Quadratic roots, medium range

Percentage Accurate: 31.2% → 99.6%

Time: 13.2s

Alternatives: 6

Speedup: 3.6×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

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

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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]

Initial Program: 31.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

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

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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]

Alternative 1: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(a \cdot c\right) \cdot 4}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (/ (* (* a c) 4.0) (* a -2.0)) (+ b (sqrt (fma a (* c -4.0) (* b b))))))

C

double code(double a, double b, double c) {
	return (((a * c) * 4.0) / (a * -2.0)) / (b + sqrt(fma(a, (c * -4.0), (b * b))));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(a * c) * 4.0) / Float64(a * -2.0)) / Float64(b + sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied rewrites 33.2%

   \[\leadsto \frac{\color{blue}{\frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} - \frac{\mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{2 \cdot a} \]

6. Applied rewrites 31.0%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, -b, c \cdot \left(a \cdot -4\right)\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} \cdot \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}{2 \cdot a} \]

8. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(b, b + \left(-b\right), a \cdot \left(-4 \cdot c\right)\right)}{b \cdot \left(b + \left(-b\right)\right) - a \cdot \left(-4 \cdot c\right)} \cdot \left(b \cdot \left(b + \left(-b\right)\right) - a \cdot \left(-4 \cdot c\right)\right)}{b + \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}}}}{2 \cdot a} \]

10. Applied rewrites 99.5%

    \[\leadsto \color{blue}{\frac{\frac{\left(a \cdot c\right) \cdot 4}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}}} \]

11. Final simplification 99.5%

    \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot 4}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} \]
12. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (/ (* a (* c -4.0)) (+ b (sqrt (fma a (* c -4.0) (* b b))))) (* a 2.0)))

C

double code(double a, double b, double c) {
	return ((a * (c * -4.0)) / (b + sqrt(fma(a, (c * -4.0), (b * b))))) / (a * 2.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(a * Float64(c * -4.0)) / Float64(b + sqrt(fma(a, Float64(c * -4.0), Float64(b * b))))) / Float64(a * 2.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied rewrites 33.2%

   \[\leadsto \frac{\color{blue}{\frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} - \frac{\mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{2 \cdot a} \]
5. Step-by-step derivation

6. Applied rewrites 35.2%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, -b, c \cdot \left(a \cdot -4\right)\right)\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2}} \]

8. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\frac{a \cdot \left(-4 \cdot c\right)}{b + \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}}}}{a \cdot 2} \]

9. Final simplification 99.4%

   \[\leadsto \frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a \cdot 2} \]
10. Add Preprocessing

Alternative 3: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a, c \cdot -4, 0\right)}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma a (* c -4.0) 0.0)
  (* (+ b (sqrt (fma a (* c -4.0) (* b b)))) (* a 2.0))))

C

double code(double a, double b, double c) {
	return fma(a, (c * -4.0), 0.0) / ((b + sqrt(fma(a, (c * -4.0), (b * b)))) * (a * 2.0));
}

Julia

function code(a, b, c)
	return Float64(fma(a, Float64(c * -4.0), 0.0) / Float64(Float64(b + sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))) * Float64(a * 2.0)))
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(c * -4.0), $MachinePrecision] + 0.0), $MachinePrecision] / N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied rewrites 33.2%

   \[\leadsto \frac{\color{blue}{\frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} - \frac{\mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{2 \cdot a} \]

6. Applied rewrites 31.0%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, -b, c \cdot \left(a \cdot -4\right)\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} \cdot \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}{2 \cdot a} \]

8. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(b, b + \left(-b\right), a \cdot \left(-4 \cdot c\right)\right)}{b \cdot \left(b + \left(-b\right)\right) - a \cdot \left(-4 \cdot c\right)} \cdot \left(b \cdot \left(b + \left(-b\right)\right) - a \cdot \left(-4 \cdot c\right)\right)}{b + \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}}}}{2 \cdot a} \]

10. Applied rewrites 99.4%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, -4 \cdot c, 0\right)}{\left(b + \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}} \]

11. Final simplification 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot -4, 0\right)}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)} \]
12. Add Preprocessing

Alternative 4: 91.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (fma (* c c) (/ a (* b b)) c) (- b)))

C

double code(double a, double b, double c) {
	return fma((c * c), (a / (b * b)), c) / -b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]

Derivation

1. Initial program 32.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{-1 \cdot \frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

   3. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\right)} \]

   4. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\right)} \]

   5. lower-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}}\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{a \cdot {c}^{2}}{{b}^{2}} + c}}{b}\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\frac{\frac{\color{blue}{{c}^{2} \cdot a}}{{b}^{2}} + c}{b}\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{c}^{2} \cdot \frac{a}{{b}^{2}}} + c}{b}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\mathsf{fma}\left({c}^{2}, \frac{a}{{b}^{2}}, c\right)}}{b}\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(\color{blue}{c \cdot c}, \frac{a}{{b}^{2}}, c\right)}{b}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(\color{blue}{c \cdot c}, \frac{a}{{b}^{2}}, c\right)}{b}\right) \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(c \cdot c, \color{blue}{\frac{a}{{b}^{2}}}, c\right)}{b}\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\color{blue}{b \cdot b}}, c\right)}{b}\right) \]

   14. lower-*.f64 90.6

       \[\leadsto -\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{\color{blue}{b \cdot b}}, c\right)}{b} \]

5. Applied rewrites 90.6%

   \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}} \]

6. Final simplification 90.6%

   \[\leadsto \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b} \]
7. Add Preprocessing

Alternative 5: 81.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c (- b)))

C

double code(double a, double b, double c) {
	return c / -b;
}

Fortran

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

Java

public static double code(double a, double b, double c) {
	return c / -b;
}

Python

def code(a, b, c):
	return c / -b

Julia

function code(a, b, c)
	return Float64(c / Float64(-b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / -b;
end

Wolfram

code[a_, b_, c_] := N[(c / (-b)), $MachinePrecision]

Derivation

1. Initial program 32.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

   4. lower-neg.f64 80.5

      \[\leadsto \frac{c}{\color{blue}{-b}} \]

5. Applied rewrites 80.5%

   \[\leadsto \color{blue}{\frac{c}{-b}} \]
6. Add Preprocessing

Alternative 6: 3.2% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 32.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied rewrites 33.2%

   \[\leadsto \frac{\color{blue}{\frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} - \frac{\mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{2 \cdot a} \]

6. Applied rewrites 31.0%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(b, -b, c \cdot \left(a \cdot -4\right)\right)\right)}{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} \cdot \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}{2 \cdot a} \]

7. Taylor expanded in c around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{-1 \cdot {b}^{2} + {b}^{2}}{a \cdot b}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot {b}^{2} + {b}^{2}}{a \cdot b} \cdot \frac{1}{4}} \]

   2. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{-1 \cdot {b}^{2} + {b}^{2}}{a}}{b}} \cdot \frac{1}{4} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-1 \cdot {b}^{2} + {b}^{2}}{a} \cdot \frac{1}{4}}{b}} \]

   4. distribute-lft1-in N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(-1 + 1\right) \cdot {b}^{2}}}{a} \cdot \frac{1}{4}}{b} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{0} \cdot {b}^{2}}{a} \cdot \frac{1}{4}}{b} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\left(0 \cdot \frac{{b}^{2}}{a}\right)} \cdot \frac{1}{4}}{b} \]

   7. mul0-lft N/A

      \[\leadsto \frac{\color{blue}{0} \cdot \frac{1}{4}}{b} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{b} \]

   9. lower-/.f64 3.2

      \[\leadsto \color{blue}{\frac{0}{b}} \]

9. Applied rewrites 3.2%

   \[\leadsto \color{blue}{\frac{0}{b}} \]

10. Taylor expanded in b around 0

    \[\leadsto 0 \]
11. Step-by-step derivation

12. Applied rewrites 3.2%

    \[\leadsto 0 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024237 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))