) t_0 = x - (x / y); tmp = 0.0; if (y <= -1.0) tmp = t_0; elseif (y <= 1.0) tmp = (x - (y * x)) * y; else tmp = t_0; end tmp_2 = tmp; end
code[x_, y_] := Block[{t$95$0 = N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.0], N[(N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := x - \frac{x}{y}\\
\mathbf{if}\;y \leq -1:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;y \leq 1:\\
\;\;\;\;\left(x - y \cdot x\right) \cdot y\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
if y < -1 or 1 < y Initial program 77.4%
Taylor expanded in y around inf
mul-1-negN/A
unsub-negN/A
lower--.f64N/A
lower-/.f6499.9
Applied rewrites99.9%
if -1 < y < 1Initial program 100.0%
Taylor expanded in y around 0
*-commutativeN/A
lower-*.f64N/A
mul-1-negN/A
unsub-negN/A
lower--.f64N/A
*-commutativeN/A
lower-*.f6499.3
Applied rewrites99.3%
(FPCore (x y) :precision binary64 (* y x))
double code(double x, double y) {
return y * x;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
code = y * x
end function
public static double code(double x, double y) {
return y * x;
}
def code(x, y): return y * x
function code(x, y) return Float64(y * x) end
function tmp = code(x, y) tmp = y * x; end
code[x_, y_] := N[(y * x), $MachinePrecision]
\begin{array}{l}
\\
y \cdot x
\end{array}
Initial program 89.4%
Taylor expanded in y around 0
*-commutativeN/A
lower-*.f6454.1
Applied rewrites54.1%
(FPCore (x y)
:precision binary64
(let* ((t_0 (- (/ x (* y y)) (- (/ x y) x))))
(if (< y -3693.8482788297247)
t_0
(if (< y 6799310503.41891) (/ (* x y) (+ y 1.0)) t_0))))
double code(double x, double y) {
double t_0 = (x / (y * y)) - ((x / y) - x);
double tmp;
if (y < -3693.8482788297247) {
tmp = t_0;
} else if (y < 6799310503.41891) {
tmp = (x * y) / (y + 1.0);
} 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 = (x / (y * y)) - ((x / y) - x)
if (y < (-3693.8482788297247d0)) then
tmp = t_0
else if (y < 6799310503.41891d0) then
tmp = (x * y) / (y + 1.0d0)
else
tmp = t_0
end if
code = tmp
end function
public static double code(double x, double y) {
double t_0 = (x / (y * y)) - ((x / y) - x);
double tmp;
if (y < -3693.8482788297247) {
tmp = t_0;
} else if (y < 6799310503.41891) {
tmp = (x * y) / (y + 1.0);
} else {
tmp = t_0;
}
return tmp;
}
def code(x, y): t_0 = (x / (y * y)) - ((x / y) - x) tmp = 0 if y < -3693.8482788297247: tmp = t_0 elif y < 6799310503.41891: tmp = (x * y) / (y + 1.0) else: tmp = t_0 return tmp
function code(x, y) t_0 = Float64(Float64(x / Float64(y * y)) - Float64(Float64(x / y) - x)) tmp = 0.0 if (y < -3693.8482788297247) tmp = t_0; elseif (y < 6799310503.41891) tmp = Float64(Float64(x * y) / Float64(y + 1.0)); else tmp = t_0; end return tmp end
function tmp_2 = code(x, y) t_0 = (x / (y * y)) - ((x / y) - x); tmp = 0.0; if (y < -3693.8482788297247) tmp = t_0; elseif (y < 6799310503.41891) tmp = (x * y) / (y + 1.0); else tmp = t_0; end tmp_2 = tmp; end
code[x_, y_] := Block[{t$95$0 = N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -3693.8482788297247], t$95$0, If[Less[y, 6799310503.41891], N[(N[(x * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\
\mathbf{if}\;y < -3693.8482788297247:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;y < 6799310503.41891:\\
\;\;\;\;\frac{x \cdot y}{y + 1}\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
herbie shell --seed 2024235
(FPCore (x y)
:name "Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, B"
:precision binary64
:alt
(! :herbie-platform default (if (< y -36938482788297247/10000000000000) (- (/ x (* y y)) (- (/ x y) x)) (if (< y 679931050341891/100000) (/ (* x y) (+ y 1)) (- (/ x (* y y)) (- (/ x y) x)))))
(/ (* x y) (+ y 1.0)))