Average Error: 34.2 → 10.6
Time: 13.2s
Precision: binary64
Cost: 7368
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.3 \cdot 10^{+93}:\\ \;\;\;\;\frac{b_2}{\frac{a}{-2}}\\ \mathbf{elif}\;b_2 \leq 2.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]
(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))
↓
(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.3e+93)
   (/ b_2 (/ a -2.0))
   (if (<= b_2 2.8e-117)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (* -0.5 (/ c b_2)))))
double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
}
↓
double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.3e+93) {
		tmp = b_2 / (a / -2.0);
	} else if (b_2 <= 2.8e-117) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
end function
↓
real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.3d+93)) then
        tmp = b_2 / (a / (-2.0d0))
    else if (b_2 <= 2.8d-117) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}
↓
public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.3e+93) {
		tmp = b_2 / (a / -2.0);
	} else if (b_2 <= 2.8e-117) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a
↓
def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.3e+93:
		tmp = b_2 / (a / -2.0)
	elif b_2 <= 2.8e-117:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end
↓
function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.3e+93)
		tmp = Float64(b_2 / Float64(a / -2.0));
	elseif (b_2 <= 2.8e-117)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
end
↓
function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.3e+93)
		tmp = b_2 / (a / -2.0);
	elseif (b_2 <= 2.8e-117)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]
↓
code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.3e+93], N[(b$95$2 / N[(a / -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.8e-117], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -2.3 \cdot 10^{+93}:\\
\;\;\;\;\frac{b_2}{\frac{a}{-2}}\\

\mathbf{elif}\;b_2 \leq 2.8 \cdot 10^{-117}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b_2}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -2.3000000000000002e93

    1. Initial program 45.6

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

    2. Taylor expanded in b_2 around -inf 4.4

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

    3. Simplified4.4

      \[\leadsto \color{blue}{\frac{b_2}{\frac{a}{-2}}} \]
      Proof
      (/.f64 b_2 (/.f64 a -2)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 b_2 a) -2)): 1 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -2 (/.f64 b_2 a))): 0 points increase in error, 0 points decrease in error

    if -2.3000000000000002e93 < b_2 < 2.8e-117

    1. Initial program 12.4

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

    if 2.8e-117 < b_2

    1. Initial program 51.7

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

    2. Taylor expanded in b_2 around inf 22.5

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{c \cdot a}{b_2}}}{a} \]

    3. Simplified24.0

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}}{a} \]
      Proof
      (*.f64 -1/2 (/.f64 c (/.f64 b_2 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c a) b_2))): 42 points increase in error, 41 points decrease in error

    4. Taylor expanded in c around 0 11.2

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.3 \cdot 10^{+93}:\\ \;\;\;\;\frac{b_2}{\frac{a}{-2}}\\ \mathbf{elif}\;b_2 \leq 2.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error13.6
Cost7176
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -8.2 \cdot 10^{-52}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \mathbf{elif}\;b_2 \leq 2.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]
Alternative 2
Error53.7
Cost452
\[\begin{array}{l} \mathbf{if}\;b_2 \leq 0.00106:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot 0.5\\ \end{array} \]
Alternative 3
Error53.7
Cost452
\[\begin{array}{l} \mathbf{if}\;b_2 \leq 0.00106:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{b_2}{c}}\\ \end{array} \]
Alternative 4
Error36.7
Cost452
\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.55 \cdot 10^{-262}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \end{array} \]
Alternative 5
Error36.7
Cost452
\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.55 \cdot 10^{-262}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]
Alternative 6
Error22.6
Cost452
\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.55 \cdot 10^{-262}:\\ \;\;\;\;\frac{b_2}{\frac{a}{-2}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]
Alternative 7
Error59.2
Cost256
\[\frac{-b_2}{a} \]

Error

Reproduce

herbie shell --seed 2022300 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))