Quadratic roots, wide range

Percentage Accurate: 17.8% → 97.8%

Time: 34.1s

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: 17.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (-
   (-
    (* -0.25 (* (/ (pow (* a c) 4.0) a) (/ 20.0 (pow b 7.0))))
    (/ (* a (pow c 2.0)) (pow b 3.0)))
   (/ c b))))

C

double code(double a, double b, double c) {
	return (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (((-0.25 * ((pow((a * c), 4.0) / a) * (20.0 / pow(b, 7.0)))) - ((a * pow(c, 2.0)) / pow(b, 3.0))) - (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 = ((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + ((((-0.25d0) * ((((a * c) ** 4.0d0) / a) * (20.0d0 / (b ** 7.0d0)))) - ((a * (c ** 2.0d0)) / (b ** 3.0d0))) - (c / b))
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + (((-0.25 * ((Math.pow((a * c), 4.0) / a) * (20.0 / Math.pow(b, 7.0)))) - ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) - (c / b));
}

Python

def code(a, b, c):
	return (-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (((-0.25 * ((math.pow((a * c), 4.0) / a) * (20.0 / math.pow(b, 7.0)))) - ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) - (c / b))

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(Float64(-0.25 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(20.0 / (b ^ 7.0)))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) - Float64(c / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + (((-0.25 * ((((a * c) ^ 4.0) / a) * (20.0 / (b ^ 7.0)))) - ((a * (c ^ 2.0)) / (b ^ 3.0))) - (c / b));
end

Wolfram

code[a_, b_, c_] := 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[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.7%

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

5. Taylor expanded in b around inf 98.6%

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

6. Taylor expanded in c around 0 98.6%

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

   1. distribute-rgt-out 98.6%

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

   2. associate-*r* 98.6%

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

   3. *-commutative 98.6%

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

   4. times-frac 98.6%

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

8. Simplified 98.6%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right)}\right)\right) \]

9. Final simplification 98.6%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \]
10. Add Preprocessing

Alternative 2: 97.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.7%

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

5. Taylor expanded in b around inf 97.8%

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

6. Final simplification 97.8%

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

Alternative 3: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.7%

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

5. Taylor expanded in b around inf 96.3%

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

   1. mul-1-neg 96.3%

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

   2. unsub-neg 96.3%

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

   3. mul-1-neg 96.3%

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

   4. distribute-neg-frac 96.3%

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

   5. associate-/l* 96.3%

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

   6. associate-/r/ 96.3%

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

7. Simplified 96.3%

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

   1. unpow2 96.3%

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

9. Applied egg-rr 96.3%

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

10. Final simplification 96.3%

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

Alternative 4: 90.4% 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 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.7%

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

5. Taylor expanded in b around inf 91.5%

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

   1. mul-1-neg 91.5%

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

   2. distribute-neg-frac 91.5%

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

7. Simplified 91.5%

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

8. Final simplification 91.5%

   \[\leadsto \frac{-c}{b} \]
9. Add Preprocessing

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 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.7%

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

5. Taylor expanded in b around inf 91.1%

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

   1. associate-/l* 91.1%

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

   2. associate-*r/ 91.1%

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

   3. *-commutative 91.1%

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

7. Simplified 91.1%

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

   1. expm1-log1p-u 81.0%

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

   2. expm1-udef 21.8%

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

   3. associate-/l/ 21.8%

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

   4. *-commutative 21.8%

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

9. Applied egg-rr 21.8%

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

    1. expm1-def 80.9%

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

    2. expm1-log1p 91.0%

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

    3. *-commutative 91.0%

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

    4. associate-/r* 91.1%

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

    5. associate-/l* 91.1%

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

    6. associate-*l/ 91.1%

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

    7. *-commutative 91.1%

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

    8. associate-/l* 91.3%

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

    9. associate-/l* 91.1%

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

    10. associate-/r/ 91.4%

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

    11. associate-/l* 91.4%

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

    12. metadata-eval 91.4%

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

11. Simplified 91.4%

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

    1. frac-2neg 91.4%

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

    2. metadata-eval 91.4%

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

    3. frac-times 91.3%

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

    4. *-un-lft-identity 91.3%

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

13. Applied egg-rr 91.3%

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

    1. expm1-log1p-u 13.4%

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

    2. expm1-udef 10.0%

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

    3. associate-/l* 10.0%

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

    4. add-sqr-sqrt 0.0%

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

    5. sqrt-unprod 1.7%

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

    6. sqr-neg 1.7%

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

    7. sqrt-unprod 1.7%

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

    8. add-sqr-sqrt 1.7%

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

15. Applied egg-rr 1.7%

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

    1. expm1-def 1.7%

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

    2. expm1-log1p 1.7%

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

    3. associate-/r/ 1.7%

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

    4. *-inverses 1.7%

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

    5. *-lft-identity 1.7%

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

17. Simplified 1.7%

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

18. Final simplification 1.7%

    \[\leadsto \frac{c}{b} \]
19. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(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)))