
(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 -1.0)
(* y x)
(if (<= y 7e-143)
x
(if (<= y 2.4e-47)
(* y z)
(if (<= y 2.8e-14)
x
(if (or (<= y 1650000.0) (not (<= y 1.6e+124))) (* y z) (* y x)))))))
double code(double x, double y, double z) {
double tmp;
if (y <= -1.0) {
tmp = y * x;
} else if (y <= 7e-143) {
tmp = x;
} else if (y <= 2.4e-47) {
tmp = y * z;
} else if (y <= 2.8e-14) {
tmp = x;
} else if ((y <= 1650000.0) || !(y <= 1.6e+124)) {
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 <= (-1.0d0)) then
tmp = y * x
else if (y <= 7d-143) then
tmp = x
else if (y <= 2.4d-47) then
tmp = y * z
else if (y <= 2.8d-14) then
tmp = x
else if ((y <= 1650000.0d0) .or. (.not. (y <= 1.6d+124))) 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 <= -1.0) {
tmp = y * x;
} else if (y <= 7e-143) {
tmp = x;
} else if (y <= 2.4e-47) {
tmp = y * z;
} else if (y <= 2.8e-14) {
tmp = x;
} else if ((y <= 1650000.0) || !(y <= 1.6e+124)) {
tmp = y * z;
} else {
tmp = y * x;
}
return tmp;
}
def code(x, y, z): tmp = 0 if y <= -1.0: tmp = y * x elif y <= 7e-143: tmp = x elif y <= 2.4e-47: tmp = y * z elif y <= 2.8e-14: tmp = x elif (y <= 1650000.0) or not (y <= 1.6e+124): tmp = y * z else: tmp = y * x return tmp
function code(x, y, z) tmp = 0.0 if (y <= -1.0) tmp = Float64(y * x); elseif (y <= 7e-143) tmp = x; elseif (y <= 2.4e-47) tmp = Float64(y * z); elseif (y <= 2.8e-14) tmp = x; elseif ((y <= 1650000.0) || !(y <= 1.6e+124)) 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 <= -1.0) tmp = y * x; elseif (y <= 7e-143) tmp = x; elseif (y <= 2.4e-47) tmp = y * z; elseif (y <= 2.8e-14) tmp = x; elseif ((y <= 1650000.0) || ~((y <= 1.6e+124))) tmp = y * z; else tmp = y * x; end tmp_2 = tmp; end
code[x_, y_, z_] := If[LessEqual[y, -1.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 7e-143], x, If[LessEqual[y, 2.4e-47], N[(y * z), $MachinePrecision], If[LessEqual[y, 2.8e-14], x, If[Or[LessEqual[y, 1650000.0], N[Not[LessEqual[y, 1.6e+124]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -1:\\
\;\;\;\;y \cdot x\\
\mathbf{elif}\;y \leq 7 \cdot 10^{-143}:\\
\;\;\;\;x\\
\mathbf{elif}\;y \leq 2.4 \cdot 10^{-47}:\\
\;\;\;\;y \cdot z\\
\mathbf{elif}\;y \leq 2.8 \cdot 10^{-14}:\\
\;\;\;\;x\\
\mathbf{elif}\;y \leq 1650000 \lor \neg \left(y \leq 1.6 \cdot 10^{+124}\right):\\
\;\;\;\;y \cdot z\\
\mathbf{else}:\\
\;\;\;\;y \cdot x\\
\end{array}
\end{array}
if y < -1 or 1.65e6 < y < 1.59999999999999996e124Initial program 100.0%
Taylor expanded in y around inf 98.3%
+-commutative98.3%
Simplified98.3%
Taylor expanded in z around 0 63.5%
*-commutative63.5%
Simplified63.5%
if -1 < y < 7.00000000000000011e-143 or 2.3999999999999999e-47 < y < 2.8000000000000001e-14Initial program 100.0%
Taylor expanded in y around 0 72.4%
if 7.00000000000000011e-143 < y < 2.3999999999999999e-47 or 2.8000000000000001e-14 < y < 1.65e6 or 1.59999999999999996e124 < y Initial program 100.0%
Taylor expanded in x around 0 64.8%
Final simplification67.4%
(FPCore (x y z) :precision binary64 (if (or (<= x -1.35e-57) (not (<= x 4.2e-38))) (* x (+ y 1.0)) (* y z)))
double code(double x, double y, double z) {
double tmp;
if ((x <= -1.35e-57) || !(x <= 4.2e-38)) {
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 <= (-1.35d-57)) .or. (.not. (x <= 4.2d-38))) 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 <= -1.35e-57) || !(x <= 4.2e-38)) {
tmp = x * (y + 1.0);
} else {
tmp = y * z;
}
return tmp;
}
def code(x, y, z): tmp = 0 if (x <= -1.35e-57) or not (x <= 4.2e-38): tmp = x * (y + 1.0) else: tmp = y * z return tmp
function code(x, y, z) tmp = 0.0 if ((x <= -1.35e-57) || !(x <= 4.2e-38)) 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 <= -1.35e-57) || ~((x <= 4.2e-38))) tmp = x * (y + 1.0); else tmp = y * z; end tmp_2 = tmp; end
code[x_, y_, z_] := If[Or[LessEqual[x, -1.35e-57], N[Not[LessEqual[x, 4.2e-38]], $MachinePrecision]], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;x \leq -1.35 \cdot 10^{-57} \lor \neg \left(x \leq 4.2 \cdot 10^{-38}\right):\\
\;\;\;\;x \cdot \left(y + 1\right)\\
\mathbf{else}:\\
\;\;\;\;y \cdot z\\
\end{array}
\end{array}
if x < -1.3500000000000001e-57 or 4.20000000000000026e-38 < x Initial program 100.0%
Taylor expanded in x around inf 85.6%
+-commutative85.6%
Simplified85.6%
if -1.3500000000000001e-57 < x < 4.20000000000000026e-38Initial program 100.0%
Taylor expanded in x around 0 78.4%
Final simplification82.6%
(FPCore (x y z) :precision binary64 (if (or (<= x -2.8e+27) (not (<= x 1.5e-40))) (* x (+ y 1.0)) (* y (+ x z))))
double code(double x, double y, double z) {
double tmp;
if ((x <= -2.8e+27) || !(x <= 1.5e-40)) {
tmp = x * (y + 1.0);
} 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 <= (-2.8d+27)) .or. (.not. (x <= 1.5d-40))) then
tmp = x * (y + 1.0d0)
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 <= -2.8e+27) || !(x <= 1.5e-40)) {
tmp = x * (y + 1.0);
} else {
tmp = y * (x + z);
}
return tmp;
}
def code(x, y, z): tmp = 0 if (x <= -2.8e+27) or not (x <= 1.5e-40): tmp = x * (y + 1.0) else: tmp = y * (x + z) return tmp
function code(x, y, z) tmp = 0.0 if ((x <= -2.8e+27) || !(x <= 1.5e-40)) tmp = Float64(x * Float64(y + 1.0)); else tmp = Float64(y * Float64(x + z)); end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if ((x <= -2.8e+27) || ~((x <= 1.5e-40))) tmp = x * (y + 1.0); else tmp = y * (x + z); end tmp_2 = tmp; end
code[x_, y_, z_] := If[Or[LessEqual[x, -2.8e+27], N[Not[LessEqual[x, 1.5e-40]], $MachinePrecision]], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;x \leq -2.8 \cdot 10^{+27} \lor \neg \left(x \leq 1.5 \cdot 10^{-40}\right):\\
\;\;\;\;x \cdot \left(y + 1\right)\\
\mathbf{else}:\\
\;\;\;\;y \cdot \left(x + z\right)\\
\end{array}
\end{array}
if x < -2.7999999999999999e27 or 1.5000000000000001e-40 < x Initial program 100.0%
Taylor expanded in x around inf 88.4%
+-commutative88.4%
Simplified88.4%
if -2.7999999999999999e27 < x < 1.5000000000000001e-40Initial program 100.0%
Taylor expanded in y around inf 82.9%
+-commutative82.9%
Simplified82.9%
Final simplification85.8%
(FPCore (x y z) :precision binary64 (if (or (<= y -1.0) (not (<= y 1600000.0))) (* y x) x))
double code(double x, double y, double z) {
double tmp;
if ((y <= -1.0) || !(y <= 1600000.0)) {
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 <= 1600000.0d0))) 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 <= 1600000.0)) {
tmp = y * x;
} else {
tmp = x;
}
return tmp;
}
def code(x, y, z): tmp = 0 if (y <= -1.0) or not (y <= 1600000.0): tmp = y * x else: tmp = x return tmp
function code(x, y, z) tmp = 0.0 if ((y <= -1.0) || !(y <= 1600000.0)) 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 <= 1600000.0))) 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, 1600000.0]], $MachinePrecision]], N[(y * x), $MachinePrecision], x]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1600000\right):\\
\;\;\;\;y \cdot x\\
\mathbf{else}:\\
\;\;\;\;x\\
\end{array}
\end{array}
if y < -1 or 1.6e6 < y Initial program 100.0%
Taylor expanded in y around inf 98.9%
+-commutative98.9%
Simplified98.9%
Taylor expanded in z around 0 55.9%
*-commutative55.9%
Simplified55.9%
if -1 < y < 1.6e6Initial program 100.0%
Taylor expanded in y around 0 62.3%
Final simplification59.3%
(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 34.4%
Final simplification34.4%
herbie shell --seed 2024076
(FPCore (x y z)
:name "Main:bigenough2 from A"
:precision binary64
(+ x (* y (+ z x))))