
(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%
(FPCore (x y z)
:precision binary64
(if (<= y -1.35e+213)
(* y x)
(if (<= y -1.36e+100)
(* y z)
(if (<= y -500.0)
(* y x)
(if (<= y -5.6e-56)
(* y z)
(if (<= y 9e-31) x (if (<= y 2.5e+52) (* y z) (* y x))))))))
double code(double x, double y, double z) {
double tmp;
if (y <= -1.35e+213) {
tmp = y * x;
} else if (y <= -1.36e+100) {
tmp = y * z;
} else if (y <= -500.0) {
tmp = y * x;
} else if (y <= -5.6e-56) {
tmp = y * z;
} else if (y <= 9e-31) {
tmp = x;
} else if (y <= 2.5e+52) {
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 <= (-1.35d+213)) then
tmp = y * x
else if (y <= (-1.36d+100)) then
tmp = y * z
else if (y <= (-500.0d0)) then
tmp = y * x
else if (y <= (-5.6d-56)) then
tmp = y * z
else if (y <= 9d-31) then
tmp = x
else if (y <= 2.5d+52) 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 <= -1.35e+213) {
tmp = y * x;
} else if (y <= -1.36e+100) {
tmp = y * z;
} else if (y <= -500.0) {
tmp = y * x;
} else if (y <= -5.6e-56) {
tmp = y * z;
} else if (y <= 9e-31) {
tmp = x;
} else if (y <= 2.5e+52) {
tmp = y * z;
} else {
tmp = y * x;
}
return tmp;
}
def code(x, y, z): tmp = 0 if y <= -1.35e+213: tmp = y * x elif y <= -1.36e+100: tmp = y * z elif y <= -500.0: tmp = y * x elif y <= -5.6e-56: tmp = y * z elif y <= 9e-31: tmp = x elif y <= 2.5e+52: tmp = y * z else: tmp = y * x return tmp
function code(x, y, z) tmp = 0.0 if (y <= -1.35e+213) tmp = Float64(y * x); elseif (y <= -1.36e+100) tmp = Float64(y * z); elseif (y <= -500.0) tmp = Float64(y * x); elseif (y <= -5.6e-56) tmp = Float64(y * z); elseif (y <= 9e-31) tmp = x; elseif (y <= 2.5e+52) 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 <= -1.35e+213) tmp = y * x; elseif (y <= -1.36e+100) tmp = y * z; elseif (y <= -500.0) tmp = y * x; elseif (y <= -5.6e-56) tmp = y * z; elseif (y <= 9e-31) tmp = x; elseif (y <= 2.5e+52) tmp = y * z; else tmp = y * x; end tmp_2 = tmp; end
code[x_, y_, z_] := If[LessEqual[y, -1.35e+213], N[(y * x), $MachinePrecision], If[LessEqual[y, -1.36e+100], N[(y * z), $MachinePrecision], If[LessEqual[y, -500.0], N[(y * x), $MachinePrecision], If[LessEqual[y, -5.6e-56], N[(y * z), $MachinePrecision], If[LessEqual[y, 9e-31], x, If[LessEqual[y, 2.5e+52], N[(y * z), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -1.35 \cdot 10^{+213}:\\
\;\;\;\;y \cdot x\\
\mathbf{elif}\;y \leq -1.36 \cdot 10^{+100}:\\
\;\;\;\;y \cdot z\\
\mathbf{elif}\;y \leq -500:\\
\;\;\;\;y \cdot x\\
\mathbf{elif}\;y \leq -5.6 \cdot 10^{-56}:\\
\;\;\;\;y \cdot z\\
\mathbf{elif}\;y \leq 9 \cdot 10^{-31}:\\
\;\;\;\;x\\
\mathbf{elif}\;y \leq 2.5 \cdot 10^{+52}:\\
\;\;\;\;y \cdot z\\
\mathbf{else}:\\
\;\;\;\;y \cdot x\\
\end{array}
\end{array}
if y < -1.35e213 or -1.35999999999999994e100 < y < -500 or 2.5e52 < y Initial program 100.0%
Taylor expanded in y around inf 98.4%
+-commutative98.4%
Simplified98.4%
Taylor expanded in z around 0 63.3%
*-commutative63.3%
Simplified63.3%
if -1.35e213 < y < -1.35999999999999994e100 or -500 < y < -5.59999999999999986e-56 or 9.0000000000000008e-31 < y < 2.5e52Initial program 100.0%
Taylor expanded in x around 0 71.7%
if -5.59999999999999986e-56 < y < 9.0000000000000008e-31Initial program 100.0%
Taylor expanded in y around 0 73.2%
(FPCore (x y z)
:precision binary64
(if (or (<= x -3.6e-27)
(and (not (<= x -5e-212))
(or (<= x -1.45e-222) (not (<= x 1.7e-109)))))
(* x (+ y 1.0))
(* y z)))
double code(double x, double y, double z) {
double tmp;
if ((x <= -3.6e-27) || (!(x <= -5e-212) && ((x <= -1.45e-222) || !(x <= 1.7e-109)))) {
tmp = x * (y + 1.0);
} else {
tmp = 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 <= (-3.6d-27)) .or. (.not. (x <= (-5d-212))) .and. (x <= (-1.45d-222)) .or. (.not. (x <= 1.7d-109))) then
tmp = x * (y + 1.0d0)
else
tmp = y * z
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if ((x <= -3.6e-27) || (!(x <= -5e-212) && ((x <= -1.45e-222) || !(x <= 1.7e-109)))) {
tmp = x * (y + 1.0);
} else {
tmp = y * z;
}
return tmp;
}
def code(x, y, z): tmp = 0 if (x <= -3.6e-27) or (not (x <= -5e-212) and ((x <= -1.45e-222) or not (x <= 1.7e-109))): tmp = x * (y + 1.0) else: tmp = y * z return tmp
function code(x, y, z) tmp = 0.0 if ((x <= -3.6e-27) || (!(x <= -5e-212) && ((x <= -1.45e-222) || !(x <= 1.7e-109)))) tmp = Float64(x * Float64(y + 1.0)); else tmp = Float64(y * z); end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if ((x <= -3.6e-27) || (~((x <= -5e-212)) && ((x <= -1.45e-222) || ~((x <= 1.7e-109))))) tmp = x * (y + 1.0); else tmp = y * z; end tmp_2 = tmp; end
code[x_, y_, z_] := If[Or[LessEqual[x, -3.6e-27], And[N[Not[LessEqual[x, -5e-212]], $MachinePrecision], Or[LessEqual[x, -1.45e-222], N[Not[LessEqual[x, 1.7e-109]], $MachinePrecision]]]], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;x \leq -3.6 \cdot 10^{-27} \lor \neg \left(x \leq -5 \cdot 10^{-212}\right) \land \left(x \leq -1.45 \cdot 10^{-222} \lor \neg \left(x \leq 1.7 \cdot 10^{-109}\right)\right):\\
\;\;\;\;x \cdot \left(y + 1\right)\\
\mathbf{else}:\\
\;\;\;\;y \cdot z\\
\end{array}
\end{array}
if x < -3.5999999999999999e-27 or -5.00000000000000043e-212 < x < -1.4500000000000001e-222 or 1.70000000000000006e-109 < x Initial program 100.0%
Taylor expanded in x around inf 84.7%
+-commutative84.7%
Simplified84.7%
if -3.5999999999999999e-27 < x < -5.00000000000000043e-212 or -1.4500000000000001e-222 < x < 1.70000000000000006e-109Initial program 100.0%
Taylor expanded in x around 0 76.1%
Final simplification81.5%
(FPCore (x y z) :precision binary64 (if (or (<= y -4.8e-56) (not (<= y 1.16e-37))) (* y (+ x z)) x))
double code(double x, double y, double z) {
double tmp;
if ((y <= -4.8e-56) || !(y <= 1.16e-37)) {
tmp = y * (x + z);
} 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 <= (-4.8d-56)) .or. (.not. (y <= 1.16d-37))) then
tmp = y * (x + z)
else
tmp = x
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if ((y <= -4.8e-56) || !(y <= 1.16e-37)) {
tmp = y * (x + z);
} else {
tmp = x;
}
return tmp;
}
def code(x, y, z): tmp = 0 if (y <= -4.8e-56) or not (y <= 1.16e-37): tmp = y * (x + z) else: tmp = x return tmp
function code(x, y, z) tmp = 0.0 if ((y <= -4.8e-56) || !(y <= 1.16e-37)) tmp = Float64(y * Float64(x + z)); else tmp = x; end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if ((y <= -4.8e-56) || ~((y <= 1.16e-37))) tmp = y * (x + z); else tmp = x; end tmp_2 = tmp; end
code[x_, y_, z_] := If[Or[LessEqual[y, -4.8e-56], N[Not[LessEqual[y, 1.16e-37]], $MachinePrecision]], N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision], x]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -4.8 \cdot 10^{-56} \lor \neg \left(y \leq 1.16 \cdot 10^{-37}\right):\\
\;\;\;\;y \cdot \left(x + z\right)\\
\mathbf{else}:\\
\;\;\;\;x\\
\end{array}
\end{array}
if y < -4.80000000000000001e-56 or 1.15999999999999998e-37 < y Initial program 100.0%
Taylor expanded in y around inf 95.6%
+-commutative95.6%
Simplified95.6%
if -4.80000000000000001e-56 < y < 1.15999999999999998e-37Initial program 100.0%
Taylor expanded in y around 0 73.6%
Final simplification86.1%
(FPCore (x y z) :precision binary64 (if (or (<= y -1.0) (not (<= y 1.05e-9))) (* y x) x))
double code(double x, double y, double z) {
double tmp;
if ((y <= -1.0) || !(y <= 1.05e-9)) {
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 <= 1.05d-9))) 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 <= 1.05e-9)) {
tmp = y * x;
} else {
tmp = x;
}
return tmp;
}
def code(x, y, z): tmp = 0 if (y <= -1.0) or not (y <= 1.05e-9): tmp = y * x else: tmp = x return tmp
function code(x, y, z) tmp = 0.0 if ((y <= -1.0) || !(y <= 1.05e-9)) 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 <= 1.05e-9))) 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, 1.05e-9]], $MachinePrecision]], N[(y * x), $MachinePrecision], x]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.05 \cdot 10^{-9}\right):\\
\;\;\;\;y \cdot x\\
\mathbf{else}:\\
\;\;\;\;x\\
\end{array}
\end{array}
if y < -1 or 1.0500000000000001e-9 < y Initial program 100.0%
Taylor expanded in y around inf 98.6%
+-commutative98.6%
Simplified98.6%
Taylor expanded in z around 0 53.2%
*-commutative53.2%
Simplified53.2%
if -1 < y < 1.0500000000000001e-9Initial program 100.0%
Taylor expanded in y around 0 68.8%
Final simplification60.8%
(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 34.7%
herbie shell --seed 2024103
(FPCore (x y z)
:name "Main:bigenough2 from A"
:precision binary64
(+ x (* y (+ z x))))