
(FPCore (x y z) :precision binary64 (/ (+ x (* y (- z x))) z))
double code(double x, double y, double z) {
return (x + (y * (z - x))) / z;
}
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 - x))) / z
end function
public static double code(double x, double y, double z) {
return (x + (y * (z - x))) / z;
}
def code(x, y, z): return (x + (y * (z - x))) / z
function code(x, y, z) return Float64(Float64(x + Float64(y * Float64(z - x))) / z) end
function tmp = code(x, y, z) tmp = (x + (y * (z - x))) / z; end
code[x_, y_, z_] := N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]
\begin{array}{l}
\\
\frac{x + y \cdot \left(z - x\right)}{z}
\end{array}
Sampling outcomes in binary64 precision:
Herbie found 6 alternatives:
| Alternative | Accuracy | Speedup |
|---|
(FPCore (x y z) :precision binary64 (/ (+ x (* y (- z x))) z))
double code(double x, double y, double z) {
return (x + (y * (z - x))) / z;
}
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 - x))) / z
end function
public static double code(double x, double y, double z) {
return (x + (y * (z - x))) / z;
}
def code(x, y, z): return (x + (y * (z - x))) / z
function code(x, y, z) return Float64(Float64(x + Float64(y * Float64(z - x))) / z) end
function tmp = code(x, y, z) tmp = (x + (y * (z - x))) / z; end
code[x_, y_, z_] := N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]
\begin{array}{l}
\\
\frac{x + y \cdot \left(z - x\right)}{z}
\end{array}
(FPCore (x y z) :precision binary64 (if (<= y 240.0) (- y (/ x (/ z (+ y -1.0)))) (* y (- 1.0 (/ x z)))))
double code(double x, double y, double z) {
double tmp;
if (y <= 240.0) {
tmp = y - (x / (z / (y + -1.0)));
} else {
tmp = y * (1.0 - (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 <= 240.0d0) then
tmp = y - (x / (z / (y + (-1.0d0))))
else
tmp = y * (1.0d0 - (x / z))
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if (y <= 240.0) {
tmp = y - (x / (z / (y + -1.0)));
} else {
tmp = y * (1.0 - (x / z));
}
return tmp;
}
def code(x, y, z): tmp = 0 if y <= 240.0: tmp = y - (x / (z / (y + -1.0))) else: tmp = y * (1.0 - (x / z)) return tmp
function code(x, y, z) tmp = 0.0 if (y <= 240.0) tmp = Float64(y - Float64(x / Float64(z / Float64(y + -1.0)))); else tmp = Float64(y * Float64(1.0 - Float64(x / z))); end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if (y <= 240.0) tmp = y - (x / (z / (y + -1.0))); else tmp = y * (1.0 - (x / z)); end tmp_2 = tmp; end
code[x_, y_, z_] := If[LessEqual[y, 240.0], N[(y - N[(x / N[(z / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq 240:\\
\;\;\;\;y - \frac{x}{\frac{z}{y + -1}}\\
\mathbf{else}:\\
\;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\
\end{array}
\end{array}
if y < 240Initial program 91.0%
Taylor expanded in y around 0 98.0%
Taylor expanded in z around -inf 95.2%
mul-1-neg95.2%
unsub-neg95.2%
distribute-rgt-out95.2%
associate-/l*98.5%
Simplified98.5%
if 240 < y Initial program 81.9%
Taylor expanded in y around 0 91.1%
Taylor expanded in y around inf 99.9%
Final simplification98.8%
(FPCore (x y z)
:precision binary64
(if (<= y -1900000.0)
y
(if (<= y -9.4e-35)
(/ x z)
(if (<= y -3.1e-68) y (if (<= y 2.5e-25) (/ x z) y)))))
double code(double x, double y, double z) {
double tmp;
if (y <= -1900000.0) {
tmp = y;
} else if (y <= -9.4e-35) {
tmp = x / z;
} else if (y <= -3.1e-68) {
tmp = y;
} else if (y <= 2.5e-25) {
tmp = x / z;
} else {
tmp = 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 <= (-1900000.0d0)) then
tmp = y
else if (y <= (-9.4d-35)) then
tmp = x / z
else if (y <= (-3.1d-68)) then
tmp = y
else if (y <= 2.5d-25) then
tmp = x / z
else
tmp = y
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if (y <= -1900000.0) {
tmp = y;
} else if (y <= -9.4e-35) {
tmp = x / z;
} else if (y <= -3.1e-68) {
tmp = y;
} else if (y <= 2.5e-25) {
tmp = x / z;
} else {
tmp = y;
}
return tmp;
}
def code(x, y, z): tmp = 0 if y <= -1900000.0: tmp = y elif y <= -9.4e-35: tmp = x / z elif y <= -3.1e-68: tmp = y elif y <= 2.5e-25: tmp = x / z else: tmp = y return tmp
function code(x, y, z) tmp = 0.0 if (y <= -1900000.0) tmp = y; elseif (y <= -9.4e-35) tmp = Float64(x / z); elseif (y <= -3.1e-68) tmp = y; elseif (y <= 2.5e-25) tmp = Float64(x / z); else tmp = y; end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if (y <= -1900000.0) tmp = y; elseif (y <= -9.4e-35) tmp = x / z; elseif (y <= -3.1e-68) tmp = y; elseif (y <= 2.5e-25) tmp = x / z; else tmp = y; end tmp_2 = tmp; end
code[x_, y_, z_] := If[LessEqual[y, -1900000.0], y, If[LessEqual[y, -9.4e-35], N[(x / z), $MachinePrecision], If[LessEqual[y, -3.1e-68], y, If[LessEqual[y, 2.5e-25], N[(x / z), $MachinePrecision], y]]]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -1900000:\\
\;\;\;\;y\\
\mathbf{elif}\;y \leq -9.4 \cdot 10^{-35}:\\
\;\;\;\;\frac{x}{z}\\
\mathbf{elif}\;y \leq -3.1 \cdot 10^{-68}:\\
\;\;\;\;y\\
\mathbf{elif}\;y \leq 2.5 \cdot 10^{-25}:\\
\;\;\;\;\frac{x}{z}\\
\mathbf{else}:\\
\;\;\;\;y\\
\end{array}
\end{array}
if y < -1.9e6 or -9.4e-35 < y < -3.0999999999999999e-68 or 2.49999999999999981e-25 < y Initial program 80.0%
Taylor expanded in x around 0 55.2%
if -1.9e6 < y < -9.4e-35 or -3.0999999999999999e-68 < y < 2.49999999999999981e-25Initial program 99.9%
Taylor expanded in y around 0 80.0%
Final simplification66.3%
(FPCore (x y z) :precision binary64 (if (or (<= y -1.0) (not (<= y 0.0034))) (* y (- 1.0 (/ x z))) (+ y (/ x z))))
double code(double x, double y, double z) {
double tmp;
if ((y <= -1.0) || !(y <= 0.0034)) {
tmp = y * (1.0 - (x / z));
} else {
tmp = y + (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 <= (-1.0d0)) .or. (.not. (y <= 0.0034d0))) then
tmp = y * (1.0d0 - (x / z))
else
tmp = y + (x / z)
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if ((y <= -1.0) || !(y <= 0.0034)) {
tmp = y * (1.0 - (x / z));
} else {
tmp = y + (x / z);
}
return tmp;
}
def code(x, y, z): tmp = 0 if (y <= -1.0) or not (y <= 0.0034): tmp = y * (1.0 - (x / z)) else: tmp = y + (x / z) return tmp
function code(x, y, z) tmp = 0.0 if ((y <= -1.0) || !(y <= 0.0034)) tmp = Float64(y * Float64(1.0 - Float64(x / z))); else tmp = Float64(y + Float64(x / z)); end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if ((y <= -1.0) || ~((y <= 0.0034))) tmp = y * (1.0 - (x / z)); else tmp = y + (x / z); end tmp_2 = tmp; end
code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.0034]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.0034\right):\\
\;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\
\mathbf{else}:\\
\;\;\;\;y + \frac{x}{z}\\
\end{array}
\end{array}
if y < -1 or 0.00339999999999999981 < y Initial program 77.8%
Taylor expanded in y around 0 96.0%
Taylor expanded in y around inf 98.6%
if -1 < y < 0.00339999999999999981Initial program 99.9%
Taylor expanded in y around 0 96.9%
Taylor expanded in x around 0 98.8%
Final simplification98.7%
(FPCore (x y z) :precision binary64 (if (<= y 3.2e+39) (+ y (/ x z)) (if (<= y 1.82e+179) (* y (/ (- x) z)) y)))
double code(double x, double y, double z) {
double tmp;
if (y <= 3.2e+39) {
tmp = y + (x / z);
} else if (y <= 1.82e+179) {
tmp = y * (-x / z);
} else {
tmp = 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 <= 3.2d+39) then
tmp = y + (x / z)
else if (y <= 1.82d+179) then
tmp = y * (-x / z)
else
tmp = y
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if (y <= 3.2e+39) {
tmp = y + (x / z);
} else if (y <= 1.82e+179) {
tmp = y * (-x / z);
} else {
tmp = y;
}
return tmp;
}
def code(x, y, z): tmp = 0 if y <= 3.2e+39: tmp = y + (x / z) elif y <= 1.82e+179: tmp = y * (-x / z) else: tmp = y return tmp
function code(x, y, z) tmp = 0.0 if (y <= 3.2e+39) tmp = Float64(y + Float64(x / z)); elseif (y <= 1.82e+179) tmp = Float64(y * Float64(Float64(-x) / z)); else tmp = y; end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if (y <= 3.2e+39) tmp = y + (x / z); elseif (y <= 1.82e+179) tmp = y * (-x / z); else tmp = y; end tmp_2 = tmp; end
code[x_, y_, z_] := If[LessEqual[y, 3.2e+39], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.82e+179], N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision], y]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq 3.2 \cdot 10^{+39}:\\
\;\;\;\;y + \frac{x}{z}\\
\mathbf{elif}\;y \leq 1.82 \cdot 10^{+179}:\\
\;\;\;\;y \cdot \frac{-x}{z}\\
\mathbf{else}:\\
\;\;\;\;y\\
\end{array}
\end{array}
if y < 3.19999999999999993e39Initial program 91.2%
Taylor expanded in y around 0 98.0%
Taylor expanded in x around 0 87.9%
if 3.19999999999999993e39 < y < 1.81999999999999997e179Initial program 90.3%
Taylor expanded in y around 0 86.5%
Taylor expanded in y around inf 99.8%
Taylor expanded in x around inf 62.1%
mul-1-neg62.1%
distribute-frac-neg62.1%
Simplified62.1%
if 1.81999999999999997e179 < y Initial program 64.8%
Taylor expanded in x around 0 60.8%
Final simplification82.7%
(FPCore (x y z) :precision binary64 (+ y (/ x z)))
double code(double x, double y, double z) {
return y + (x / z);
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
code = y + (x / z)
end function
public static double code(double x, double y, double z) {
return y + (x / z);
}
def code(x, y, z): return y + (x / z)
function code(x, y, z) return Float64(y + Float64(x / z)) end
function tmp = code(x, y, z) tmp = y + (x / z); end
code[x_, y_, z_] := N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
y + \frac{x}{z}
\end{array}
Initial program 88.9%
Taylor expanded in y around 0 96.4%
Taylor expanded in x around 0 79.3%
Final simplification79.3%
(FPCore (x y z) :precision binary64 y)
double code(double x, double y, double z) {
return y;
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
code = y
end function
public static double code(double x, double y, double z) {
return y;
}
def code(x, y, z): return y
function code(x, y, z) return y end
function tmp = code(x, y, z) tmp = y; end
code[x_, y_, z_] := y
\begin{array}{l}
\\
y
\end{array}
Initial program 88.9%
Taylor expanded in x around 0 40.1%
Final simplification40.1%
(FPCore (x y z) :precision binary64 (- (+ y (/ x z)) (/ y (/ z x))))
double code(double x, double y, double z) {
return (y + (x / z)) - (y / (z / x));
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
code = (y + (x / z)) - (y / (z / x))
end function
public static double code(double x, double y, double z) {
return (y + (x / z)) - (y / (z / x));
}
def code(x, y, z): return (y + (x / z)) - (y / (z / x))
function code(x, y, z) return Float64(Float64(y + Float64(x / z)) - Float64(y / Float64(z / x))) end
function tmp = code(x, y, z) tmp = (y + (x / z)) - (y / (z / x)); end
code[x_, y_, z_] := N[(N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}
\end{array}
herbie shell --seed 2023262
(FPCore (x y z)
:name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
:precision binary64
:herbie-target
(- (+ y (/ x z)) (/ y (/ z x)))
(/ (+ x (* y (- z x))) z))