Quadratic roots, wide range

Percentage Accurate: 18.4% → 99.4%

Time: 14.3s

Alternatives: 4

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: 18.4% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[(N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(c * N[(N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision] + N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $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 c around inf 16.6%

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

   1. flip-+ 16.5%

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

   2. pow2 16.5%

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

   3. add-sqr-sqrt 17.0%

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

   4. cancel-sign-sub-inv 17.0%

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

   5. metadata-eval 17.0%

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

   6. cancel-sign-sub-inv 17.0%

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

   7. metadata-eval 17.0%

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

7. Applied egg-rr 17.0%

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

8. Taylor expanded in b around 0 99.4%

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

   1. *-commutative 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 2: 95.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[(N[((-a) * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / 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 c around 0 96.1%

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

   1. associate-*r/ 96.1%

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

   2. neg-mul-1 96.1%

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

   3. distribute-rgt-neg-in 96.1%

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

7. Simplified 96.1%

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

8. Taylor expanded in a around inf 95.8%

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

   1. pow1 95.8%

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

   2. mul-1-neg 95.8%

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

10. Applied egg-rr 95.8%

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

    1. unpow1 95.8%

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

    2. distribute-neg-frac2 95.8%

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

12. Simplified 95.8%

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

13. Taylor expanded in b around inf 96.4%

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

    1. neg-mul-1 96.4%

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

    2. +-commutative 96.4%

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

    3. unsub-neg 96.4%

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

    4. mul-1-neg 96.4%

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

    5. associate-/l* 96.4%

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

    6. unpow2 96.4%

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

    7. unpow2 96.4%

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

    8. times-frac 96.4%

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

    9. sqr-neg 96.4%

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

    10. unpow2 96.4%

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

    11. distribute-lft-neg-in 96.4%

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

    12. unpow2 96.4%

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

    13. sqr-neg 96.4%

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

    14. unpow2 96.4%

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

15. Simplified 96.4%

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

Alternative 3: 90.0% 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(c / Float64(-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 91.4%

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

   1. associate-*r/ 91.4%

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

   2. mul-1-neg 91.4%

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

7. Simplified 91.4%

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

8. Final simplification 91.4%

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

Alternative 4: 3.3% 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 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 c around 0 96.1%

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

   1. associate-*r/ 96.1%

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

   2. neg-mul-1 96.1%

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

   3. distribute-rgt-neg-in 96.1%

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

7. Simplified 96.1%

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

8. Taylor expanded in a around 0 91.1%

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

   1. expm1-log1p-u 81.2%

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

   2. expm1-undefine 19.6%

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

10. Applied egg-rr 19.6%

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

    1. sub-neg 19.6%

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

    2. log1p-undefine 19.6%

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

    3. rem-exp-log 29.6%

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

    4. associate-*r/ 29.6%

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

    5. *-commutative 29.6%

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

    6. associate-*r/ 29.6%

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

    7. mul-1-neg 29.6%

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

    8. unsub-neg 29.6%

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

    9. metadata-eval 29.6%

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

12. Simplified 29.6%

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

13. Taylor expanded in c around 0 3.3%

    \[\leadsto \color{blue}{1} + -1 \]

14. Final simplification 3.3%

    \[\leadsto 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024092 
(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)))