Quadratic roots, wide range

Percentage Accurate: 17.8% → 97.5%

Time: 9.0s

Alternatives: 5

Speedup: 3.6×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.8% 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: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{b \cdot b}, \frac{a}{b}, \left(\left(\left(\mathsf{fma}\left(c \cdot a, -5, -2 \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot a\right) \cdot {b}^{-7}\right) \cdot c\right), c, \frac{-1}{b}\right) \cdot c \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (fma
   (fma
    (/ -1.0 (* b b))
    (/ a b)
    (* (* (* (* (fma (* c a) -5.0 (* -2.0 (* b b))) a) a) (pow b -7.0)) c))
   c
   (/ -1.0 b))
  c))

C

double code(double a, double b, double c) {
	return fma(fma((-1.0 / (b * b)), (a / b), ((((fma((c * a), -5.0, (-2.0 * (b * b))) * a) * a) * pow(b, -7.0)) * c)), c, (-1.0 / b)) * c;
}

Julia

function code(a, b, c)
	return Float64(fma(fma(Float64(-1.0 / Float64(b * b)), Float64(a / b), Float64(Float64(Float64(Float64(fma(Float64(c * a), -5.0, Float64(-2.0 * Float64(b * b))) * a) * a) * (b ^ -7.0)) * c)), c, Float64(-1.0 / b)) * c)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(a / b), $MachinePrecision] + N[(N[(N[(N[(N[(N[(c * a), $MachinePrecision] * -5.0 + N[(-2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -7.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * c + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 18.8%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{b}\right)} \]

5. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.25}{a}, \frac{{a}^{4} \cdot 20}{{b}^{6}} \cdot \frac{c}{b}, \frac{\left(a \cdot -2\right) \cdot a}{{b}^{5}}\right), c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c} \]

6. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-5 \cdot \left({a}^{3} \cdot c\right) + -2 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{b}^{7}}, c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c \]
7. Step-by-step derivation

8. Applied rewrites 97.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot -2\right) \cdot b, b, \left({a}^{3} \cdot c\right) \cdot -5\right)}{{b}^{7}}, c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c \]

9. Taylor expanded in a around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{{a}^{2} \cdot \left(-5 \cdot \left(a \cdot c\right) + -2 \cdot {b}^{2}\right)}{{b}^{7}}, c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c \]
10. Step-by-step derivation

11. Applied rewrites 97.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(c \cdot a, -5, \left(b \cdot b\right) \cdot -2\right) \cdot \left(a \cdot a\right)}{{b}^{7}}, c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c \]
12. Step-by-step derivation

13. Applied rewrites 97.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{b \cdot b}, \frac{a}{b}, \left({b}^{-7} \cdot \left(\left(\mathsf{fma}\left(c \cdot a, -5, \left(b \cdot b\right) \cdot -2\right) \cdot a\right) \cdot a\right)\right) \cdot c\right), c, \frac{-1}{b}\right) \cdot c \]

14. Final simplification 97.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{b \cdot b}, \frac{a}{b}, \left(\left(\left(\mathsf{fma}\left(c \cdot a, -5, -2 \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot a\right) \cdot {b}^{-7}\right) \cdot c\right), c, \frac{-1}{b}\right) \cdot c \]
15. Add Preprocessing

Alternative 2: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot -2}{b}, \frac{c}{b}, -a\right)}{{b}^{3}}, c, \frac{-1}{b}\right) \cdot c \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (fma (/ (fma (/ (* (* a a) -2.0) b) (/ c b) (- a)) (pow b 3.0)) c (/ -1.0 b))
  c))

C

double code(double a, double b, double c) {
	return fma((fma((((a * a) * -2.0) / b), (c / b), -a) / pow(b, 3.0)), c, (-1.0 / b)) * c;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(fma(Float64(Float64(Float64(a * a) * -2.0) / b), Float64(c / b), Float64(-a)) / (b ^ 3.0)), c, Float64(-1.0 / b)) * c)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision] / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + (-a)), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * c + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 18.8%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{b}\right)} \]

5. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.25}{a}, \frac{{a}^{4} \cdot 20}{{b}^{6}} \cdot \frac{c}{b}, \frac{\left(a \cdot -2\right) \cdot a}{{b}^{5}}\right), c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c} \]

6. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-5 \cdot \left({a}^{3} \cdot c\right) + -2 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{b}^{7}}, c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c \]
7. Step-by-step derivation

8. Applied rewrites 97.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot -2\right) \cdot b, b, \left({a}^{3} \cdot c\right) \cdot -5\right)}{{b}^{7}}, c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c \]

9. Taylor expanded in a around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{{a}^{2} \cdot \left(-5 \cdot \left(a \cdot c\right) + -2 \cdot {b}^{2}\right)}{{b}^{7}}, c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c \]
10. Step-by-step derivation

11. Applied rewrites 97.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(c \cdot a, -5, \left(b \cdot b\right) \cdot -2\right) \cdot \left(a \cdot a\right)}{{b}^{7}}, c, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c \]

12. Taylor expanded in b around inf

    \[\leadsto \mathsf{fma}\left(\frac{-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + -1 \cdot a}{{b}^{3}}, c, \frac{-1}{b}\right) \cdot c \]
13. Step-by-step derivation

14. Applied rewrites 97.2%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-2 \cdot \left(a \cdot a\right)}{b}, \frac{c}{b}, -a\right)}{{b}^{3}}, c, \frac{-1}{b}\right) \cdot c \]

15. Final simplification 97.2%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot -2}{b}, \frac{c}{b}, -a\right)}{{b}^{3}}, c, \frac{-1}{b}\right) \cdot c \]
16. Add Preprocessing

Alternative 3: 95.3% accurate, 1.2× speedup

Math

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

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(c, Float64(Float64(c * a) / Float64(b * b)), c) / Float64(-b))
end

Wolfram

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

Derivation

1. Initial program 18.8%

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

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. times-frac N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 96.1

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

5. Applied rewrites 96.1%

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

7. Applied rewrites 96.1%

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

8. Final simplification 96.1%

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

Alternative 4: 90.4% 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(Float64(-c) / 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 18.8%

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

3. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

   3. mul-1-neg N/A

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

   4. lower-neg.f64 90.2

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

5. Applied rewrites 90.2%

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

Alternative 5: 3.3% 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 18.8%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 18.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. distribute-lft-neg-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. metadata-eval N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 19.8

       \[\leadsto \mathsf{fma}\left(-b, \color{blue}{\frac{0.5}{a}}, \frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}{2 \cdot a}\right) \]

   15. lift-/.f64 N/A

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

   16. div-inv N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 20.3%

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

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 3.3

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

9. Applied rewrites 3.3%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))