
(FPCore (x y z) :precision binary64 (+ x (* y (+ z x))))
double code(double x, double y, double z) {
return x + (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 = x + (y * (z + x))
end function
public static double code(double x, double y, double z) {
return x + (y * (z + x));
}
def code(x, y, z): return x + (y * (z + x))
function code(x, y, z) return Float64(x + Float64(y * Float64(z + x))) end
function tmp = code(x, y, z) tmp = x + (y * (z + x)); end
code[x_, y_, z_] := N[(x + N[(y * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
x + y \cdot \left(z + x\right)
\end{array}
Sampling outcomes in binary64 precision:
Herbie found 7 alternatives:
| Alternative | Accuracy | Speedup |
|---|
(FPCore (x y z) :precision binary64 (+ x (* y (+ z x))))
double code(double x, double y, double z) {
return x + (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 = x + (y * (z + x))
end function
public static double code(double x, double y, double z) {
return x + (y * (z + x));
}
def code(x, y, z): return x + (y * (z + x))
function code(x, y, z) return Float64(x + Float64(y * Float64(z + x))) end
function tmp = code(x, y, z) tmp = x + (y * (z + x)); end
code[x_, y_, z_] := N[(x + N[(y * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
x + y \cdot \left(z + x\right)
\end{array}
(FPCore (x y z) :precision binary64 (fma y (+ x z) x))
double code(double x, double y, double z) {
return fma(y, (x + z), x);
}
function code(x, y, z) return fma(y, Float64(x + z), x) end
code[x_, y_, z_] := N[(y * N[(x + z), $MachinePrecision] + x), $MachinePrecision]
\begin{array}{l}
\\
\mathsf{fma}\left(y, x + z, x\right)
\end{array}
Initial program 100.0%
+-commutative100.0%
fma-define100.0%
+-commutative100.0%
Simplified100.0%
Final simplification100.0%
(FPCore (x y z)
:precision binary64
(if (<= y -2e+285)
(* y z)
(if (<= y -2.5e+25)
(* y x)
(if (<= y -3e-9)
(* y z)
(if (<= y 1.55e-50)
x
(if (or (<= y 2.3e+18) (and (not (<= y 7.5e+68)) (<= y 7.8e+213)))
(* y z)
(* y x)))))))
double code(double x, double y, double z) {
double tmp;
if (y <= -2e+285) {
tmp = y * z;
} else if (y <= -2.5e+25) {
tmp = y * x;
} else if (y <= -3e-9) {
tmp = y * z;
} else if (y <= 1.55e-50) {
tmp = x;
} else if ((y <= 2.3e+18) || (!(y <= 7.5e+68) && (y <= 7.8e+213))) {
tmp = y * z;
} else {
tmp = y * x;
}
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 <= (-2d+285)) then
tmp = y * z
else if (y <= (-2.5d+25)) then
tmp = y * x
else if (y <= (-3d-9)) then
tmp = y * z
else if (y <= 1.55d-50) then
tmp = x
else if ((y <= 2.3d+18) .or. (.not. (y <= 7.5d+68)) .and. (y <= 7.8d+213)) then
tmp = y * z
else
tmp = y * x
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if (y <= -2e+285) {
tmp = y * z;
} else if (y <= -2.5e+25) {
tmp = y * x;
} else if (y <= -3e-9) {
tmp = y * z;
} else if (y <= 1.55e-50) {
tmp = x;
} else if ((y <= 2.3e+18) || (!(y <= 7.5e+68) && (y <= 7.8e+213))) {
tmp = y * z;
} else {
tmp = y * x;
}
return tmp;
}
def code(x, y, z): tmp = 0 if y <= -2e+285: tmp = y * z elif y <= -2.5e+25: tmp = y * x elif y <= -3e-9: tmp = y * z elif y <= 1.55e-50: tmp = x elif (y <= 2.3e+18) or (not (y <= 7.5e+68) and (y <= 7.8e+213)): tmp = y * z else: tmp = y * x return tmp
function code(x, y, z) tmp = 0.0 if (y <= -2e+285) tmp = Float64(y * z); elseif (y <= -2.5e+25) tmp = Float64(y * x); elseif (y <= -3e-9) tmp = Float64(y * z); elseif (y <= 1.55e-50) tmp = x; elseif ((y <= 2.3e+18) || (!(y <= 7.5e+68) && (y <= 7.8e+213))) tmp = Float64(y * z); else tmp = Float64(y * x); end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if (y <= -2e+285) tmp = y * z; elseif (y <= -2.5e+25) tmp = y * x; elseif (y <= -3e-9) tmp = y * z; elseif (y <= 1.55e-50) tmp = x; elseif ((y <= 2.3e+18) || (~((y <= 7.5e+68)) && (y <= 7.8e+213))) tmp = y * z; else tmp = y * x; end tmp_2 = tmp; end
code[x_, y_, z_] := If[LessEqual[y, -2e+285], N[(y * z), $MachinePrecision], If[LessEqual[y, -2.5e+25], N[(y * x), $MachinePrecision], If[LessEqual[y, -3e-9], N[(y * z), $MachinePrecision], If[LessEqual[y, 1.55e-50], x, If[Or[LessEqual[y, 2.3e+18], And[N[Not[LessEqual[y, 7.5e+68]], $MachinePrecision], LessEqual[y, 7.8e+213]]], N[(y * z), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -2 \cdot 10^{+285}:\\
\;\;\;\;y \cdot z\\
\mathbf{elif}\;y \leq -2.5 \cdot 10^{+25}:\\
\;\;\;\;y \cdot x\\
\mathbf{elif}\;y \leq -3 \cdot 10^{-9}:\\
\;\;\;\;y \cdot z\\
\mathbf{elif}\;y \leq 1.55 \cdot 10^{-50}:\\
\;\;\;\;x\\
\mathbf{elif}\;y \leq 2.3 \cdot 10^{+18} \lor \neg \left(y \leq 7.5 \cdot 10^{+68}\right) \land y \leq 7.8 \cdot 10^{+213}:\\
\;\;\;\;y \cdot z\\
\mathbf{else}:\\
\;\;\;\;y \cdot x\\
\end{array}
\end{array}
if y < -2e285 or -2.50000000000000012e25 < y < -2.99999999999999998e-9 or 1.5500000000000001e-50 < y < 2.3e18 or 7.49999999999999959e68 < y < 7.8000000000000003e213Initial program 99.9%
Taylor expanded in x around 0 75.2%
if -2e285 < y < -2.50000000000000012e25 or 2.3e18 < y < 7.49999999999999959e68 or 7.8000000000000003e213 < y Initial program 100.0%
Taylor expanded in x around inf 68.3%
+-commutative68.3%
Simplified68.3%
Taylor expanded in y around inf 68.3%
if -2.99999999999999998e-9 < y < 1.5500000000000001e-50Initial program 100.0%
Taylor expanded in y around 0 78.2%
Final simplification74.4%
(FPCore (x y z) :precision binary64 (if (or (<= z -5.5e+72) (not (<= z 3.1e+31))) (* y z) (* x (+ y 1.0))))
double code(double x, double y, double z) {
double tmp;
if ((z <= -5.5e+72) || !(z <= 3.1e+31)) {
tmp = y * z;
} else {
tmp = x * (y + 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 ((z <= (-5.5d+72)) .or. (.not. (z <= 3.1d+31))) then
tmp = y * z
else
tmp = x * (y + 1.0d0)
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if ((z <= -5.5e+72) || !(z <= 3.1e+31)) {
tmp = y * z;
} else {
tmp = x * (y + 1.0);
}
return tmp;
}
def code(x, y, z): tmp = 0 if (z <= -5.5e+72) or not (z <= 3.1e+31): tmp = y * z else: tmp = x * (y + 1.0) return tmp
function code(x, y, z) tmp = 0.0 if ((z <= -5.5e+72) || !(z <= 3.1e+31)) tmp = Float64(y * z); else tmp = Float64(x * Float64(y + 1.0)); end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if ((z <= -5.5e+72) || ~((z <= 3.1e+31))) tmp = y * z; else tmp = x * (y + 1.0); end tmp_2 = tmp; end
code[x_, y_, z_] := If[Or[LessEqual[z, -5.5e+72], N[Not[LessEqual[z, 3.1e+31]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;z \leq -5.5 \cdot 10^{+72} \lor \neg \left(z \leq 3.1 \cdot 10^{+31}\right):\\
\;\;\;\;y \cdot z\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(y + 1\right)\\
\end{array}
\end{array}
if z < -5.5e72 or 3.1000000000000002e31 < z Initial program 100.0%
Taylor expanded in x around 0 76.7%
if -5.5e72 < z < 3.1000000000000002e31Initial program 100.0%
Taylor expanded in x around inf 82.4%
+-commutative82.4%
Simplified82.4%
Final simplification80.4%
(FPCore (x y z) :precision binary64 (if (or (<= y -19.0) (not (<= y 1700000.0))) (* y (+ x z)) (* x (+ y 1.0))))
double code(double x, double y, double z) {
double tmp;
if ((y <= -19.0) || !(y <= 1700000.0)) {
tmp = y * (x + z);
} else {
tmp = x * (y + 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 <= (-19.0d0)) .or. (.not. (y <= 1700000.0d0))) then
tmp = y * (x + z)
else
tmp = x * (y + 1.0d0)
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if ((y <= -19.0) || !(y <= 1700000.0)) {
tmp = y * (x + z);
} else {
tmp = x * (y + 1.0);
}
return tmp;
}
def code(x, y, z): tmp = 0 if (y <= -19.0) or not (y <= 1700000.0): tmp = y * (x + z) else: tmp = x * (y + 1.0) return tmp
function code(x, y, z) tmp = 0.0 if ((y <= -19.0) || !(y <= 1700000.0)) tmp = Float64(y * Float64(x + z)); else tmp = Float64(x * Float64(y + 1.0)); end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if ((y <= -19.0) || ~((y <= 1700000.0))) tmp = y * (x + z); else tmp = x * (y + 1.0); end tmp_2 = tmp; end
code[x_, y_, z_] := If[Or[LessEqual[y, -19.0], N[Not[LessEqual[y, 1700000.0]], $MachinePrecision]], N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -19 \lor \neg \left(y \leq 1700000\right):\\
\;\;\;\;y \cdot \left(x + z\right)\\
\mathbf{else}:\\
\;\;\;\;x \cdot \left(y + 1\right)\\
\end{array}
\end{array}
if y < -19 or 1.7e6 < y Initial program 100.0%
Taylor expanded in y around inf 99.4%
+-commutative99.4%
Simplified99.4%
if -19 < y < 1.7e6Initial program 100.0%
Taylor expanded in x around inf 76.6%
+-commutative76.6%
Simplified76.6%
Final simplification88.3%
(FPCore (x y z) :precision binary64 (if (or (<= y -1.0) (not (<= y 1.0))) (* y x) x))
double code(double x, double y, double z) {
double tmp;
if ((y <= -1.0) || !(y <= 1.0)) {
tmp = y * x;
} else {
tmp = x;
}
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 <= 1.0d0))) then
tmp = y * x
else
tmp = x
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if ((y <= -1.0) || !(y <= 1.0)) {
tmp = y * x;
} else {
tmp = x;
}
return tmp;
}
def code(x, y, z): tmp = 0 if (y <= -1.0) or not (y <= 1.0): tmp = y * x else: tmp = x return tmp
function code(x, y, z) tmp = 0.0 if ((y <= -1.0) || !(y <= 1.0)) tmp = Float64(y * x); else tmp = x; end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if ((y <= -1.0) || ~((y <= 1.0))) tmp = y * x; else tmp = x; end tmp_2 = tmp; end
code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * x), $MachinePrecision], x]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\
\;\;\;\;y \cdot x\\
\mathbf{else}:\\
\;\;\;\;x\\
\end{array}
\end{array}
if y < -1 or 1 < y Initial program 100.0%
Taylor expanded in x around inf 50.7%
+-commutative50.7%
Simplified50.7%
Taylor expanded in y around inf 49.2%
if -1 < y < 1Initial program 100.0%
Taylor expanded in y around 0 73.8%
Final simplification61.0%
(FPCore (x y z) :precision binary64 (+ x (* y (+ x z))))
double code(double x, double y, double z) {
return x + (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 = x + (y * (x + z))
end function
public static double code(double x, double y, double z) {
return x + (y * (x + z));
}
def code(x, y, z): return x + (y * (x + z))
function code(x, y, z) return Float64(x + Float64(y * Float64(x + z))) end
function tmp = code(x, y, z) tmp = x + (y * (x + z)); end
code[x_, y_, z_] := N[(x + N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
x + y \cdot \left(x + z\right)
\end{array}
Initial program 100.0%
Final simplification100.0%
(FPCore (x y z) :precision binary64 x)
double code(double x, double y, double z) {
return x;
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
code = x
end function
public static double code(double x, double y, double z) {
return x;
}
def code(x, y, z): return x
function code(x, y, z) return x end
function tmp = code(x, y, z) tmp = x; end
code[x_, y_, z_] := x
\begin{array}{l}
\\
x
\end{array}
Initial program 100.0%
Taylor expanded in y around 0 37.0%
Final simplification37.0%
herbie shell --seed 2024034
(FPCore (x y z)
:name "Main:bigenough2 from A"
:precision binary64
(+ x (* y (+ z x))))