Quadratic roots, medium range

Percentage Accurate: 31.8% → 99.5%

Time: 13.5s

Alternatives: 5

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: 31.8% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \left(c \cdot -2\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (/ 1.0 (+ b (sqrt (fma c (* a -4.0) (* b b))))) (* c -2.0)))

C

double code(double a, double b, double c) {
	return (1.0 / (b + sqrt(fma(c, (a * -4.0), (b * b))))) * (c * -2.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * b))))) * Float64(c * -2.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 31.7

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

   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. distribute-lft-neg-in N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

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

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 31.7

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

4. Applied rewrites 31.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. div-inv N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 32.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

8. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{\mathsf{fma}\left(c, a \cdot -4, 0\right)}{a \cdot 2}} \]

9. Taylor expanded in c around 0

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

    1. lower-*.f64 99.6

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 2: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -500000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -500000.0)
   (/ (- (sqrt (fma b b (* -4.0 (* c a)))) b) (* a 2.0))
   (- (fma a (/ (* c c) (* b (* b b))) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -500000.0) {
		tmp = (sqrt(fma(b, b, (-4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = -fma(a, ((c * c) / (b * (b * b))), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -500000.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-fma(a, Float64(Float64(c * c) / Float64(b * Float64(b * b))), Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -500000.0], N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], (-N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5e5

   1. Initial program 76.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 76.6

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

      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. distribute-lft-neg-in N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 76.6

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

   4. Applied rewrites 76.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 76.7

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.7

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

   6. Applied rewrites 76.7%

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

   if -5e5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 29.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 29.9

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

      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. distribute-lft-neg-in N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 29.9

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

   4. Applied rewrites 29.9%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\left(\mathsf{neg}\left(\frac{c}{b}\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right)} + \left(\mathsf{neg}\left(\frac{c}{b}\right)\right) \]

      4. distribute-neg-out N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{c}{b}\right)\right)} \]

      5. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{c}{b}\right)\right)} \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} + \frac{c}{b}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(a, \frac{{c}^{2}}{{b}^{3}}, \frac{c}{b}\right)}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(a, \color{blue}{\frac{{c}^{2}}{{b}^{3}}}, \frac{c}{b}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(a, \frac{\color{blue}{c \cdot c}}{{b}^{3}}, \frac{c}{b}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(a, \frac{\color{blue}{c \cdot c}}{{b}^{3}}, \frac{c}{b}\right)\right) \]

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 91.8

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

   7. Applied rewrites 91.8%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -500000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(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)))