
(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 5e+26) (+ y (* x (/ (- 1.0 y) z))) (* y (- 1.0 (/ x z)))))
double code(double x, double y, double z) {
double tmp;
if (y <= 5e+26) {
tmp = y + (x * ((1.0 - y) / z));
} 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 <= 5d+26) then
tmp = y + (x * ((1.0d0 - y) / z))
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 <= 5e+26) {
tmp = y + (x * ((1.0 - y) / z));
} else {
tmp = y * (1.0 - (x / z));
}
return tmp;
}
def code(x, y, z): tmp = 0 if y <= 5e+26: tmp = y + (x * ((1.0 - y) / z)) else: tmp = y * (1.0 - (x / z)) return tmp
function code(x, y, z) tmp = 0.0 if (y <= 5e+26) tmp = Float64(y + Float64(x * Float64(Float64(1.0 - y) / z))); 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 <= 5e+26) tmp = y + (x * ((1.0 - y) / z)); else tmp = y * (1.0 - (x / z)); end tmp_2 = tmp; end
code[x_, y_, z_] := If[LessEqual[y, 5e+26], N[(y + N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq 5 \cdot 10^{+26}:\\
\;\;\;\;y + x \cdot \frac{1 - y}{z}\\
\mathbf{else}:\\
\;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\
\end{array}
\end{array}
if y < 5.0000000000000001e26Initial program 90.5%
Taylor expanded in y around 0 95.9%
Taylor expanded in x around 0 98.5%
+-commutative98.5%
neg-mul-198.5%
sub-neg98.5%
div-sub98.5%
Simplified98.5%
if 5.0000000000000001e26 < y Initial program 74.8%
Taylor expanded in y around inf 74.8%
associate-/l*100.0%
div-sub100.0%
*-inverses100.0%
Simplified100.0%
Final simplification98.8%
(FPCore (x y z)
:precision binary64
(if (<= y -2.65e-54)
y
(if (or (<= y 2e-147) (and (not (<= y 6.8e-112)) (<= y 1.35e-68)))
(/ x z)
y)))
double code(double x, double y, double z) {
double tmp;
if (y <= -2.65e-54) {
tmp = y;
} else if ((y <= 2e-147) || (!(y <= 6.8e-112) && (y <= 1.35e-68))) {
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 <= (-2.65d-54)) then
tmp = y
else if ((y <= 2d-147) .or. (.not. (y <= 6.8d-112)) .and. (y <= 1.35d-68)) 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 <= -2.65e-54) {
tmp = y;
} else if ((y <= 2e-147) || (!(y <= 6.8e-112) && (y <= 1.35e-68))) {
tmp = x / z;
} else {
tmp = y;
}
return tmp;
}
def code(x, y, z): tmp = 0 if y <= -2.65e-54: tmp = y elif (y <= 2e-147) or (not (y <= 6.8e-112) and (y <= 1.35e-68)): tmp = x / z else: tmp = y return tmp
function code(x, y, z) tmp = 0.0 if (y <= -2.65e-54) tmp = y; elseif ((y <= 2e-147) || (!(y <= 6.8e-112) && (y <= 1.35e-68))) tmp = Float64(x / z); else tmp = y; end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if (y <= -2.65e-54) tmp = y; elseif ((y <= 2e-147) || (~((y <= 6.8e-112)) && (y <= 1.35e-68))) tmp = x / z; else tmp = y; end tmp_2 = tmp; end
code[x_, y_, z_] := If[LessEqual[y, -2.65e-54], y, If[Or[LessEqual[y, 2e-147], And[N[Not[LessEqual[y, 6.8e-112]], $MachinePrecision], LessEqual[y, 1.35e-68]]], N[(x / z), $MachinePrecision], y]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -2.65 \cdot 10^{-54}:\\
\;\;\;\;y\\
\mathbf{elif}\;y \leq 2 \cdot 10^{-147} \lor \neg \left(y \leq 6.8 \cdot 10^{-112}\right) \land y \leq 1.35 \cdot 10^{-68}:\\
\;\;\;\;\frac{x}{z}\\
\mathbf{else}:\\
\;\;\;\;y\\
\end{array}
\end{array}
if y < -2.65000000000000028e-54 or 1.9999999999999999e-147 < y < 6.7999999999999996e-112 or 1.3500000000000001e-68 < y Initial program 78.8%
Taylor expanded in x around 0 56.3%
if -2.65000000000000028e-54 < y < 1.9999999999999999e-147 or 6.7999999999999996e-112 < y < 1.3500000000000001e-68Initial program 100.0%
Taylor expanded in y around 0 79.6%
Final simplification65.4%
(FPCore (x y z) :precision binary64 (if (or (<= x -3.7e+39) (not (<= x 2.6e-26))) (* x (/ (- 1.0 y) z)) (+ y (/ x z))))
double code(double x, double y, double z) {
double tmp;
if ((x <= -3.7e+39) || !(x <= 2.6e-26)) {
tmp = x * ((1.0 - y) / 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 ((x <= (-3.7d+39)) .or. (.not. (x <= 2.6d-26))) then
tmp = x * ((1.0d0 - y) / 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 ((x <= -3.7e+39) || !(x <= 2.6e-26)) {
tmp = x * ((1.0 - y) / z);
} else {
tmp = y + (x / z);
}
return tmp;
}
def code(x, y, z): tmp = 0 if (x <= -3.7e+39) or not (x <= 2.6e-26): tmp = x * ((1.0 - y) / z) else: tmp = y + (x / z) return tmp
function code(x, y, z) tmp = 0.0 if ((x <= -3.7e+39) || !(x <= 2.6e-26)) tmp = Float64(x * Float64(Float64(1.0 - y) / z)); else tmp = Float64(y + Float64(x / z)); end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if ((x <= -3.7e+39) || ~((x <= 2.6e-26))) tmp = x * ((1.0 - y) / z); else tmp = y + (x / z); end tmp_2 = tmp; end
code[x_, y_, z_] := If[Or[LessEqual[x, -3.7e+39], N[Not[LessEqual[x, 2.6e-26]], $MachinePrecision]], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;x \leq -3.7 \cdot 10^{+39} \lor \neg \left(x \leq 2.6 \cdot 10^{-26}\right):\\
\;\;\;\;x \cdot \frac{1 - y}{z}\\
\mathbf{else}:\\
\;\;\;\;y + \frac{x}{z}\\
\end{array}
\end{array}
if x < -3.70000000000000012e39 or 2.6000000000000001e-26 < x Initial program 88.6%
Taylor expanded in x around inf 83.0%
associate-/l*86.3%
mul-1-neg86.3%
unsub-neg86.3%
Simplified86.3%
if -3.70000000000000012e39 < x < 2.6000000000000001e-26Initial program 86.0%
Taylor expanded in z around inf 75.5%
Taylor expanded in x around 0 88.8%
Final simplification87.8%
(FPCore (x y z) :precision binary64 (if (or (<= y -250000000.0) (not (<= y 0.085))) (* y (- 1.0 (/ x z))) (+ y (/ x z))))
double code(double x, double y, double z) {
double tmp;
if ((y <= -250000000.0) || !(y <= 0.085)) {
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 <= (-250000000.0d0)) .or. (.not. (y <= 0.085d0))) 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 <= -250000000.0) || !(y <= 0.085)) {
tmp = y * (1.0 - (x / z));
} else {
tmp = y + (x / z);
}
return tmp;
}
def code(x, y, z): tmp = 0 if (y <= -250000000.0) or not (y <= 0.085): tmp = y * (1.0 - (x / z)) else: tmp = y + (x / z) return tmp
function code(x, y, z) tmp = 0.0 if ((y <= -250000000.0) || !(y <= 0.085)) 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 <= -250000000.0) || ~((y <= 0.085))) tmp = y * (1.0 - (x / z)); else tmp = y + (x / z); end tmp_2 = tmp; end
code[x_, y_, z_] := If[Or[LessEqual[y, -250000000.0], N[Not[LessEqual[y, 0.085]], $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 -250000000 \lor \neg \left(y \leq 0.085\right):\\
\;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\
\mathbf{else}:\\
\;\;\;\;y + \frac{x}{z}\\
\end{array}
\end{array}
if y < -2.5e8 or 0.0850000000000000061 < y Initial program 73.9%
Taylor expanded in y around inf 73.4%
associate-/l*99.4%
div-sub99.4%
*-inverses99.4%
Simplified99.4%
if -2.5e8 < y < 0.0850000000000000061Initial program 99.9%
Taylor expanded in z around inf 97.8%
Taylor expanded in x around 0 97.9%
Final simplification98.6%
(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 87.1%
Taylor expanded in z around inf 68.2%
Taylor expanded in x around 0 78.1%
Final simplification78.1%
(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 87.1%
Taylor expanded in x around 0 43.4%
Final simplification43.4%
(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 2024067
(FPCore (x y z)
:name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
:precision binary64
:alt
(- (+ y (/ x z)) (/ y (/ z x)))
(/ (+ x (* y (- z x))) z))