Quadratic roots, medium range

Percentage Accurate: 31.6% → 99.3%

Time: 18.6s

Alternatives: 4

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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(a \cdot a\right)\\ \frac{\frac{-c}{a}}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{\frac{t_0}{a \cdot 2}} + \frac{a}{\frac{t_0}{b \cdot 2}}} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 4.0 (* a a))))
   (/
    (/ (- c) a)
    (+
     (/ (sqrt (fma a (* c -4.0) (* b b))) (/ t_0 (* a 2.0)))
     (/ a (/ t_0 (* b 2.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = 4.0 * (a * a);
	return (-c / a) / ((sqrt(fma(a, (c * -4.0), (b * b))) / (t_0 / (a * 2.0))) + (a / (t_0 / (b * 2.0))));
}

Julia

function code(a, b, c)
	t_0 = Float64(4.0 * Float64(a * a))
	return Float64(Float64(Float64(-c) / a) / Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) / Float64(t_0 / Float64(a * 2.0))) + Float64(a / Float64(t_0 / Float64(b * 2.0)))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, N[(N[((-c) / a), $MachinePrecision] / N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[(t$95$0 / N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 33.0%

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

3. Applied egg-rr 32.8%

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

   1. un-div-inv 32.8%

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

   2. div-sub 32.6%

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

   3. flip-- 32.3%

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

5. Applied egg-rr 32.3%

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

7. Simplified 34.3%

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

8. Taylor expanded in a around 0 99.3%

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

   1. associate-*r/ 99.3%

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

   2. neg-mul-1 99.3%

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

10. Simplified 99.3%

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

11. Final simplification 99.3%

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

Alternative 2: 90.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{-2 \cdot \frac{b}{c} + 2 \cdot \frac{a}{b}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

3. Applied egg-rr 32.8%

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

4. Taylor expanded in a around 0 32.8%

   \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \left(a \cdot 2\right) - b \cdot \left(a \cdot 2\right)\right) \cdot \color{blue}{\frac{0.25}{{a}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 32.8%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \left(a \cdot 2\right) - b \cdot \left(a \cdot 2\right)\right) \cdot \frac{0.25}{\color{blue}{a \cdot a}} \]

6. Simplified 32.8%

   \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \left(a \cdot 2\right) - b \cdot \left(a \cdot 2\right)\right) \cdot \color{blue}{\frac{0.25}{a \cdot a}} \]
7. Step-by-step derivation

   1. associate-*r/ 32.8%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \left(a \cdot 2\right) - b \cdot \left(a \cdot 2\right)\right) \cdot 0.25}{a \cdot a}} \]

   2. associate-/l* 32.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \left(a \cdot 2\right) - b \cdot \left(a \cdot 2\right)}{\frac{a \cdot a}{0.25}}} \]

   3. associate-*r* 32.8%

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot a\right) \cdot 2} - b \cdot \left(a \cdot 2\right)}{\frac{a \cdot a}{0.25}} \]

   4. associate-*r* 32.8%

      \[\leadsto \frac{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot a\right) \cdot 2 - \color{blue}{\left(b \cdot a\right) \cdot 2}}{\frac{a \cdot a}{0.25}} \]

   5. *-commutative 32.8%

      \[\leadsto \frac{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot a\right) \cdot 2 - \color{blue}{\left(a \cdot b\right)} \cdot 2}{\frac{a \cdot a}{0.25}} \]

   6. distribute-rgt-out-- 32.8%

      \[\leadsto \frac{\color{blue}{2 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot a - a \cdot b\right)}}{\frac{a \cdot a}{0.25}} \]

   7. associate-/l* 32.8%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{a \cdot a}{0.25}}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot a - a \cdot b}}} \]

   8. div-inv 32.8%

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

   9. metadata-eval 32.8%

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

   10. *-commutative 32.8%

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

8. Applied egg-rr 32.8%

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

9. Taylor expanded in a around 0 91.4%

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

10. Final simplification 91.4%

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

Alternative 3: 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 33.0%

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

2. Taylor expanded in b around inf 79.9%

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

   1. associate-*r/ 79.9%

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

   2. neg-mul-1 79.9%

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

4. Simplified 79.9%

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

5. Final simplification 79.9%

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

Alternative 4: 3.2% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 33.0%

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

3. Applied egg-rr 32.8%

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

   1. *-commutative 32.8%

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

   2. sub-neg 32.8%

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

   3. distribute-lft-in 32.5%

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

   4. fma-def 34.2%

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

5. Applied egg-rr 34.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.25}{a \cdot a}, \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} \cdot \left(a \cdot 2\right), \frac{0.25}{a \cdot a} \cdot \left(-2 \cdot \left(a \cdot b\right)\right)\right)} \]

6. Taylor expanded in c around 0 3.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{b}{a}} \]
7. Step-by-step derivation

   1. distribute-rgt-out 3.2%

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

   2. metadata-eval 3.2%

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

   3. mul0-rgt 3.2%

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

8. Simplified 3.2%

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

9. Final simplification 3.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023297 
(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)))