Quadratic roots, wide range

Percentage Accurate: 17.7% → 97.7%

Time: 11.2s

Alternatives: 5

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot c\right)}^{4}\\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{16 \cdot t_0 + 4 \cdot t_0}{a \cdot {b}^{7}} - {\left(\frac{c}{b}\right)}^{2} \cdot \frac{a}{b}\right) - \frac{c}{b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* a c) 4.0)))
   (+
    (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (-
     (-
      (* -0.25 (/ (+ (* 16.0 t_0) (* 4.0 t_0)) (* a (pow b 7.0))))
      (* (pow (/ c b) 2.0) (/ a b)))
     (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = pow((a * c), 4.0);
	return (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (((-0.25 * (((16.0 * t_0) + (4.0 * t_0)) / (a * pow(b, 7.0)))) - (pow((c / b), 2.0) * (a / b))) - (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
    real(8) :: t_0
    t_0 = (a * c) ** 4.0d0
    code = ((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + ((((-0.25d0) * (((16.0d0 * t_0) + (4.0d0 * t_0)) / (a * (b ** 7.0d0)))) - (((c / b) ** 2.0d0) * (a / b))) - (c / b))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.pow((a * c), 4.0);
	return (-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + (((-0.25 * (((16.0 * t_0) + (4.0 * t_0)) / (a * Math.pow(b, 7.0)))) - (Math.pow((c / b), 2.0) * (a / b))) - (c / b));
}

Python

def code(a, b, c):
	t_0 = math.pow((a * c), 4.0)
	return (-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (((-0.25 * (((16.0 * t_0) + (4.0 * t_0)) / (a * math.pow(b, 7.0)))) - (math.pow((c / b), 2.0) * (a / b))) - (c / b))

Julia

function code(a, b, c)
	t_0 = Float64(a * c) ^ 4.0
	return Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(Float64(-0.25 * Float64(Float64(Float64(16.0 * t_0) + Float64(4.0 * t_0)) / Float64(a * (b ^ 7.0)))) - Float64((Float64(c / b) ^ 2.0) * Float64(a / b))) - Float64(c / b)))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = (a * c) ^ 4.0;
	tmp = (-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + (((-0.25 * (((16.0 * t_0) + (4.0 * t_0)) / (a * (b ^ 7.0)))) - (((c / b) ^ 2.0) * (a / b))) - (c / b));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[(N[(16.0 * t$95$0), $MachinePrecision] + N[(4.0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision] * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 20.6%

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

   1. *-commutative 20.6%

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

3. Simplified 20.6%

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

4. Taylor expanded in b around inf 96.7%

   \[\leadsto \color{blue}{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]
5. Step-by-step derivation

   1. *-commutative 96.7%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot -2\right)}}^{2}}{a \cdot {b}^{7}}\right)\right) \]

   2. unpow-prod-down 96.7%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{{\left({a}^{2} \cdot {c}^{2}\right)}^{2} \cdot {-2}^{2}}}{a \cdot {b}^{7}}\right)\right) \]

   3. pow-prod-down 96.7%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\color{blue}{\left({\left(a \cdot c\right)}^{2}\right)}}^{2} \cdot {-2}^{2}}{a \cdot {b}^{7}}\right)\right) \]

   4. pow-pow 96.7%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{{\left(a \cdot c\right)}^{\left(2 \cdot 2\right)}} \cdot {-2}^{2}}{a \cdot {b}^{7}}\right)\right) \]

   5. metadata-eval 96.7%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(a \cdot c\right)}^{\color{blue}{4}} \cdot {-2}^{2}}{a \cdot {b}^{7}}\right)\right) \]

   6. metadata-eval 96.7%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(a \cdot c\right)}^{4} \cdot \color{blue}{4}}{a \cdot {b}^{7}}\right)\right) \]

6. Applied egg-rr 96.7%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{{\left(a \cdot c\right)}^{4} \cdot 4}}{a \cdot {b}^{7}}\right)\right) \]
7. Step-by-step derivation

   1. expm1-log1p-u 96.7%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({a}^{4} \cdot {c}^{4}\right)\right)} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   2. expm1-udef 96.6%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({a}^{4} \cdot {c}^{4}\right)} - 1\right)} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   3. pow-prod-down 96.6%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(a \cdot c\right)}^{4}}\right)} - 1\right) + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

8. Applied egg-rr 96.6%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(a \cdot c\right)}^{4}\right)} - 1\right)} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]
9. Step-by-step derivation

   1. expm1-def 96.7%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(a \cdot c\right)}^{4}\right)\right)} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   2. expm1-log1p 96.7%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{{\left(a \cdot c\right)}^{4}} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

10. Simplified 96.7%

    \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{{\left(a \cdot c\right)}^{4}} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]
11. Step-by-step derivation

    1. *-commutative 96.7%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{\color{blue}{{c}^{2} \cdot a}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot {\left(a \cdot c\right)}^{4} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

    2. unpow3 96.7%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{{c}^{2} \cdot a}{\color{blue}{\left(b \cdot b\right) \cdot b}} + -0.25 \cdot \frac{16 \cdot {\left(a \cdot c\right)}^{4} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

    3. times-frac 96.7%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \color{blue}{\left(\frac{{c}^{2}}{b \cdot b} \cdot \frac{a}{b}\right)} + -0.25 \cdot \frac{16 \cdot {\left(a \cdot c\right)}^{4} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

    4. unpow2 96.7%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\frac{\color{blue}{c \cdot c}}{b \cdot b} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{16 \cdot {\left(a \cdot c\right)}^{4} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

    5. frac-times 96.7%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{16 \cdot {\left(a \cdot c\right)}^{4} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

    6. pow2 96.7%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{2}} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{16 \cdot {\left(a \cdot c\right)}^{4} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

12. Applied egg-rr 96.7%

    \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \color{blue}{\left({\left(\frac{c}{b}\right)}^{2} \cdot \frac{a}{b}\right)} + -0.25 \cdot \frac{16 \cdot {\left(a \cdot c\right)}^{4} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

13. Final simplification 96.7%

    \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{16 \cdot {\left(a \cdot c\right)}^{4} + 4 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} - {\left(\frac{c}{b}\right)}^{2} \cdot \frac{a}{b}\right) - \frac{c}{b}\right) \]

Alternative 2: 96.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{-2}{\frac{{b}^{5}}{{a}^{2} \cdot {c}^{3}}} - \frac{c}{b}\right) - \frac{a}{{b}^{3}} \cdot {c}^{2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (- (/ -2.0 (/ (pow b 5.0) (* (pow a 2.0) (pow c 3.0)))) (/ c b))
  (* (/ a (pow b 3.0)) (pow c 2.0))))

C

double code(double a, double b, double c) {
	return ((-2.0 / (pow(b, 5.0) / (pow(a, 2.0) * pow(c, 3.0)))) - (c / b)) - ((a / pow(b, 3.0)) * pow(c, 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 = (((-2.0d0) / ((b ** 5.0d0) / ((a ** 2.0d0) * (c ** 3.0d0)))) - (c / b)) - ((a / (b ** 3.0d0)) * (c ** 2.0d0))
end function

Java

public static double code(double a, double b, double c) {
	return ((-2.0 / (Math.pow(b, 5.0) / (Math.pow(a, 2.0) * Math.pow(c, 3.0)))) - (c / b)) - ((a / Math.pow(b, 3.0)) * Math.pow(c, 2.0));
}

Python

def code(a, b, c):
	return ((-2.0 / (math.pow(b, 5.0) / (math.pow(a, 2.0) * math.pow(c, 3.0)))) - (c / b)) - ((a / math.pow(b, 3.0)) * math.pow(c, 2.0))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-2.0 / Float64((b ^ 5.0) / Float64((a ^ 2.0) * (c ^ 3.0)))) - Float64(c / b)) - Float64(Float64(a / (b ^ 3.0)) * (c ^ 2.0)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-2.0 / ((b ^ 5.0) / ((a ^ 2.0) * (c ^ 3.0)))) - (c / b)) - ((a / (b ^ 3.0)) * (c ^ 2.0));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-2.0 / N[(N[Power[b, 5.0], $MachinePrecision] / N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.6%

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

   1. *-commutative 20.6%

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

3. Simplified 20.6%

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

4. Taylor expanded in b around inf 95.7%

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

   1. associate-+r+ 95.7%

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

   2. mul-1-neg 95.7%

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

   3. unsub-neg 95.7%

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

   4. mul-1-neg 95.7%

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

   5. unsub-neg 95.7%

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

   6. associate-*r/ 95.7%

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

   7. associate-/l* 95.7%

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

   8. *-commutative 95.7%

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

   9. associate-/l* 95.7%

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

   10. associate-/r/ 95.7%

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

6. Simplified 95.7%

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

7. Final simplification 95.7%

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

Alternative 3: 95.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / b) - (c * (c * (a * (b ** (-3.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.6%

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

   1. *-commutative 20.6%

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

3. Simplified 20.6%

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

4. Taylor expanded in b around inf 93.4%

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

   1. mul-1-neg 93.4%

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

   2. unsub-neg 93.4%

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

   3. mul-1-neg 93.4%

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

   4. distribute-neg-frac 93.4%

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

   5. associate-/l* 93.4%

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

   6. associate-/r/ 93.4%

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

6. Simplified 93.4%

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

   1. expm1-log1p-u 93.4%

      \[\leadsto \frac{-c}{b} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{{b}^{3}} \cdot {c}^{2}\right)\right)} \]

   2. expm1-udef 91.5%

      \[\leadsto \frac{-c}{b} - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{a}{{b}^{3}} \cdot {c}^{2}\right)} - 1\right)} \]

   3. *-commutative 91.5%

      \[\leadsto \frac{-c}{b} - \left(e^{\mathsf{log1p}\left(\color{blue}{{c}^{2} \cdot \frac{a}{{b}^{3}}}\right)} - 1\right) \]

   4. div-inv 91.5%

      \[\leadsto \frac{-c}{b} - \left(e^{\mathsf{log1p}\left({c}^{2} \cdot \color{blue}{\left(a \cdot \frac{1}{{b}^{3}}\right)}\right)} - 1\right) \]

   5. pow-flip 91.5%

      \[\leadsto \frac{-c}{b} - \left(e^{\mathsf{log1p}\left({c}^{2} \cdot \left(a \cdot \color{blue}{{b}^{\left(-3\right)}}\right)\right)} - 1\right) \]

   6. metadata-eval 91.5%

      \[\leadsto \frac{-c}{b} - \left(e^{\mathsf{log1p}\left({c}^{2} \cdot \left(a \cdot {b}^{\color{blue}{-3}}\right)\right)} - 1\right) \]

8. Applied egg-rr 91.5%

   \[\leadsto \frac{-c}{b} - \color{blue}{\left(e^{\mathsf{log1p}\left({c}^{2} \cdot \left(a \cdot {b}^{-3}\right)\right)} - 1\right)} \]
9. Step-by-step derivation

   1. expm1-def 93.4%

      \[\leadsto \frac{-c}{b} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({c}^{2} \cdot \left(a \cdot {b}^{-3}\right)\right)\right)} \]

   2. expm1-log1p-u 93.4%

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

   3. *-commutative 93.4%

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

   4. unpow2 93.4%

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

   5. associate-*r* 93.4%

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

10. Applied egg-rr 93.4%

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

11. Final simplification 93.4%

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

Alternative 4: 90.5% 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 20.6%

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

   1. *-commutative 20.6%

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

3. Simplified 20.6%

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

4. Taylor expanded in b around inf 88.1%

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

   1. mul-1-neg 88.1%

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

   2. distribute-neg-frac 88.1%

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

6. Simplified 88.1%

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

7. Final simplification 88.1%

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

Alternative 5: 1.7% accurate, 38.7× 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(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 20.6%

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

   1. *-commutative 20.6%

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

3. Simplified 20.6%

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

4. Taylor expanded in b around inf 87.6%

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

   1. associate-/l* 87.7%

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

   2. associate-*r/ 87.7%

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

   3. *-commutative 87.7%

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

6. Simplified 87.7%

   \[\leadsto \frac{\color{blue}{\frac{a \cdot -2}{\frac{b}{c}}}}{a \cdot 2} \]
7. Step-by-step derivation

   1. expm1-log1p-u 71.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{a \cdot -2}{\frac{b}{c}}}{a \cdot 2}\right)\right)} \]

   2. expm1-udef 19.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{a \cdot -2}{\frac{b}{c}}}{a \cdot 2}\right)} - 1} \]

   3. associate-/l/ 19.7%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a \cdot -2}{\left(a \cdot 2\right) \cdot \frac{b}{c}}}\right)} - 1 \]

   4. *-commutative 19.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{-2 \cdot a}}{\left(a \cdot 2\right) \cdot \frac{b}{c}}\right)} - 1 \]

8. Applied egg-rr 19.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2 \cdot a}{\left(a \cdot 2\right) \cdot \frac{b}{c}}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 71.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-2 \cdot a}{\left(a \cdot 2\right) \cdot \frac{b}{c}}\right)\right)} \]

   2. expm1-log1p 87.7%

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

   3. times-frac 87.6%

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

   4. associate-/l* 87.5%

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

   5. associate-*l/ 87.6%

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

   6. *-commutative 87.6%

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

   7. *-commutative 87.6%

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

   8. *-commutative 87.6%

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

   9. associate-/r* 87.6%

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

   10. metadata-eval 87.6%

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

10. Simplified 87.6%

    \[\leadsto \color{blue}{\left(c \cdot \frac{a}{b}\right) \cdot \frac{-1}{a}} \]
11. Step-by-step derivation

    1. expm1-log1p-u 71.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(c \cdot \frac{a}{b}\right) \cdot \frac{-1}{a}\right)\right)} \]

    2. expm1-udef 19.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(c \cdot \frac{a}{b}\right) \cdot \frac{-1}{a}\right)} - 1} \]

    3. associate-*l* 19.7%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{c \cdot \left(\frac{a}{b} \cdot \frac{-1}{a}\right)}\right)} - 1 \]

    4. frac-2neg 19.7%

       \[\leadsto e^{\mathsf{log1p}\left(c \cdot \left(\frac{a}{b} \cdot \color{blue}{\frac{--1}{-a}}\right)\right)} - 1 \]

    5. metadata-eval 19.7%

       \[\leadsto e^{\mathsf{log1p}\left(c \cdot \left(\frac{a}{b} \cdot \frac{\color{blue}{1}}{-a}\right)\right)} - 1 \]

    6. un-div-inv 19.7%

       \[\leadsto e^{\mathsf{log1p}\left(c \cdot \color{blue}{\frac{\frac{a}{b}}{-a}}\right)} - 1 \]

12. Applied egg-rr 19.7%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{\frac{a}{b}}{-a}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 71.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{\frac{a}{b}}{-a}\right)\right)} \]

    2. expm1-log1p 87.7%

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

    3. associate-/l/ 87.7%

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

    4. distribute-lft-neg-out 87.7%

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

14. Simplified 87.7%

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

    1. expm1-log1p-u 71.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{a}{-a \cdot b}\right)\right)} \]

    2. expm1-udef 19.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{a}{-a \cdot b}\right)} - 1} \]

    3. add-sqr-sqrt 0.0%

       \[\leadsto e^{\mathsf{log1p}\left(c \cdot \frac{a}{\color{blue}{\sqrt{-a \cdot b} \cdot \sqrt{-a \cdot b}}}\right)} - 1 \]

    4. sqrt-unprod 2.4%

       \[\leadsto e^{\mathsf{log1p}\left(c \cdot \frac{a}{\color{blue}{\sqrt{\left(-a \cdot b\right) \cdot \left(-a \cdot b\right)}}}\right)} - 1 \]

    5. sqr-neg 2.4%

       \[\leadsto e^{\mathsf{log1p}\left(c \cdot \frac{a}{\sqrt{\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}}\right)} - 1 \]

    6. sqrt-unprod 2.4%

       \[\leadsto e^{\mathsf{log1p}\left(c \cdot \frac{a}{\color{blue}{\sqrt{a \cdot b} \cdot \sqrt{a \cdot b}}}\right)} - 1 \]

    7. add-sqr-sqrt 2.4%

       \[\leadsto e^{\mathsf{log1p}\left(c \cdot \frac{a}{\color{blue}{a \cdot b}}\right)} - 1 \]

16. Applied egg-rr 2.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{a}{a \cdot b}\right)} - 1} \]
17. Step-by-step derivation

    1. expm1-def 1.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{a}{a \cdot b}\right)\right)} \]

    2. expm1-log1p 1.7%

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

    3. associate-/r* 1.7%

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

    4. *-inverses 1.7%

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

    5. associate-*r/ 1.7%

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

    6. *-rgt-identity 1.7%

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

18. Simplified 1.7%

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

19. Final simplification 1.7%

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

Reproduce

herbie shell --seed 2023339 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))