
(FPCore (x y z) :precision binary64 (/ (- x y) (- z y)))
double code(double x, double y, double z) {
return (x - y) / (z - y);
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
code = (x - y) / (z - y)
end function
public static double code(double x, double y, double z) {
return (x - y) / (z - y);
}
def code(x, y, z): return (x - y) / (z - y)
function code(x, y, z) return Float64(Float64(x - y) / Float64(z - y)) end
function tmp = code(x, y, z) tmp = (x - y) / (z - y); end
code[x_, y_, z_] := N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\frac{x - y}{z - y}
\end{array}
Sampling outcomes in binary64 precision:
Herbie found 6 alternatives:
| Alternative | Accuracy | Speedup |
|---|
(FPCore (x y z) :precision binary64 (/ (- x y) (- z y)))
double code(double x, double y, double z) {
return (x - y) / (z - y);
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
code = (x - y) / (z - y)
end function
public static double code(double x, double y, double z) {
return (x - y) / (z - y);
}
def code(x, y, z): return (x - y) / (z - y)
function code(x, y, z) return Float64(Float64(x - y) / Float64(z - y)) end
function tmp = code(x, y, z) tmp = (x - y) / (z - y); end
code[x_, y_, z_] := N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\frac{x - y}{z - y}
\end{array}
(FPCore (x y z) :precision binary64 (/ (- x y) (- z y)))
double code(double x, double y, double z) {
return (x - y) / (z - y);
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
code = (x - y) / (z - y)
end function
public static double code(double x, double y, double z) {
return (x - y) / (z - y);
}
def code(x, y, z): return (x - y) / (z - y)
function code(x, y, z) return Float64(Float64(x - y) / Float64(z - y)) end
function tmp = code(x, y, z) tmp = (x - y) / (z - y); end
code[x_, y_, z_] := N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\frac{x - y}{z - y}
\end{array}
Initial program 100.0%
Final simplification100.0%
(FPCore (x y z) :precision binary64 (if (or (<= y -3.7e-147) (not (<= y 1.66e-43))) (- 1.0 (/ x y)) (/ x z)))
double code(double x, double y, double z) {
double tmp;
if ((y <= -3.7e-147) || !(y <= 1.66e-43)) {
tmp = 1.0 - (x / y);
} else {
tmp = x / z;
}
return tmp;
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
real(8) :: tmp
if ((y <= (-3.7d-147)) .or. (.not. (y <= 1.66d-43))) then
tmp = 1.0d0 - (x / y)
else
tmp = x / z
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if ((y <= -3.7e-147) || !(y <= 1.66e-43)) {
tmp = 1.0 - (x / y);
} else {
tmp = x / z;
}
return tmp;
}
def code(x, y, z): tmp = 0 if (y <= -3.7e-147) or not (y <= 1.66e-43): tmp = 1.0 - (x / y) else: tmp = x / z return tmp
function code(x, y, z) tmp = 0.0 if ((y <= -3.7e-147) || !(y <= 1.66e-43)) tmp = Float64(1.0 - Float64(x / y)); else tmp = Float64(x / z); end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if ((y <= -3.7e-147) || ~((y <= 1.66e-43))) tmp = 1.0 - (x / y); else tmp = x / z; end tmp_2 = tmp; end
code[x_, y_, z_] := If[Or[LessEqual[y, -3.7e-147], N[Not[LessEqual[y, 1.66e-43]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -3.7 \cdot 10^{-147} \lor \neg \left(y \leq 1.66 \cdot 10^{-43}\right):\\
\;\;\;\;1 - \frac{x}{y}\\
\mathbf{else}:\\
\;\;\;\;\frac{x}{z}\\
\end{array}
\end{array}
if y < -3.7000000000000002e-147 or 1.66e-43 < y Initial program 100.0%
Taylor expanded in z around 0 67.5%
div-sub67.5%
sub-neg67.5%
*-inverses67.5%
metadata-eval67.5%
distribute-lft-in67.5%
metadata-eval67.5%
+-commutative67.5%
mul-1-neg67.5%
unsub-neg67.5%
Simplified67.5%
if -3.7000000000000002e-147 < y < 1.66e-43Initial program 100.0%
Taylor expanded in y around 0 82.6%
Final simplification73.1%
(FPCore (x y z) :precision binary64 (if (or (<= y -3.5e-34) (not (<= y 2.7e-43))) (- 1.0 (/ x y)) (/ x (- z y))))
double code(double x, double y, double z) {
double tmp;
if ((y <= -3.5e-34) || !(y <= 2.7e-43)) {
tmp = 1.0 - (x / y);
} else {
tmp = x / (z - y);
}
return tmp;
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
real(8) :: tmp
if ((y <= (-3.5d-34)) .or. (.not. (y <= 2.7d-43))) then
tmp = 1.0d0 - (x / y)
else
tmp = x / (z - y)
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if ((y <= -3.5e-34) || !(y <= 2.7e-43)) {
tmp = 1.0 - (x / y);
} else {
tmp = x / (z - y);
}
return tmp;
}
def code(x, y, z): tmp = 0 if (y <= -3.5e-34) or not (y <= 2.7e-43): tmp = 1.0 - (x / y) else: tmp = x / (z - y) return tmp
function code(x, y, z) tmp = 0.0 if ((y <= -3.5e-34) || !(y <= 2.7e-43)) tmp = Float64(1.0 - Float64(x / y)); else tmp = Float64(x / Float64(z - y)); end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if ((y <= -3.5e-34) || ~((y <= 2.7e-43))) tmp = 1.0 - (x / y); else tmp = x / (z - y); end tmp_2 = tmp; end
code[x_, y_, z_] := If[Or[LessEqual[y, -3.5e-34], N[Not[LessEqual[y, 2.7e-43]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -3.5 \cdot 10^{-34} \lor \neg \left(y \leq 2.7 \cdot 10^{-43}\right):\\
\;\;\;\;1 - \frac{x}{y}\\
\mathbf{else}:\\
\;\;\;\;\frac{x}{z - y}\\
\end{array}
\end{array}
if y < -3.5e-34 or 2.69999999999999991e-43 < y Initial program 100.0%
Taylor expanded in z around 0 70.4%
div-sub70.4%
sub-neg70.4%
*-inverses70.4%
metadata-eval70.4%
distribute-lft-in70.4%
metadata-eval70.4%
+-commutative70.4%
mul-1-neg70.4%
unsub-neg70.4%
Simplified70.4%
if -3.5e-34 < y < 2.69999999999999991e-43Initial program 100.0%
Taylor expanded in x around inf 84.3%
Final simplification76.5%
(FPCore (x y z) :precision binary64 (if (or (<= x -2.8e+101) (not (<= x 7e+35))) (/ x (- z y)) (/ y (- y z))))
double code(double x, double y, double z) {
double tmp;
if ((x <= -2.8e+101) || !(x <= 7e+35)) {
tmp = x / (z - y);
} else {
tmp = y / (y - z);
}
return tmp;
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
real(8) :: tmp
if ((x <= (-2.8d+101)) .or. (.not. (x <= 7d+35))) then
tmp = x / (z - y)
else
tmp = y / (y - z)
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if ((x <= -2.8e+101) || !(x <= 7e+35)) {
tmp = x / (z - y);
} else {
tmp = y / (y - z);
}
return tmp;
}
def code(x, y, z): tmp = 0 if (x <= -2.8e+101) or not (x <= 7e+35): tmp = x / (z - y) else: tmp = y / (y - z) return tmp
function code(x, y, z) tmp = 0.0 if ((x <= -2.8e+101) || !(x <= 7e+35)) tmp = Float64(x / Float64(z - y)); else tmp = Float64(y / Float64(y - z)); end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if ((x <= -2.8e+101) || ~((x <= 7e+35))) tmp = x / (z - y); else tmp = y / (y - z); end tmp_2 = tmp; end
code[x_, y_, z_] := If[Or[LessEqual[x, -2.8e+101], N[Not[LessEqual[x, 7e+35]], $MachinePrecision]], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;x \leq -2.8 \cdot 10^{+101} \lor \neg \left(x \leq 7 \cdot 10^{+35}\right):\\
\;\;\;\;\frac{x}{z - y}\\
\mathbf{else}:\\
\;\;\;\;\frac{y}{y - z}\\
\end{array}
\end{array}
if x < -2.79999999999999981e101 or 7.0000000000000001e35 < x Initial program 99.9%
Taylor expanded in x around inf 81.8%
if -2.79999999999999981e101 < x < 7.0000000000000001e35Initial program 100.0%
Taylor expanded in x around 0 78.5%
neg-mul-178.5%
distribute-neg-frac78.5%
Simplified78.5%
frac-2neg78.5%
div-inv78.3%
remove-double-neg78.3%
sub-neg78.3%
distribute-neg-in78.3%
remove-double-neg78.3%
Applied egg-rr78.3%
associate-*r/78.5%
*-rgt-identity78.5%
neg-mul-178.5%
+-commutative78.5%
neg-mul-178.5%
unsub-neg78.5%
Simplified78.5%
Final simplification79.9%
(FPCore (x y z) :precision binary64 (if (<= y -4.6e-36) 1.0 (if (<= y 1.12e-43) (/ x z) 1.0)))
double code(double x, double y, double z) {
double tmp;
if (y <= -4.6e-36) {
tmp = 1.0;
} else if (y <= 1.12e-43) {
tmp = x / z;
} else {
tmp = 1.0;
}
return tmp;
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
real(8) :: tmp
if (y <= (-4.6d-36)) then
tmp = 1.0d0
else if (y <= 1.12d-43) then
tmp = x / z
else
tmp = 1.0d0
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if (y <= -4.6e-36) {
tmp = 1.0;
} else if (y <= 1.12e-43) {
tmp = x / z;
} else {
tmp = 1.0;
}
return tmp;
}
def code(x, y, z): tmp = 0 if y <= -4.6e-36: tmp = 1.0 elif y <= 1.12e-43: tmp = x / z else: tmp = 1.0 return tmp
function code(x, y, z) tmp = 0.0 if (y <= -4.6e-36) tmp = 1.0; elseif (y <= 1.12e-43) tmp = Float64(x / z); else tmp = 1.0; end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if (y <= -4.6e-36) tmp = 1.0; elseif (y <= 1.12e-43) tmp = x / z; else tmp = 1.0; end tmp_2 = tmp; end
code[x_, y_, z_] := If[LessEqual[y, -4.6e-36], 1.0, If[LessEqual[y, 1.12e-43], N[(x / z), $MachinePrecision], 1.0]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -4.6 \cdot 10^{-36}:\\
\;\;\;\;1\\
\mathbf{elif}\;y \leq 1.12 \cdot 10^{-43}:\\
\;\;\;\;\frac{x}{z}\\
\mathbf{else}:\\
\;\;\;\;1\\
\end{array}
\end{array}
if y < -4.59999999999999993e-36 or 1.12e-43 < y Initial program 100.0%
Taylor expanded in y around inf 55.1%
if -4.59999999999999993e-36 < y < 1.12e-43Initial program 100.0%
Taylor expanded in y around 0 74.6%
Final simplification63.7%
(FPCore (x y z) :precision binary64 1.0)
double code(double x, double y, double z) {
return 1.0;
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
code = 1.0d0
end function
public static double code(double x, double y, double z) {
return 1.0;
}
def code(x, y, z): return 1.0
function code(x, y, z) return 1.0 end
function tmp = code(x, y, z) tmp = 1.0; end
code[x_, y_, z_] := 1.0
\begin{array}{l}
\\
1
\end{array}
Initial program 100.0%
Taylor expanded in y around inf 34.7%
Final simplification34.7%
(FPCore (x y z) :precision binary64 (- (/ x (- z y)) (/ y (- z y))))
double code(double x, double y, double z) {
return (x / (z - y)) - (y / (z - y));
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
code = (x / (z - y)) - (y / (z - y))
end function
public static double code(double x, double y, double z) {
return (x / (z - y)) - (y / (z - y));
}
def code(x, y, z): return (x / (z - y)) - (y / (z - y))
function code(x, y, z) return Float64(Float64(x / Float64(z - y)) - Float64(y / Float64(z - y))) end
function tmp = code(x, y, z) tmp = (x / (z - y)) - (y / (z - y)); end
code[x_, y_, z_] := N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] - N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\frac{x}{z - y} - \frac{y}{z - y}
\end{array}
herbie shell --seed 2024043
(FPCore (x y z)
:name "Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1"
:precision binary64
:alt
(- (/ x (- z y)) (/ y (- z y)))
(/ (- x y) (- z y)))