Quadratic roots, wide range

Percentage Accurate: 17.5% → 97.6%

Time: 13.3s

Alternatives: 9

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: 17.5% 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: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + {c}^{4} \cdot \left(-10 \cdot \frac{{a}^{2}}{{b}^{7}} + -4 \cdot \frac{a}{c \cdot {b}^{5}}\right)\right)\right)}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   a
   (+
    (* -2.0 (/ c b))
    (*
     a
     (+
      (* -2.0 (/ (pow c 2.0) (pow b 3.0)))
      (*
       (pow c 4.0)
       (+
        (* -10.0 (/ (pow a 2.0) (pow b 7.0)))
        (* -4.0 (/ a (* c (pow b 5.0))))))))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (a * ((-2.0 * (c / b)) + (a * ((-2.0 * (pow(c, 2.0) / pow(b, 3.0))) + (pow(c, 4.0) * ((-10.0 * (pow(a, 2.0) / pow(b, 7.0))) + (-4.0 * (a / (c * pow(b, 5.0)))))))))) / (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 = (a * (((-2.0d0) * (c / b)) + (a * (((-2.0d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + ((c ** 4.0d0) * (((-10.0d0) * ((a ** 2.0d0) / (b ** 7.0d0))) + ((-4.0d0) * (a / (c * (b ** 5.0d0)))))))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((-2.0 * (c / b)) + (a * ((-2.0 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (Math.pow(c, 4.0) * ((-10.0 * (Math.pow(a, 2.0) / Math.pow(b, 7.0))) + (-4.0 * (a / (c * Math.pow(b, 5.0)))))))))) / (a * 2.0);
}

Python

def code(a, b, c):
	return (a * ((-2.0 * (c / b)) + (a * ((-2.0 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (math.pow(c, 4.0) * ((-10.0 * (math.pow(a, 2.0) / math.pow(b, 7.0))) + (-4.0 * (a / (c * math.pow(b, 5.0)))))))))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(-2.0 * Float64(c / b)) + Float64(a * Float64(Float64(-2.0 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64((c ^ 4.0) * Float64(Float64(-10.0 * Float64((a ^ 2.0) / (b ^ 7.0))) + Float64(-4.0 * Float64(a / Float64(c * (b ^ 5.0)))))))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((-2.0 * (c / b)) + (a * ((-2.0 * ((c ^ 2.0) / (b ^ 3.0))) + ((c ^ 4.0) * ((-10.0 * ((a ^ 2.0) / (b ^ 7.0))) + (-4.0 * (a / (c * (b ^ 5.0)))))))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-2.0 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(-10.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(a / N[(c * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in a around 0 97.0%

   \[\leadsto \frac{\color{blue}{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.5 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)\right)}}{a \cdot 2} \]

6. Taylor expanded in b around 0 97.0%

   \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.5 \cdot \color{blue}{\frac{a \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right)\right)}{a \cdot 2} \]
7. Step-by-step derivation

   1. distribute-rgt-out 97.0%

      \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.5 \cdot \frac{a \cdot \color{blue}{\left({c}^{4} \cdot \left(4 + 16\right)\right)}}{{b}^{7}}\right)\right)\right)}{a \cdot 2} \]

   2. metadata-eval 97.0%

      \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.5 \cdot \frac{a \cdot \left({c}^{4} \cdot \color{blue}{20}\right)}{{b}^{7}}\right)\right)\right)}{a \cdot 2} \]

8. Simplified 97.0%

   \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.5 \cdot \color{blue}{\frac{a \cdot \left({c}^{4} \cdot 20\right)}{{b}^{7}}}\right)\right)\right)}{a \cdot 2} \]

9. Taylor expanded in c around inf 97.0%

   \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + \color{blue}{{c}^{4} \cdot \left(-10 \cdot \frac{{a}^{2}}{{b}^{7}} + -4 \cdot \frac{a}{{b}^{5} \cdot c}\right)}\right)\right)}{a \cdot 2} \]

10. Final simplification 97.0%

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

Alternative 2: 97.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in c around 0 96.1%

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

6. Taylor expanded in a around 0 96.4%

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

   1. mul-1-neg 96.4%

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

   2. distribute-frac-neg 96.4%

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

   3. +-commutative 96.4%

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

   4. distribute-frac-neg 96.4%

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

   5. unsub-neg 96.4%

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

   6. mul-1-neg 96.4%

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

   7. unsub-neg 96.4%

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

   8. *-commutative 96.4%

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

   9. associate-/l* 96.4%

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

   10. associate-*l* 96.4%

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

8. Simplified 96.4%

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

9. Final simplification 96.4%

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

Alternative 3: 97.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in a around 0 96.4%

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

6. Taylor expanded in c around inf 96.4%

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

   1. mul-1-neg 96.4%

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

   2. unsub-neg 96.4%

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

   3. *-commutative 96.4%

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

8. Simplified 96.4%

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

9. Final simplification 96.4%

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

Alternative 4: 96.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{{a}^{2} \cdot \left(-2 \cdot {\left(\frac{c}{b}\right)}^{2}\right) - a \cdot c}{{b}^{3}} + \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (/ (- (* (pow a 2.0) (* -2.0 (pow (/ c b) 2.0))) (* a c)) (pow b 3.0))
   (/ -1.0 b))))

C

double code(double a, double b, double c) {
	return c * ((((pow(a, 2.0) * (-2.0 * pow((c / b), 2.0))) - (a * c)) / pow(b, 3.0)) + (-1.0 / 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 * (((((a ** 2.0d0) * ((-2.0d0) * ((c / b) ** 2.0d0))) - (a * c)) / (b ** 3.0d0)) + ((-1.0d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((((Math.pow(a, 2.0) * (-2.0 * Math.pow((c / b), 2.0))) - (a * c)) / Math.pow(b, 3.0)) + (-1.0 / b));
}

Python

def code(a, b, c):
	return c * ((((math.pow(a, 2.0) * (-2.0 * math.pow((c / b), 2.0))) - (a * c)) / math.pow(b, 3.0)) + (-1.0 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(Float64(Float64((a ^ 2.0) * Float64(-2.0 * (Float64(c / b) ^ 2.0))) - Float64(a * c)) / (b ^ 3.0)) + Float64(-1.0 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * (((((a ^ 2.0) * (-2.0 * ((c / b) ^ 2.0))) - (a * c)) / (b ^ 3.0)) + (-1.0 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[(-2.0 * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in c around 0 96.1%

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

6. Taylor expanded in b around inf 96.1%

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

8. Simplified 96.1%

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

9. Final simplification 96.1%

   \[\leadsto c \cdot \left(\frac{{a}^{2} \cdot \left(-2 \cdot {\left(\frac{c}{b}\right)}^{2}\right) - a \cdot c}{{b}^{3}} + \frac{-1}{b}\right) \]
10. Add Preprocessing

Alternative 5: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in a around 0 95.4%

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

   1. mul-1-neg 95.4%

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

   2. unsub-neg 95.4%

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

   3. mul-1-neg 95.4%

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

   4. distribute-neg-frac2 95.4%

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

   5. associate-/l* 95.4%

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

7. Simplified 95.4%

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

8. Final simplification 95.4%

   \[\leadsto \frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}} \]
9. Add Preprocessing

Alternative 6: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ -1.0 b) (* a (/ c (pow b 3.0))))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (a * (c / 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 * (((-1.0d0) / b) - (a * (c / (b ** 3.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (a * (c / Math.pow(b, 3.0))));
}

Python

def code(a, b, c):
	return c * ((-1.0 / b) - (a * (c / math.pow(b, 3.0))))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(a * Float64(c / (b ^ 3.0)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - (a * (c / (b ^ 3.0))));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in a around 0 95.4%

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

   1. mul-1-neg 95.4%

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

   2. unsub-neg 95.4%

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

   3. mul-1-neg 95.4%

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

   4. distribute-neg-frac2 95.4%

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

   5. associate-/l* 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in c around 0 95.0%

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

10. Simplified 95.3%

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

11. Taylor expanded in c around 0 95.0%

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

    1. sub-neg 95.0%

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

    2. mul-1-neg 95.0%

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

    3. *-commutative 95.0%

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

    4. associate-*r/ 95.0%

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

    5. distribute-rgt-neg-out 95.0%

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

    6. +-commutative 95.0%

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

    7. distribute-rgt-neg-out 95.0%

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

    8. associate-*r/ 95.0%

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

    9. *-commutative 95.0%

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

    10. unsub-neg 95.0%

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

    11. distribute-neg-frac 95.0%

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

    12. metadata-eval 95.0%

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

    13. associate-/l* 95.0%

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

13. Simplified 95.0%

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

14. Final simplification 95.0%

    \[\leadsto c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right) \]
15. Add Preprocessing

Alternative 7: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a, \frac{c}{b} \cdot \frac{c}{b}, c\right)}{-b} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in a around 0 95.4%

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

   1. mul-1-neg 95.4%

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

   2. unsub-neg 95.4%

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

   3. mul-1-neg 95.4%

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

   4. distribute-neg-frac2 95.4%

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

   5. associate-/l* 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in c around 0 95.0%

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

10. Simplified 95.3%

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

    1. unpow2 95.3%

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

12. Applied egg-rr 95.3%

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

13. Final simplification 95.3%

    \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c}{b} \cdot \frac{c}{b}, c\right)}{-b} \]
14. Add Preprocessing

Alternative 8: 90.7% 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 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in b around inf 91.0%

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

   1. associate-*r/ 91.0%

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

   2. mul-1-neg 91.0%

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

7. Simplified 91.0%

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

8. Final simplification 91.0%

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

Alternative 9: 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 17.0%

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

   1. *-commutative 17.0%

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

3. Simplified 17.0%

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

5. Taylor expanded in b around -inf 8.6%

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

   1. mul-1-neg 8.6%

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

   2. *-commutative 8.6%

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

   3. distribute-rgt-neg-in 8.6%

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

   4. +-commutative 8.6%

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

   5. mul-1-neg 8.6%

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

   6. unsub-neg 8.6%

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

7. Simplified 8.6%

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

8. Taylor expanded in a around inf 1.7%

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

9. Final simplification 1.7%

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

Reproduce

herbie shell --seed 2024072 
(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)))