
(FPCore (a b c) :precision binary64 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
double code(double a, double b, double c) {
return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}
real(8) function code(a, b, c)
real(8), intent (in) :: a
real(8), intent (in) :: b
real(8), intent (in) :: c
code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function
public static double code(double a, double b, double c) {
return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}
def code(a, b, c): return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)
function code(a, b, c) return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) end
function tmp = code(a, b, c) tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a); end
code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
\end{array}
Sampling outcomes in binary64 precision:
Herbie found 9 alternatives:
| Alternative | Accuracy | Speedup |
|---|
(FPCore (a b c) :precision binary64 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
double code(double a, double b, double c) {
return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}
real(8) function code(a, b, c)
real(8), intent (in) :: a
real(8), intent (in) :: b
real(8), intent (in) :: c
code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function
public static double code(double a, double b, double c) {
return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}
def code(a, b, c): return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)
function code(a, b, c) return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) end
function tmp = code(a, b, c) tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a); end
code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
\end{array}
(FPCore (a b c)
:precision binary64
(if (<= b -1.05e+125)
(- (/ b a))
(if (<= b 2.5e-74)
(/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* a 2.0))
(/ (- c) b))))
double code(double a, double b, double c) {
double tmp;
if (b <= -1.05e+125) {
tmp = -(b / a);
} else if (b <= 2.5e-74) {
tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (a * 2.0);
} else {
tmp = -c / b;
}
return tmp;
}
function code(a, b, c) tmp = 0.0 if (b <= -1.05e+125) tmp = Float64(-Float64(b / a)); elseif (b <= 2.5e-74) tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(a * 2.0)); else tmp = Float64(Float64(-c) / b); end return tmp end
code[a_, b_, c_] := If[LessEqual[b, -1.05e+125], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 2.5e-74], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;b \leq -1.05 \cdot 10^{+125}:\\
\;\;\;\;-\frac{b}{a}\\
\mathbf{elif}\;b \leq 2.5 \cdot 10^{-74}:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\
\mathbf{else}:\\
\;\;\;\;\frac{-c}{b}\\
\end{array}
\end{array}
(FPCore (a b c)
:precision binary64
(if (<= b -7e+124)
(- (/ b a))
(if (<= b 2.4e-68)
(/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
(/ (- c) b))))
double code(double a, double b, double c) {
double tmp;
if (b <= -7e+124) {
tmp = -(b / a);
} else if (b <= 2.4e-68) {
tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
} else {
tmp = -c / b;
}
return tmp;
}
real(8) function code(a, b, c)
real(8), intent (in) :: a
real(8), intent (in) :: b
real(8), intent (in) :: c
real(8) :: tmp
if (b <= (-7d+124)) then
tmp = -(b / a)
else if (b <= 2.4d-68) then
tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
else
tmp = -c / b
end if
code = tmp
end function
public static double code(double a, double b, double c) {
double tmp;
if (b <= -7e+124) {
tmp = -(b / a);
} else if (b <= 2.4e-68) {
tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
} else {
tmp = -c / b;
}
return tmp;
}
def code(a, b, c): tmp = 0 if b <= -7e+124: tmp = -(b / a) elif b <= 2.4e-68: tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0) else: tmp = -c / b return tmp
function code(a, b, c) tmp = 0.0 if (b <= -7e+124) tmp = Float64(-Float64(b / a)); elseif (b <= 2.4e-68) tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)); else tmp = Float64(Float64(-c) / b); end return tmp end
function tmp_2 = code(a, b, c) tmp = 0.0; if (b <= -7e+124) tmp = -(b / a); elseif (b <= 2.4e-68) tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0); else tmp = -c / b; end tmp_2 = tmp; end
code[a_, b_, c_] := If[LessEqual[b, -7e+124], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 2.4e-68], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;b \leq -7 \cdot 10^{+124}:\\
\;\;\;\;-\frac{b}{a}\\
\mathbf{elif}\;b \leq 2.4 \cdot 10^{-68}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\
\mathbf{else}:\\
\;\;\;\;\frac{-c}{b}\\
\end{array}
\end{array}
(FPCore (a b c)
:precision binary64
(if (<= b -2.65e-149)
(- (/ c b) (/ b a))
(if (<= b 1.7e-75)
(* (- b (sqrt (* a (* c -4.0)))) (/ -0.5 a))
(/ (- c) b))))
double code(double a, double b, double c) {
double tmp;
if (b <= -2.65e-149) {
tmp = (c / b) - (b / a);
} else if (b <= 1.7e-75) {
tmp = (b - sqrt((a * (c * -4.0)))) * (-0.5 / a);
} else {
tmp = -c / b;
}
return tmp;
}
real(8) function code(a, b, c)
real(8), intent (in) :: a
real(8), intent (in) :: b
real(8), intent (in) :: c
real(8) :: tmp
if (b <= (-2.65d-149)) then
tmp = (c / b) - (b / a)
else if (b <= 1.7d-75) then
tmp = (b - sqrt((a * (c * (-4.0d0))))) * ((-0.5d0) / a)
else
tmp = -c / b
end if
code = tmp
end function
public static double code(double a, double b, double c) {
double tmp;
if (b <= -2.65e-149) {
tmp = (c / b) - (b / a);
} else if (b <= 1.7e-75) {
tmp = (b - Math.sqrt((a * (c * -4.0)))) * (-0.5 / a);
} else {
tmp = -c / b;
}
return tmp;
}
def code(a, b, c): tmp = 0 if b <= -2.65e-149: tmp = (c / b) - (b / a) elif b <= 1.7e-75: tmp = (b - math.sqrt((a * (c * -4.0)))) * (-0.5 / a) else: tmp = -c / b return tmp
function code(a, b, c) tmp = 0.0 if (b <= -2.65e-149) tmp = Float64(Float64(c / b) - Float64(b / a)); elseif (b <= 1.7e-75) tmp = Float64(Float64(b - sqrt(Float64(a * Float64(c * -4.0)))) * Float64(-0.5 / a)); else tmp = Float64(Float64(-c) / b); end return tmp end
function tmp_2 = code(a, b, c) tmp = 0.0; if (b <= -2.65e-149) tmp = (c / b) - (b / a); elseif (b <= 1.7e-75) tmp = (b - sqrt((a * (c * -4.0)))) * (-0.5 / a); else tmp = -c / b; end tmp_2 = tmp; end
code[a_, b_, c_] := If[LessEqual[b, -2.65e-149], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.7e-75], N[(N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;b \leq -2.65 \cdot 10^{-149}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\
\mathbf{elif}\;b \leq 1.7 \cdot 10^{-75}:\\
\;\;\;\;\left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\
\mathbf{else}:\\
\;\;\;\;\frac{-c}{b}\\
\end{array}
\end{array}
(FPCore (a b c)
:precision binary64
(if (<= b -2.65e-149)
(- (/ c b) (/ b a))
(if (<= b 6.6e-70)
(/ (- (sqrt (* -4.0 (* a c))) b) (* a 2.0))
(/ (- c) b))))
double code(double a, double b, double c) {
double tmp;
if (b <= -2.65e-149) {
tmp = (c / b) - (b / a);
} else if (b <= 6.6e-70) {
tmp = (sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
} else {
tmp = -c / b;
}
return tmp;
}
real(8) function code(a, b, c)
real(8), intent (in) :: a
real(8), intent (in) :: b
real(8), intent (in) :: c
real(8) :: tmp
if (b <= (-2.65d-149)) then
tmp = (c / b) - (b / a)
else if (b <= 6.6d-70) then
tmp = (sqrt(((-4.0d0) * (a * c))) - b) / (a * 2.0d0)
else
tmp = -c / b
end if
code = tmp
end function
public static double code(double a, double b, double c) {
double tmp;
if (b <= -2.65e-149) {
tmp = (c / b) - (b / a);
} else if (b <= 6.6e-70) {
tmp = (Math.sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
} else {
tmp = -c / b;
}
return tmp;
}
def code(a, b, c): tmp = 0 if b <= -2.65e-149: tmp = (c / b) - (b / a) elif b <= 6.6e-70: tmp = (math.sqrt((-4.0 * (a * c))) - b) / (a * 2.0) else: tmp = -c / b return tmp
function code(a, b, c) tmp = 0.0 if (b <= -2.65e-149) tmp = Float64(Float64(c / b) - Float64(b / a)); elseif (b <= 6.6e-70) tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b) / Float64(a * 2.0)); else tmp = Float64(Float64(-c) / b); end return tmp end
function tmp_2 = code(a, b, c) tmp = 0.0; if (b <= -2.65e-149) tmp = (c / b) - (b / a); elseif (b <= 6.6e-70) tmp = (sqrt((-4.0 * (a * c))) - b) / (a * 2.0); else tmp = -c / b; end tmp_2 = tmp; end
code[a_, b_, c_] := If[LessEqual[b, -2.65e-149], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.6e-70], N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;b \leq -2.65 \cdot 10^{-149}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\
\mathbf{elif}\;b \leq 6.6 \cdot 10^{-70}:\\
\;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\
\mathbf{else}:\\
\;\;\;\;\frac{-c}{b}\\
\end{array}
\end{array}
(FPCore (a b c) :precision binary64 (if (<= b -1e-310) (- (/ c b) (/ b a)) (/ (- c) b)))
double code(double a, double b, double c) {
double tmp;
if (b <= -1e-310) {
tmp = (c / b) - (b / a);
} else {
tmp = -c / b;
}
return tmp;
}
real(8) function code(a, b, c)
real(8), intent (in) :: a
real(8), intent (in) :: b
real(8), intent (in) :: c
real(8) :: tmp
if (b <= (-1d-310)) then
tmp = (c / b) - (b / a)
else
tmp = -c / b
end if
code = tmp
end function
public static double code(double a, double b, double c) {
double tmp;
if (b <= -1e-310) {
tmp = (c / b) - (b / a);
} else {
tmp = -c / b;
}
return tmp;
}
def code(a, b, c): tmp = 0 if b <= -1e-310: tmp = (c / b) - (b / a) else: tmp = -c / b return tmp
function code(a, b, c) tmp = 0.0 if (b <= -1e-310) tmp = Float64(Float64(c / b) - Float64(b / a)); else tmp = Float64(Float64(-c) / b); end return tmp end
function tmp_2 = code(a, b, c) tmp = 0.0; if (b <= -1e-310) tmp = (c / b) - (b / a); else tmp = -c / b; end tmp_2 = tmp; end
code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\
\mathbf{else}:\\
\;\;\;\;\frac{-c}{b}\\
\end{array}
\end{array}
(FPCore (a b c) :precision binary64 (if (<= b 1.5e-28) (- (/ b a)) (/ c b)))
double code(double a, double b, double c) {
double tmp;
if (b <= 1.5e-28) {
tmp = -(b / a);
} else {
tmp = c / b;
}
return tmp;
}
real(8) function code(a, b, c)
real(8), intent (in) :: a
real(8), intent (in) :: b
real(8), intent (in) :: c
real(8) :: tmp
if (b <= 1.5d-28) then
tmp = -(b / a)
else
tmp = c / b
end if
code = tmp
end function
public static double code(double a, double b, double c) {
double tmp;
if (b <= 1.5e-28) {
tmp = -(b / a);
} else {
tmp = c / b;
}
return tmp;
}
def code(a, b, c): tmp = 0 if b <= 1.5e-28: tmp = -(b / a) else: tmp = c / b return tmp
function code(a, b, c) tmp = 0.0 if (b <= 1.5e-28) tmp = Float64(-Float64(b / a)); else tmp = Float64(c / b); end return tmp end
function tmp_2 = code(a, b, c) tmp = 0.0; if (b <= 1.5e-28) tmp = -(b / a); else tmp = c / b; end tmp_2 = tmp; end
code[a_, b_, c_] := If[LessEqual[b, 1.5e-28], (-N[(b / a), $MachinePrecision]), N[(c / b), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;b \leq 1.5 \cdot 10^{-28}:\\
\;\;\;\;-\frac{b}{a}\\
\mathbf{else}:\\
\;\;\;\;\frac{c}{b}\\
\end{array}
\end{array}
(FPCore (a b c) :precision binary64 (if (<= b -1e-310) (- (/ b a)) (/ (- c) b)))
double code(double a, double b, double c) {
double tmp;
if (b <= -1e-310) {
tmp = -(b / a);
} else {
tmp = -c / b;
}
return tmp;
}
real(8) function code(a, b, c)
real(8), intent (in) :: a
real(8), intent (in) :: b
real(8), intent (in) :: c
real(8) :: tmp
if (b <= (-1d-310)) then
tmp = -(b / a)
else
tmp = -c / b
end if
code = tmp
end function
public static double code(double a, double b, double c) {
double tmp;
if (b <= -1e-310) {
tmp = -(b / a);
} else {
tmp = -c / b;
}
return tmp;
}
def code(a, b, c): tmp = 0 if b <= -1e-310: tmp = -(b / a) else: tmp = -c / b return tmp
function code(a, b, c) tmp = 0.0 if (b <= -1e-310) tmp = Float64(-Float64(b / a)); else tmp = Float64(Float64(-c) / b); end return tmp end
function tmp_2 = code(a, b, c) tmp = 0.0; if (b <= -1e-310) tmp = -(b / a); else tmp = -c / b; end tmp_2 = tmp; end
code[a_, b_, c_] := If[LessEqual[b, -1e-310], (-N[(b / a), $MachinePrecision]), N[((-c) / b), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\
\;\;\;\;-\frac{b}{a}\\
\mathbf{else}:\\
\;\;\;\;\frac{-c}{b}\\
\end{array}
\end{array}
(FPCore (a b c) :precision binary64 (/ b a))
double code(double a, double b, double c) {
return b / a;
}
real(8) function code(a, b, c)
real(8), intent (in) :: a
real(8), intent (in) :: b
real(8), intent (in) :: c
code = b / a
end function
public static double code(double a, double b, double c) {
return b / a;
}
def code(a, b, c): return b / a
function code(a, b, c) return Float64(b / a) end
function tmp = code(a, b, c) tmp = b / a; end
code[a_, b_, c_] := N[(b / a), $MachinePrecision]
\begin{array}{l}
\\
\frac{b}{a}
\end{array}
(FPCore (a b c) :precision binary64 (/ c b))
double code(double a, double b, double c) {
return c / b;
}
real(8) function code(a, b, c)
real(8), intent (in) :: a
real(8), intent (in) :: b
real(8), intent (in) :: c
code = c / b
end function
public static double code(double a, double b, double c) {
return c / b;
}
def code(a, b, c): return c / b
function code(a, b, c) return Float64(c / b) end
function tmp = code(a, b, c) tmp = c / b; end
code[a_, b_, c_] := N[(c / b), $MachinePrecision]
\begin{array}{l}
\\
\frac{c}{b}
\end{array}
herbie shell --seed 2024008
(FPCore (a b c)
:name "Quadratic roots, full range"
:precision binary64
(/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))