Quadratic roots, medium range

Percentage Accurate: 31.6% → 99.4%

Time: 9.6s

Alternatives: 6

Speedup: 29.0×

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: 31.6% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \frac{t_0 \cdot \frac{0.5}{a}}{b + \sqrt{\mathsf{fma}\left(b, b, t_0\right)}} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a -4.0))))
   (/ (* t_0 (/ 0.5 a)) (+ b (sqrt (fma b b t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * -4.0);
	return (t_0 * (0.5 / a)) / (b + sqrt(fma(b, b, t_0)));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * -4.0))
	return Float64(Float64(t_0 * Float64(0.5 / a)) / Float64(b + sqrt(fma(b, b, t_0))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(b * b + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 30.4%

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

   1. *-commutative 30.4%

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

   2. +-commutative 30.4%

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

   3. unsub-neg 30.4%

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

   4. fma-neg 30.4%

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

   5. associate-*l* 30.4%

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

   6. *-commutative 30.4%

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

   7. distribute-rgt-neg-in 30.4%

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

   8. metadata-eval 30.4%

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

3. Simplified 30.4%

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

   1. fma-udef 30.4%

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

   2. *-commutative 30.4%

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

   3. metadata-eval 30.4%

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

   4. cancel-sign-sub-inv 30.4%

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

   5. associate-*l* 30.4%

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

   6. *-un-lft-identity 30.4%

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

   7. prod-diff 30.4%

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

5. Applied egg-rr 30.4%

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

   1. +-commutative 30.4%

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

   2. fma-udef 30.4%

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

   3. *-rgt-identity 30.4%

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

   4. *-rgt-identity 30.4%

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

   5. count-2 30.4%

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

   6. *-commutative 30.4%

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

   7. *-commutative 30.4%

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

   8. associate-*r* 30.4%

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

   9. *-rgt-identity 30.4%

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

   10. fma-neg 30.3%

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

   11. *-commutative 30.3%

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

   12. *-commutative 30.3%

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

   13. associate-*r* 30.3%

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

7. Simplified 30.3%

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

   1. flip-- 30.3%

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

   2. add-sqr-sqrt 30.8%

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

   3. associate-*r* 30.8%

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

   4. metadata-eval 30.8%

      \[\leadsto \frac{\frac{\left(\color{blue}{-8} \cdot \left(c \cdot a\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

   5. *-commutative 30.8%

      \[\leadsto \frac{\frac{\left(-8 \cdot \color{blue}{\left(a \cdot c\right)} + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

   6. associate-*r* 30.8%

      \[\leadsto \frac{\frac{\left(-8 \cdot \left(a \cdot c\right) + \left(b \cdot b - \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

9. Applied egg-rr 30.8%

   \[\leadsto \frac{\color{blue}{\frac{\left(-8 \cdot \left(a \cdot c\right) + \left(b \cdot b - \left(-4 \cdot c\right) \cdot a\right)\right) - b \cdot b}{\sqrt{-8 \cdot \left(a \cdot c\right) + \left(b \cdot b - \left(-4 \cdot c\right) \cdot a\right)} + b}}}{a \cdot 2} \]

11. Simplified 99.4%

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

    1. div-inv 99.3%

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

    2. fma-def 99.3%

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

    3. mul0-lft 99.3%

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

13. Applied egg-rr 99.3%

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

    1. associate-*l/ 99.4%

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

    2. fma-udef 99.4%

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

    3. +-rgt-identity 99.4%

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

    4. *-commutative 99.4%

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

    5. associate-/r* 99.4%

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

    6. metadata-eval 99.4%

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

15. Simplified 99.4%

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

16. Final simplification 99.4%

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

Alternative 2: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return -(c / b) - ((c * c) / (pow(b, 3.0) / 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 = -(c / b) - ((c * c) / ((b ** 3.0d0) / a))
end function

Java

public static double code(double a, double b, double c) {
	return -(c / b) - ((c * c) / (Math.pow(b, 3.0) / a));
}

Python

def code(a, b, c):
	return -(c / b) - ((c * c) / (math.pow(b, 3.0) / a))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = -(c / b) - ((c * c) / ((b ^ 3.0) / a));
end

Wolfram

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

Derivation

1. Initial program 30.4%

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

   1. /-rgt-identity 30.4%

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

   2. metadata-eval 30.4%

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

   3. associate-/l* 30.4%

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

   4. associate-*r/ 30.4%

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

   5. +-commutative 30.4%

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

   6. unsub-neg 30.4%

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

   7. fma-neg 30.4%

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

   8. associate-*l* 30.4%

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

   9. *-commutative 30.4%

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

   10. distribute-rgt-neg-in 30.4%

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

   11. metadata-eval 30.4%

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

   12. associate-/r* 30.4%

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

   13. metadata-eval 30.4%

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

   14. metadata-eval 30.4%

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

3. Simplified 30.4%

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

4. Taylor expanded in b around inf 92.5%

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

   1. +-commutative 92.5%

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

   2. mul-1-neg 92.5%

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

   3. unsub-neg 92.5%

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

   4. mul-1-neg 92.5%

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

   5. distribute-neg-frac 92.5%

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

   6. associate-/l* 92.5%

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

   7. unpow2 92.5%

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

6. Simplified 92.5%

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

7. Final simplification 92.5%

   \[\leadsto \left(-\frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

Alternative 3: 90.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((c * (a * (-4.0d0))) + (0.0d0 * (b * b))) / (b + (b + ((-2.0d0) * ((c * a) / b))))) / (a * 2.0d0)
end function

Java

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

Python

def code(a, b, c):
	return (((c * (a * -4.0)) + (0.0 * (b * b))) / (b + (b + (-2.0 * ((c * a) / b))))) / (a * 2.0)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (((c * (a * -4.0)) + (0.0 * (b * b))) / (b + (b + (-2.0 * ((c * a) / b))))) / (a * 2.0);
end

Wolfram

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

Derivation

1. Initial program 30.4%

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

   1. *-commutative 30.4%

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

   2. +-commutative 30.4%

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

   3. unsub-neg 30.4%

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

   4. fma-neg 30.4%

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

   5. associate-*l* 30.4%

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

   6. *-commutative 30.4%

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

   7. distribute-rgt-neg-in 30.4%

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

   8. metadata-eval 30.4%

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

3. Simplified 30.4%

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

   1. fma-udef 30.4%

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

   2. *-commutative 30.4%

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

   3. metadata-eval 30.4%

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

   4. cancel-sign-sub-inv 30.4%

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

   5. associate-*l* 30.4%

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

   6. *-un-lft-identity 30.4%

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

   7. prod-diff 30.4%

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

5. Applied egg-rr 30.4%

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

   1. +-commutative 30.4%

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

   2. fma-udef 30.4%

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

   3. *-rgt-identity 30.4%

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

   4. *-rgt-identity 30.4%

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

   5. count-2 30.4%

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

   6. *-commutative 30.4%

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

   7. *-commutative 30.4%

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

   8. associate-*r* 30.4%

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

   9. *-rgt-identity 30.4%

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

   10. fma-neg 30.3%

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

   11. *-commutative 30.3%

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

   12. *-commutative 30.3%

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

   13. associate-*r* 30.3%

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

7. Simplified 30.3%

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

   1. flip-- 30.3%

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

   2. add-sqr-sqrt 30.8%

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

   3. associate-*r* 30.8%

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

   4. metadata-eval 30.8%

      \[\leadsto \frac{\frac{\left(\color{blue}{-8} \cdot \left(c \cdot a\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

   5. *-commutative 30.8%

      \[\leadsto \frac{\frac{\left(-8 \cdot \color{blue}{\left(a \cdot c\right)} + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

   6. associate-*r* 30.8%

      \[\leadsto \frac{\frac{\left(-8 \cdot \left(a \cdot c\right) + \left(b \cdot b - \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

9. Applied egg-rr 30.8%

   \[\leadsto \frac{\color{blue}{\frac{\left(-8 \cdot \left(a \cdot c\right) + \left(b \cdot b - \left(-4 \cdot c\right) \cdot a\right)\right) - b \cdot b}{\sqrt{-8 \cdot \left(a \cdot c\right) + \left(b \cdot b - \left(-4 \cdot c\right) \cdot a\right)} + b}}}{a \cdot 2} \]

11. Simplified 99.4%

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

12. Taylor expanded in b around inf 92.4%

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

13. Final simplification 92.4%

    \[\leadsto \frac{\frac{c \cdot \left(a \cdot -4\right) + 0 \cdot \left(b \cdot b\right)}{b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)}}{a \cdot 2} \]

Alternative 4: 90.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((c * (a * (-4.0d0))) + (0.0d0 * (b * b))) / (((-2.0d0) * ((c * a) / b)) + (b * 2.0d0))) / (a * 2.0d0)
end function

Java

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

Python

def code(a, b, c):
	return (((c * (a * -4.0)) + (0.0 * (b * b))) / ((-2.0 * ((c * a) / b)) + (b * 2.0))) / (a * 2.0)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (((c * (a * -4.0)) + (0.0 * (b * b))) / ((-2.0 * ((c * a) / b)) + (b * 2.0))) / (a * 2.0);
end

Wolfram

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

Derivation

1. Initial program 30.4%

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

   1. *-commutative 30.4%

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

   2. +-commutative 30.4%

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

   3. unsub-neg 30.4%

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

   4. fma-neg 30.4%

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

   5. associate-*l* 30.4%

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

   6. *-commutative 30.4%

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

   7. distribute-rgt-neg-in 30.4%

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

   8. metadata-eval 30.4%

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

3. Simplified 30.4%

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

   1. fma-udef 30.4%

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

   2. *-commutative 30.4%

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

   3. metadata-eval 30.4%

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

   4. cancel-sign-sub-inv 30.4%

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

   5. associate-*l* 30.4%

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

   6. *-un-lft-identity 30.4%

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

   7. prod-diff 30.4%

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

5. Applied egg-rr 30.4%

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

   1. +-commutative 30.4%

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

   2. fma-udef 30.4%

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

   3. *-rgt-identity 30.4%

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

   4. *-rgt-identity 30.4%

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

   5. count-2 30.4%

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

   6. *-commutative 30.4%

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

   7. *-commutative 30.4%

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

   8. associate-*r* 30.4%

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

   9. *-rgt-identity 30.4%

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

   10. fma-neg 30.3%

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

   11. *-commutative 30.3%

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

   12. *-commutative 30.3%

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

   13. associate-*r* 30.3%

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

7. Simplified 30.3%

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

   1. flip-- 30.3%

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

   2. add-sqr-sqrt 30.8%

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

   3. associate-*r* 30.8%

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

   4. metadata-eval 30.8%

      \[\leadsto \frac{\frac{\left(\color{blue}{-8} \cdot \left(c \cdot a\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

   5. *-commutative 30.8%

      \[\leadsto \frac{\frac{\left(-8 \cdot \color{blue}{\left(a \cdot c\right)} + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

   6. associate-*r* 30.8%

      \[\leadsto \frac{\frac{\left(-8 \cdot \left(a \cdot c\right) + \left(b \cdot b - \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

9. Applied egg-rr 30.8%

   \[\leadsto \frac{\color{blue}{\frac{\left(-8 \cdot \left(a \cdot c\right) + \left(b \cdot b - \left(-4 \cdot c\right) \cdot a\right)\right) - b \cdot b}{\sqrt{-8 \cdot \left(a \cdot c\right) + \left(b \cdot b - \left(-4 \cdot c\right) \cdot a\right)} + b}}}{a \cdot 2} \]

11. Simplified 99.4%

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

12. Taylor expanded in b around inf 92.4%

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

13. Final simplification 92.4%

    \[\leadsto \frac{\frac{c \cdot \left(a \cdot -4\right) + 0 \cdot \left(b \cdot b\right)}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2}}{a \cdot 2} \]

Alternative 5: 90.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{a} \cdot \frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + \frac{-2}{\frac{b}{c \cdot a}}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (/ 0.5 a) (/ (* c (* a -4.0)) (+ b (+ b (/ -2.0 (/ b (* c a))))))))

C

double code(double a, double b, double c) {
	return (0.5 / a) * ((c * (a * -4.0)) / (b + (b + (-2.0 / (b / (c * 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 = (0.5d0 / a) * ((c * (a * (-4.0d0))) / (b + (b + ((-2.0d0) / (b / (c * a))))))
end function

Java

public static double code(double a, double b, double c) {
	return (0.5 / a) * ((c * (a * -4.0)) / (b + (b + (-2.0 / (b / (c * a))))));
}

Python

def code(a, b, c):
	return (0.5 / a) * ((c * (a * -4.0)) / (b + (b + (-2.0 / (b / (c * a))))))

Julia

function code(a, b, c)
	return Float64(Float64(0.5 / a) * Float64(Float64(c * Float64(a * -4.0)) / Float64(b + Float64(b + Float64(-2.0 / Float64(b / Float64(c * a)))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (0.5 / a) * ((c * (a * -4.0)) / (b + (b + (-2.0 / (b / (c * a))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(0.5 / a), $MachinePrecision] * N[(N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[(b + N[(-2.0 / N[(b / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.4%

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

   1. /-rgt-identity 30.4%

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

   2. metadata-eval 30.4%

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

   3. associate-/l* 30.4%

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

   4. associate-*r/ 30.4%

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

   5. +-commutative 30.4%

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

   6. unsub-neg 30.4%

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

   7. fma-neg 30.4%

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

   8. associate-*l* 30.4%

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

   9. *-commutative 30.4%

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

   10. distribute-rgt-neg-in 30.4%

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

   11. metadata-eval 30.4%

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

   12. associate-/r* 30.4%

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

   13. metadata-eval 30.4%

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

   14. metadata-eval 30.4%

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

3. Simplified 30.4%

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

4. Taylor expanded in b around inf 22.1%

   \[\leadsto \left(\color{blue}{\left(b + -2 \cdot \frac{c \cdot a}{b}\right)} - b\right) \cdot \frac{0.5}{a} \]
5. Step-by-step derivation

   1. associate-*r/ 22.1%

      \[\leadsto \left(\left(b + \color{blue}{\frac{-2 \cdot \left(c \cdot a\right)}{b}}\right) - b\right) \cdot \frac{0.5}{a} \]

6. Simplified 22.1%

   \[\leadsto \left(\color{blue}{\left(b + \frac{-2 \cdot \left(c \cdot a\right)}{b}\right)} - b\right) \cdot \frac{0.5}{a} \]
7. Step-by-step derivation

   1. flip-- 22.1%

      \[\leadsto \color{blue}{\frac{\left(b + \frac{-2 \cdot \left(c \cdot a\right)}{b}\right) \cdot \left(b + \frac{-2 \cdot \left(c \cdot a\right)}{b}\right) - b \cdot b}{\left(b + \frac{-2 \cdot \left(c \cdot a\right)}{b}\right) + b}} \cdot \frac{0.5}{a} \]

   2. associate-/l* 22.1%

      \[\leadsto \frac{\left(b + \color{blue}{\frac{-2}{\frac{b}{c \cdot a}}}\right) \cdot \left(b + \frac{-2 \cdot \left(c \cdot a\right)}{b}\right) - b \cdot b}{\left(b + \frac{-2 \cdot \left(c \cdot a\right)}{b}\right) + b} \cdot \frac{0.5}{a} \]

   3. associate-/l* 22.1%

      \[\leadsto \frac{\left(b + \frac{-2}{\frac{b}{c \cdot a}}\right) \cdot \left(b + \color{blue}{\frac{-2}{\frac{b}{c \cdot a}}}\right) - b \cdot b}{\left(b + \frac{-2 \cdot \left(c \cdot a\right)}{b}\right) + b} \cdot \frac{0.5}{a} \]

   4. associate-/l* 22.1%

      \[\leadsto \frac{\left(b + \frac{-2}{\frac{b}{c \cdot a}}\right) \cdot \left(b + \frac{-2}{\frac{b}{c \cdot a}}\right) - b \cdot b}{\left(b + \color{blue}{\frac{-2}{\frac{b}{c \cdot a}}}\right) + b} \cdot \frac{0.5}{a} \]

8. Applied egg-rr 22.1%

   \[\leadsto \color{blue}{\frac{\left(b + \frac{-2}{\frac{b}{c \cdot a}}\right) \cdot \left(b + \frac{-2}{\frac{b}{c \cdot a}}\right) - b \cdot b}{\left(b + \frac{-2}{\frac{b}{c \cdot a}}\right) + b}} \cdot \frac{0.5}{a} \]

9. Taylor expanded in b around inf 92.3%

   \[\leadsto \frac{\color{blue}{-4 \cdot \left(c \cdot a\right)}}{\left(b + \frac{-2}{\frac{b}{c \cdot a}}\right) + b} \cdot \frac{0.5}{a} \]
10. Step-by-step derivation

    1. *-commutative 92.3%

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

    2. *-commutative 92.3%

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

    3. *-commutative 92.3%

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

    4. associate-*l* 92.3%

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

11. Simplified 92.3%

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

12. Final simplification 92.3%

    \[\leadsto \frac{0.5}{a} \cdot \frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + \frac{-2}{\frac{b}{c \cdot a}}\right)} \]

Alternative 6: 81.2% accurate, 29.0× 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 30.4%

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

   1. /-rgt-identity 30.4%

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

   2. metadata-eval 30.4%

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

   3. associate-/l* 30.4%

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

   4. associate-*r/ 30.4%

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

   5. +-commutative 30.4%

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

   6. unsub-neg 30.4%

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

   7. fma-neg 30.4%

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

   8. associate-*l* 30.4%

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

   9. *-commutative 30.4%

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

   10. distribute-rgt-neg-in 30.4%

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

   11. metadata-eval 30.4%

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

   12. associate-/r* 30.4%

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

   13. metadata-eval 30.4%

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

   14. metadata-eval 30.4%

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

3. Simplified 30.4%

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

4. Taylor expanded in b around inf 82.7%

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

   1. mul-1-neg 82.7%

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

   2. distribute-neg-frac 82.7%

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

6. Simplified 82.7%

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

7. Final simplification 82.7%

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

Reproduce

herbie shell --seed 2023207 
(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)))