
(FPCore (x y) :precision binary64 (let* ((t_0 (/ x (* y 2.0)))) (/ (tan t_0) (sin t_0))))
double code(double x, double y) {
double t_0 = x / (y * 2.0);
return tan(t_0) / sin(t_0);
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: t_0
t_0 = x / (y * 2.0d0)
code = tan(t_0) / sin(t_0)
end function
public static double code(double x, double y) {
double t_0 = x / (y * 2.0);
return Math.tan(t_0) / Math.sin(t_0);
}
def code(x, y): t_0 = x / (y * 2.0) return math.tan(t_0) / math.sin(t_0)
function code(x, y) t_0 = Float64(x / Float64(y * 2.0)) return Float64(tan(t_0) / sin(t_0)) end
function tmp = code(x, y) t_0 = x / (y * 2.0); tmp = tan(t_0) / sin(t_0); end
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \frac{x}{y \cdot 2}\\
\frac{\tan t\_0}{\sin t\_0}
\end{array}
\end{array}
Sampling outcomes in binary64 precision:
Herbie found 3 alternatives:
| Alternative | Accuracy | Speedup |
|---|
(FPCore (x y) :precision binary64 (let* ((t_0 (/ x (* y 2.0)))) (/ (tan t_0) (sin t_0))))
double code(double x, double y) {
double t_0 = x / (y * 2.0);
return tan(t_0) / sin(t_0);
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: t_0
t_0 = x / (y * 2.0d0)
code = tan(t_0) / sin(t_0)
end function
public static double code(double x, double y) {
double t_0 = x / (y * 2.0);
return Math.tan(t_0) / Math.sin(t_0);
}
def code(x, y): t_0 = x / (y * 2.0) return math.tan(t_0) / math.sin(t_0)
function code(x, y) t_0 = Float64(x / Float64(y * 2.0)) return Float64(tan(t_0) / sin(t_0)) end
function tmp = code(x, y) t_0 = x / (y * 2.0); tmp = tan(t_0) / sin(t_0); end
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \frac{x}{y \cdot 2}\\
\frac{\tan t\_0}{\sin t\_0}
\end{array}
\end{array}
y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
:precision binary64
(if (<= (/ x_m (* 2.0 y_m)) 5e+289)
(/
1.0
(cos (* (* (sqrt 0.5) (sqrt (/ x_m y_m))) (pow (/ (* 2.0 y_m) x_m) -0.5))))
1.0))y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
double tmp;
if ((x_m / (2.0 * y_m)) <= 5e+289) {
tmp = 1.0 / cos(((sqrt(0.5) * sqrt((x_m / y_m))) * pow(((2.0 * y_m) / x_m), -0.5)));
} else {
tmp = 1.0;
}
return tmp;
}
y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
real(8), intent (in) :: x_m
real(8), intent (in) :: y_m
real(8) :: tmp
if ((x_m / (2.0d0 * y_m)) <= 5d+289) then
tmp = 1.0d0 / cos(((sqrt(0.5d0) * sqrt((x_m / y_m))) * (((2.0d0 * y_m) / x_m) ** (-0.5d0))))
else
tmp = 1.0d0
end if
code = tmp
end function
y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
double tmp;
if ((x_m / (2.0 * y_m)) <= 5e+289) {
tmp = 1.0 / Math.cos(((Math.sqrt(0.5) * Math.sqrt((x_m / y_m))) * Math.pow(((2.0 * y_m) / x_m), -0.5)));
} else {
tmp = 1.0;
}
return tmp;
}
y_m = math.fabs(y) x_m = math.fabs(x) def code(x_m, y_m): tmp = 0 if (x_m / (2.0 * y_m)) <= 5e+289: tmp = 1.0 / math.cos(((math.sqrt(0.5) * math.sqrt((x_m / y_m))) * math.pow(((2.0 * y_m) / x_m), -0.5))) else: tmp = 1.0 return tmp
y_m = abs(y) x_m = abs(x) function code(x_m, y_m) tmp = 0.0 if (Float64(x_m / Float64(2.0 * y_m)) <= 5e+289) tmp = Float64(1.0 / cos(Float64(Float64(sqrt(0.5) * sqrt(Float64(x_m / y_m))) * (Float64(Float64(2.0 * y_m) / x_m) ^ -0.5)))); else tmp = 1.0; end return tmp end
y_m = abs(y); x_m = abs(x); function tmp_2 = code(x_m, y_m) tmp = 0.0; if ((x_m / (2.0 * y_m)) <= 5e+289) tmp = 1.0 / cos(((sqrt(0.5) * sqrt((x_m / y_m))) * (((2.0 * y_m) / x_m) ^ -0.5))); else tmp = 1.0; end tmp_2 = tmp; end
y_m = N[Abs[y], $MachinePrecision] x_m = N[Abs[x], $MachinePrecision] code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 5e+289], N[(1.0 / N[Cos[N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[N[(x$95$m / y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]
\begin{array}{l}
y_m = \left|y\right|
\\
x_m = \left|x\right|
\\
\begin{array}{l}
\mathbf{if}\;\frac{x\_m}{2 \cdot y\_m} \leq 5 \cdot 10^{+289}:\\
\;\;\;\;\frac{1}{\cos \left(\left(\sqrt{0.5} \cdot \sqrt{\frac{x\_m}{y\_m}}\right) \cdot {\left(\frac{2 \cdot y\_m}{x\_m}\right)}^{-0.5}\right)}\\
\mathbf{else}:\\
\;\;\;\;1\\
\end{array}
\end{array}
if (/.f64 x (*.f64 y #s(literal 2 binary64))) < 5.00000000000000031e289Initial program 48.4%
Taylor expanded in x around inf
lower-/.f64N/A
associate-*r/N/A
*-commutativeN/A
associate-*r/N/A
metadata-evalN/A
associate-*r/N/A
lower-cos.f64N/A
*-commutativeN/A
lower-*.f64N/A
associate-*r/N/A
metadata-evalN/A
lower-/.f6459.2
Applied rewrites59.2%
Applied rewrites31.5%
Taylor expanded in x around 0
Applied rewrites31.3%
if 5.00000000000000031e289 < (/.f64 x (*.f64 y #s(literal 2 binary64))) Initial program 0.2%
Taylor expanded in x around 0
Applied rewrites9.4%
Final simplification29.6%
y_m = (fabs.f64 y) x_m = (fabs.f64 x) (FPCore (x_m y_m) :precision binary64 (if (<= x_m 7.5e+16) (/ 1.0 (cos (/ 0.5 (/ y_m x_m)))) 1.0))
y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
double tmp;
if (x_m <= 7.5e+16) {
tmp = 1.0 / cos((0.5 / (y_m / x_m)));
} else {
tmp = 1.0;
}
return tmp;
}
y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
real(8), intent (in) :: x_m
real(8), intent (in) :: y_m
real(8) :: tmp
if (x_m <= 7.5d+16) then
tmp = 1.0d0 / cos((0.5d0 / (y_m / x_m)))
else
tmp = 1.0d0
end if
code = tmp
end function
y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
double tmp;
if (x_m <= 7.5e+16) {
tmp = 1.0 / Math.cos((0.5 / (y_m / x_m)));
} else {
tmp = 1.0;
}
return tmp;
}
y_m = math.fabs(y) x_m = math.fabs(x) def code(x_m, y_m): tmp = 0 if x_m <= 7.5e+16: tmp = 1.0 / math.cos((0.5 / (y_m / x_m))) else: tmp = 1.0 return tmp
y_m = abs(y) x_m = abs(x) function code(x_m, y_m) tmp = 0.0 if (x_m <= 7.5e+16) tmp = Float64(1.0 / cos(Float64(0.5 / Float64(y_m / x_m)))); else tmp = 1.0; end return tmp end
y_m = abs(y); x_m = abs(x); function tmp_2 = code(x_m, y_m) tmp = 0.0; if (x_m <= 7.5e+16) tmp = 1.0 / cos((0.5 / (y_m / x_m))); else tmp = 1.0; end tmp_2 = tmp; end
y_m = N[Abs[y], $MachinePrecision] x_m = N[Abs[x], $MachinePrecision] code[x$95$m_, y$95$m_] := If[LessEqual[x$95$m, 7.5e+16], N[(1.0 / N[Cos[N[(0.5 / N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]
\begin{array}{l}
y_m = \left|y\right|
\\
x_m = \left|x\right|
\\
\begin{array}{l}
\mathbf{if}\;x\_m \leq 7.5 \cdot 10^{+16}:\\
\;\;\;\;\frac{1}{\cos \left(\frac{0.5}{\frac{y\_m}{x\_m}}\right)}\\
\mathbf{else}:\\
\;\;\;\;1\\
\end{array}
\end{array}
if x < 7.5e16Initial program 47.3%
Taylor expanded in x around inf
lower-/.f64N/A
associate-*r/N/A
*-commutativeN/A
associate-*r/N/A
metadata-evalN/A
associate-*r/N/A
lower-cos.f64N/A
*-commutativeN/A
lower-*.f64N/A
associate-*r/N/A
metadata-evalN/A
lower-/.f6460.5
Applied rewrites60.5%
Applied rewrites60.5%
if 7.5e16 < x Initial program 36.1%
Taylor expanded in x around 0
Applied rewrites39.4%
y_m = (fabs.f64 y) x_m = (fabs.f64 x) (FPCore (x_m y_m) :precision binary64 1.0)
y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
return 1.0;
}
y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
real(8), intent (in) :: x_m
real(8), intent (in) :: y_m
code = 1.0d0
end function
y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
return 1.0;
}
y_m = math.fabs(y) x_m = math.fabs(x) def code(x_m, y_m): return 1.0
y_m = abs(y) x_m = abs(x) function code(x_m, y_m) return 1.0 end
y_m = abs(y); x_m = abs(x); function tmp = code(x_m, y_m) tmp = 1.0; end
y_m = N[Abs[y], $MachinePrecision] x_m = N[Abs[x], $MachinePrecision] code[x$95$m_, y$95$m_] := 1.0
\begin{array}{l}
y_m = \left|y\right|
\\
x_m = \left|x\right|
\\
1
\end{array}
Initial program 44.7%
Taylor expanded in x around 0
Applied rewrites54.7%
(FPCore (x y)
:precision binary64
(let* ((t_0 (/ x (* y 2.0))) (t_1 (sin t_0)))
(if (< y -1.2303690911306994e+114)
1.0
(if (< y -9.102852406811914e-222)
(/ t_1 (* t_1 (log (exp (cos t_0)))))
1.0))))
double code(double x, double y) {
double t_0 = x / (y * 2.0);
double t_1 = sin(t_0);
double tmp;
if (y < -1.2303690911306994e+114) {
tmp = 1.0;
} else if (y < -9.102852406811914e-222) {
tmp = t_1 / (t_1 * log(exp(cos(t_0))));
} else {
tmp = 1.0;
}
return tmp;
}
real(8) function code(x, y)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8) :: t_0
real(8) :: t_1
real(8) :: tmp
t_0 = x / (y * 2.0d0)
t_1 = sin(t_0)
if (y < (-1.2303690911306994d+114)) then
tmp = 1.0d0
else if (y < (-9.102852406811914d-222)) then
tmp = t_1 / (t_1 * log(exp(cos(t_0))))
else
tmp = 1.0d0
end if
code = tmp
end function
public static double code(double x, double y) {
double t_0 = x / (y * 2.0);
double t_1 = Math.sin(t_0);
double tmp;
if (y < -1.2303690911306994e+114) {
tmp = 1.0;
} else if (y < -9.102852406811914e-222) {
tmp = t_1 / (t_1 * Math.log(Math.exp(Math.cos(t_0))));
} else {
tmp = 1.0;
}
return tmp;
}
def code(x, y): t_0 = x / (y * 2.0) t_1 = math.sin(t_0) tmp = 0 if y < -1.2303690911306994e+114: tmp = 1.0 elif y < -9.102852406811914e-222: tmp = t_1 / (t_1 * math.log(math.exp(math.cos(t_0)))) else: tmp = 1.0 return tmp
function code(x, y) t_0 = Float64(x / Float64(y * 2.0)) t_1 = sin(t_0) tmp = 0.0 if (y < -1.2303690911306994e+114) tmp = 1.0; elseif (y < -9.102852406811914e-222) tmp = Float64(t_1 / Float64(t_1 * log(exp(cos(t_0))))); else tmp = 1.0; end return tmp end
function tmp_2 = code(x, y) t_0 = x / (y * 2.0); t_1 = sin(t_0); tmp = 0.0; if (y < -1.2303690911306994e+114) tmp = 1.0; elseif (y < -9.102852406811914e-222) tmp = t_1 / (t_1 * log(exp(cos(t_0)))); else tmp = 1.0; end tmp_2 = tmp; end
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[Less[y, -1.2303690911306994e+114], 1.0, If[Less[y, -9.102852406811914e-222], N[(t$95$1 / N[(t$95$1 * N[Log[N[Exp[N[Cos[t$95$0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \frac{x}{y \cdot 2}\\
t_1 := \sin t\_0\\
\mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\
\;\;\;\;1\\
\mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\
\;\;\;\;\frac{t\_1}{t\_1 \cdot \log \left(e^{\cos t\_0}\right)}\\
\mathbf{else}:\\
\;\;\;\;1\\
\end{array}
\end{array}
herbie shell --seed 2024295
(FPCore (x y)
:name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
:precision binary64
:alt
(! :herbie-platform default (if (< y -1230369091130699400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) 1 (if (< y -4551426203405957/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (sin (/ x (* y 2))) (* (sin (/ x (* y 2))) (log (exp (cos (/ x (* y 2))))))) 1)))
(/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))