Quadratic roots, wide range

Percentage Accurate: 18.6% → 99.4%

Time: 11.1s

Alternatives: 7

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: 18.6% 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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 20.0

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. distribute-rgt-neg-in N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 20.0

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

4. Applied rewrites 20.0%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. div-inv N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 20.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 20.9

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

   5. lift--.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. associate--l+ N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. +-inverses N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 99.4

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

   14. lift-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*l* N/A

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

   18. lower-fma.f64 N/A

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

   19. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

9. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 18.6

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. distribute-rgt-neg-in N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 18.6

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

4. Applied rewrites 18.6%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. div-inv N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 19.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 19.1

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

   5. lift--.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. associate--l+ N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. +-inverses N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 99.4

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

   14. lift-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*l* N/A

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

   18. lower-fma.f64 N/A

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

   19. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. +-rgt-identity N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.4

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.4

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

10. Applied rewrites 99.4%

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

Reproduce

herbie shell --seed 2024228 
(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)))