Quadratic roots, medium range

Percentage Accurate: 31.1% → 99.7%

Time: 12.1s

Alternatives: 8

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: 31.1% 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: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 30.8

      \[\leadsto \frac{0.5}{\color{blue}{\frac{a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}} \]

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 30.8

       \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}} \]

4. Applied rewrites 30.8%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{a}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{a}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}}} \]

   3. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b\right)} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{a} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b\right)} \]

   5. flip-- N/A

      \[\leadsto \frac{\frac{1}{2}}{a} \cdot \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}} \]

   6. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}} \]

6. Applied rewrites 31.4%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}} \]

7. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 99.7

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in a around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-fma.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f64 99.7

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

12. Applied rewrites 99.7%

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

13. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 31.1

      \[\leadsto \frac{0.5}{\color{blue}{\frac{a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}} \]

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 31.1

       \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}} \]

4. Applied rewrites 31.1%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{a}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{a}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}}} \]

   3. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b\right)} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{a} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b\right)} \]

   5. flip-- N/A

      \[\leadsto \frac{\frac{1}{2}}{a} \cdot \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}} \]

   6. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}} \]

6. Applied rewrites 32.0%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}} \]

7. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 99.7

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

9. Applied rewrites 99.7%

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

    1. lift-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. lower-fma.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-*l* N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 99.7

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

11. Applied rewrites 99.7%

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

12. Final simplification 99.7%

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

Reproduce

herbie shell --seed 2024234 
(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)))