Quadratic roots, narrow range

Percentage Accurate: 55.2% → 99.3%

Time: 13.3s

Alternatives: 8

Speedup: 29.0×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

   1. *-commutative 58.2%

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

3. Simplified 58.2%

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

   1. neg-sub0 58.2%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. flip-- 58.0%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - b \cdot b}{0 + b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 58.0%

      \[\leadsto \frac{\frac{\color{blue}{0} - b \cdot b}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 58.0%

      \[\leadsto \frac{\frac{0 - \color{blue}{{b}^{2}}}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. add-sqr-sqrt 56.9%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. sqrt-prod 58.0%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   7. sqr-neg 58.0%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. sqrt-unprod 0.0%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   9. add-sqr-sqrt 1.6%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\left(-b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   10. sub-neg 1.6%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{0 - b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   11. neg-sub0 1.6%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{-b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   12. add-sqr-sqrt 0.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   13. sqrt-unprod 58.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   14. sqr-neg 58.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\sqrt{\color{blue}{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   15. sqrt-prod 56.9%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   16. add-sqr-sqrt 58.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 58.0%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{2}}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. neg-sub0 58.0%

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

8. Simplified 58.0%

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

   1. flip-+ 58.0%

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

   2. pow2 58.0%

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

   3. pow2 58.0%

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

   4. distribute-frac-neg 58.0%

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

   5. pow2 58.0%

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

   6. pow1 58.0%

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

   7. pow-div 58.1%

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

   8. metadata-eval 58.1%

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

   9. pow1 58.1%

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

   10. add-sqr-sqrt 59.7%

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

   11. pow2 59.7%

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

   12. associate-*l* 59.7%

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

10. Applied egg-rr 59.7%

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

    1. associate--r- 99.3%

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

    2. unpow2 99.3%

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

    3. unpow2 99.3%

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

    4. difference-of-squares 99.3%

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

    5. +-commutative 99.3%

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

    6. neg-mul-1 99.3%

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

    7. distribute-rgt1-in 99.3%

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

    8. metadata-eval 99.3%

       \[\leadsto \frac{\frac{\left(\color{blue}{0} \cdot b\right) \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

    9. mul0-lft 99.3%

       \[\leadsto \frac{\frac{\color{blue}{0} \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

    10. unpow2 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{b \cdot b} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

    11. fmm-def 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

    12. *-commutative 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)}}}{a \cdot 2} \]

    13. distribute-rgt-neg-in 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)}}}{a \cdot 2} \]

    14. metadata-eval 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)}}}{a \cdot 2} \]

12. Simplified 99.3%

    \[\leadsto \frac{\color{blue}{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}}}{a \cdot 2} \]

13. Taylor expanded in b around 0 99.3%

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

14. Final simplification 99.3%

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

Alternative 2: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -46:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right) + 0 \cdot \left(b + b\right)}{\left(c \cdot \left(-2 \cdot \left(a \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right)\right)\right) - b\right) - b}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -46.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/
    (/
     (+ (* 4.0 (* a c)) (* 0.0 (+ b b)))
     (- (- (* c (* -2.0 (* a (- (/ -1.0 b) (/ (* a c) (pow b 3.0)))))) b) b))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -46.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (((4.0 * (a * c)) + (0.0 * (b + b))) / (((c * (-2.0 * (a * ((-1.0 / b) - ((a * c) / pow(b, 3.0)))))) - b) - b)) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -46.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(4.0 * Float64(a * c)) + Float64(0.0 * Float64(b + b))) / Float64(Float64(Float64(c * Float64(-2.0 * Float64(a * Float64(Float64(-1.0 / b) - Float64(Float64(a * c) / (b ^ 3.0)))))) - b) - b)) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -46.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] + N[(0.0 * N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(c * N[(-2.0 * N[(a * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -46

   1. Initial program 89.2%

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

      1. *-commutative 89.2%

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

   3. Simplified 89.2%

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

   if -46 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 54.9%

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

      1. *-commutative 54.9%

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

   3. Simplified 54.9%

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

      1. neg-sub0 54.9%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. flip-- 54.6%

         \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - b \cdot b}{0 + b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. metadata-eval 54.6%

         \[\leadsto \frac{\frac{\color{blue}{0} - b \cdot b}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 54.6%

         \[\leadsto \frac{\frac{0 - \color{blue}{{b}^{2}}}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. add-sqr-sqrt 53.5%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. sqrt-prod 54.6%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      7. sqr-neg 54.6%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      8. sqrt-unprod 0.0%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      9. add-sqr-sqrt 1.6%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\left(-b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      10. sub-neg 1.6%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{0 - b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      11. neg-sub0 1.6%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{-b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      12. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      13. sqrt-unprod 54.6%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      14. sqr-neg 54.6%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\sqrt{\color{blue}{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      15. sqrt-prod 53.5%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      16. add-sqr-sqrt 54.6%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 54.6%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{2}}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. neg-sub0 54.6%

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

   8. Simplified 54.6%

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

      1. flip-+ 54.6%

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

      2. pow2 54.6%

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

      3. pow2 54.6%

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

      4. distribute-frac-neg 54.6%

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

      5. pow2 54.6%

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

      6. pow1 54.6%

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

      7. pow-div 54.8%

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

      8. metadata-eval 54.8%

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

      9. pow1 54.8%

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

      10. add-sqr-sqrt 56.3%

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

      11. pow2 56.3%

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

      12. associate-*l* 56.3%

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

   10. Applied egg-rr 56.3%

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

       1. associate--r- 99.3%

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

       2. unpow2 99.3%

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

       3. unpow2 99.3%

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

       4. difference-of-squares 99.3%

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

       5. +-commutative 99.3%

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

       6. neg-mul-1 99.3%

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

       7. distribute-rgt1-in 99.3%

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

       8. metadata-eval 99.3%

          \[\leadsto \frac{\frac{\left(\color{blue}{0} \cdot b\right) \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

       9. mul0-lft 99.3%

          \[\leadsto \frac{\frac{\color{blue}{0} \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

       10. unpow2 99.3%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{b \cdot b} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

       11. fmm-def 99.3%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

       12. *-commutative 99.3%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)}}}{a \cdot 2} \]

       13. distribute-rgt-neg-in 99.3%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)}}}{a \cdot 2} \]

       14. metadata-eval 99.3%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)}}}{a \cdot 2} \]

   12. Simplified 99.3%

       \[\leadsto \frac{\color{blue}{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}}}{a \cdot 2} \]

   13. Taylor expanded in c around 0 90.5%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \color{blue}{\left(b + c \cdot \left(-2 \cdot \frac{a}{b} + -2 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)\right)}}}{a \cdot 2} \]
   14. Step-by-step derivation

       1. distribute-lft-out 90.5%

          \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \left(b + c \cdot \color{blue}{\left(-2 \cdot \left(\frac{a}{b} + \frac{{a}^{2} \cdot c}{{b}^{3}}\right)\right)}\right)}}{a \cdot 2} \]

       2. associate-/l* 90.5%

          \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \left(b + c \cdot \left(-2 \cdot \left(\frac{a}{b} + \color{blue}{{a}^{2} \cdot \frac{c}{{b}^{3}}}\right)\right)\right)}}{a \cdot 2} \]

   15. Simplified 90.5%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \color{blue}{\left(b + c \cdot \left(-2 \cdot \left(\frac{a}{b} + {a}^{2} \cdot \frac{c}{{b}^{3}}\right)\right)\right)}}}{a \cdot 2} \]

   16. Taylor expanded in a around 0 90.5%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \left(b + c \cdot \left(-2 \cdot \color{blue}{\left(a \cdot \left(\frac{1}{b} + \frac{a \cdot c}{{b}^{3}}\right)\right)}\right)\right)}}{a \cdot 2} \]
   17. Step-by-step derivation

       1. +-commutative 90.5%

          \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \left(b + c \cdot \left(-2 \cdot \left(a \cdot \color{blue}{\left(\frac{a \cdot c}{{b}^{3}} + \frac{1}{b}\right)}\right)\right)\right)}}{a \cdot 2} \]

   18. Simplified 90.5%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \left(b + c \cdot \left(-2 \cdot \color{blue}{\left(a \cdot \left(\frac{a \cdot c}{{b}^{3}} + \frac{1}{b}\right)\right)}\right)\right)}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -46:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right) + 0 \cdot \left(b + b\right)}{\left(c \cdot \left(-2 \cdot \left(a \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right)\right)\right) - b\right) - b}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -46:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right) + 0 \cdot \left(b + b\right)}{\left(c \cdot \left(-2 \cdot \left(a \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right)\right)\right) - b\right) - b}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))))
   (if (<= t_0 -46.0)
     t_0
     (/
      (/
       (+ (* 4.0 (* a c)) (* 0.0 (+ b b)))
       (- (- (* c (* -2.0 (* a (- (/ -1.0 b) (/ (* a c) (pow b 3.0)))))) b) b))
      (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -46.0) {
		tmp = t_0;
	} else {
		tmp = (((4.0 * (a * c)) + (0.0 * (b + b))) / (((c * (-2.0 * (a * ((-1.0 / b) - ((a * c) / pow(b, 3.0)))))) - b) - b)) / (a * 2.0);
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (a * 2.0d0)
    if (t_0 <= (-46.0d0)) then
        tmp = t_0
    else
        tmp = (((4.0d0 * (a * c)) + (0.0d0 * (b + b))) / (((c * ((-2.0d0) * (a * (((-1.0d0) / b) - ((a * c) / (b ** 3.0d0)))))) - b) - b)) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -46.0) {
		tmp = t_0;
	} else {
		tmp = (((4.0 * (a * c)) + (0.0 * (b + b))) / (((c * (-2.0 * (a * ((-1.0 / b) - ((a * c) / Math.pow(b, 3.0)))))) - b) - b)) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -46.0:
		tmp = t_0
	else:
		tmp = (((4.0 * (a * c)) + (0.0 * (b + b))) / (((c * (-2.0 * (a * ((-1.0 / b) - ((a * c) / math.pow(b, 3.0)))))) - b) - b)) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -46.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(4.0 * Float64(a * c)) + Float64(0.0 * Float64(b + b))) / Float64(Float64(Float64(c * Float64(-2.0 * Float64(a * Float64(Float64(-1.0 / b) - Float64(Float64(a * c) / (b ^ 3.0)))))) - b) - b)) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -46.0)
		tmp = t_0;
	else
		tmp = (((4.0 * (a * c)) + (0.0 * (b + b))) / (((c * (-2.0 * (a * ((-1.0 / b) - ((a * c) / (b ^ 3.0)))))) - b) - b)) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -46.0], t$95$0, N[(N[(N[(N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] + N[(0.0 * N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(c * N[(-2.0 * N[(a * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -46

   1. Initial program 89.2%

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

   if -46 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 54.9%

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

      1. *-commutative 54.9%

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

   3. Simplified 54.9%

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

      1. neg-sub0 54.9%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. flip-- 54.6%

         \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - b \cdot b}{0 + b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. metadata-eval 54.6%

         \[\leadsto \frac{\frac{\color{blue}{0} - b \cdot b}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 54.6%

         \[\leadsto \frac{\frac{0 - \color{blue}{{b}^{2}}}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. add-sqr-sqrt 53.5%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. sqrt-prod 54.6%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      7. sqr-neg 54.6%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      8. sqrt-unprod 0.0%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      9. add-sqr-sqrt 1.6%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\left(-b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      10. sub-neg 1.6%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{0 - b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      11. neg-sub0 1.6%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{-b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      12. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      13. sqrt-unprod 54.6%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      14. sqr-neg 54.6%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\sqrt{\color{blue}{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      15. sqrt-prod 53.5%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      16. add-sqr-sqrt 54.6%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 54.6%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{2}}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. neg-sub0 54.6%

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

   8. Simplified 54.6%

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

      1. flip-+ 54.6%

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

      2. pow2 54.6%

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

      3. pow2 54.6%

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

      4. distribute-frac-neg 54.6%

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

      5. pow2 54.6%

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

      6. pow1 54.6%

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

      7. pow-div 54.8%

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

      8. metadata-eval 54.8%

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

      9. pow1 54.8%

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

      10. add-sqr-sqrt 56.3%

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

      11. pow2 56.3%

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

      12. associate-*l* 56.3%

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

   10. Applied egg-rr 56.3%

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

       1. associate--r- 99.3%

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

       2. unpow2 99.3%

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

       3. unpow2 99.3%

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

       4. difference-of-squares 99.3%

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

       5. +-commutative 99.3%

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

       6. neg-mul-1 99.3%

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

       7. distribute-rgt1-in 99.3%

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

       8. metadata-eval 99.3%

          \[\leadsto \frac{\frac{\left(\color{blue}{0} \cdot b\right) \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

       9. mul0-lft 99.3%

          \[\leadsto \frac{\frac{\color{blue}{0} \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

       10. unpow2 99.3%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{b \cdot b} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

       11. fmm-def 99.3%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

       12. *-commutative 99.3%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)}}}{a \cdot 2} \]

       13. distribute-rgt-neg-in 99.3%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)}}}{a \cdot 2} \]

       14. metadata-eval 99.3%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)}}}{a \cdot 2} \]

   12. Simplified 99.3%

       \[\leadsto \frac{\color{blue}{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}}}{a \cdot 2} \]

   13. Taylor expanded in c around 0 90.5%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \color{blue}{\left(b + c \cdot \left(-2 \cdot \frac{a}{b} + -2 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)\right)}}}{a \cdot 2} \]
   14. Step-by-step derivation

       1. distribute-lft-out 90.5%

          \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \left(b + c \cdot \color{blue}{\left(-2 \cdot \left(\frac{a}{b} + \frac{{a}^{2} \cdot c}{{b}^{3}}\right)\right)}\right)}}{a \cdot 2} \]

       2. associate-/l* 90.5%

          \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \left(b + c \cdot \left(-2 \cdot \left(\frac{a}{b} + \color{blue}{{a}^{2} \cdot \frac{c}{{b}^{3}}}\right)\right)\right)}}{a \cdot 2} \]

   15. Simplified 90.5%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \color{blue}{\left(b + c \cdot \left(-2 \cdot \left(\frac{a}{b} + {a}^{2} \cdot \frac{c}{{b}^{3}}\right)\right)\right)}}}{a \cdot 2} \]

   16. Taylor expanded in a around 0 90.5%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \left(b + c \cdot \left(-2 \cdot \color{blue}{\left(a \cdot \left(\frac{1}{b} + \frac{a \cdot c}{{b}^{3}}\right)\right)}\right)\right)}}{a \cdot 2} \]
   17. Step-by-step derivation

       1. +-commutative 90.5%

          \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \left(b + c \cdot \left(-2 \cdot \left(a \cdot \color{blue}{\left(\frac{a \cdot c}{{b}^{3}} + \frac{1}{b}\right)}\right)\right)\right)}}{a \cdot 2} \]

   18. Simplified 90.5%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \left(b + c \cdot \left(-2 \cdot \color{blue}{\left(a \cdot \left(\frac{a \cdot c}{{b}^{3}} + \frac{1}{b}\right)\right)}\right)\right)}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -46:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right) + 0 \cdot \left(b + b\right)}{\left(c \cdot \left(-2 \cdot \left(a \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right)\right)\right) - b\right) - b}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.094:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \left(b + -2 \cdot \frac{a \cdot c}{b}\right)}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))))
   (if (<= t_0 -0.094)
     t_0
     (/
      (/ (* 4.0 (* a c)) (- (- b) (+ b (* -2.0 (/ (* a c) b)))))
      (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.094) {
		tmp = t_0;
	} else {
		tmp = ((4.0 * (a * c)) / (-b - (b + (-2.0 * ((a * c) / b))))) / (a * 2.0);
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.094d0)) then
        tmp = t_0
    else
        tmp = ((4.0d0 * (a * c)) / (-b - (b + ((-2.0d0) * ((a * c) / b))))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.094) {
		tmp = t_0;
	} else {
		tmp = ((4.0 * (a * c)) / (-b - (b + (-2.0 * ((a * c) / b))))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.094:
		tmp = t_0
	else:
		tmp = ((4.0 * (a * c)) / (-b - (b + (-2.0 * ((a * c) / b))))) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.094)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(a * c)) / Float64(Float64(-b) - Float64(b + Float64(-2.0 * Float64(Float64(a * c) / b))))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.094)
		tmp = t_0;
	else
		tmp = ((4.0 * (a * c)) / (-b - (b + (-2.0 * ((a * c) / b))))) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.094], t$95$0, N[(N[(N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[(b + N[(-2.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.094

   1. Initial program 81.5%

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

   if -0.094 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 50.5%

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

      1. *-commutative 50.5%

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

   3. Simplified 50.5%

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

      1. neg-sub0 50.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. flip-- 50.2%

         \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - b \cdot b}{0 + b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. metadata-eval 50.2%

         \[\leadsto \frac{\frac{\color{blue}{0} - b \cdot b}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 50.2%

         \[\leadsto \frac{\frac{0 - \color{blue}{{b}^{2}}}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. add-sqr-sqrt 49.3%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. sqrt-prod 50.2%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      7. sqr-neg 50.2%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      8. sqrt-unprod 0.0%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      9. add-sqr-sqrt 1.6%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\left(-b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      10. sub-neg 1.6%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{0 - b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      11. neg-sub0 1.6%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{-b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      12. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      13. sqrt-unprod 50.2%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      14. sqr-neg 50.2%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\sqrt{\color{blue}{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      15. sqrt-prod 49.3%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      16. add-sqr-sqrt 50.2%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 50.2%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{2}}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. neg-sub0 50.2%

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

   8. Simplified 50.2%

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

      1. flip-+ 50.2%

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

      2. pow2 50.2%

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

      3. pow2 50.2%

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

      4. distribute-frac-neg 50.2%

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

      5. pow2 50.2%

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

      6. pow1 50.2%

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

      7. pow-div 50.3%

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

      8. metadata-eval 50.3%

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

      9. pow1 50.3%

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

      10. add-sqr-sqrt 51.8%

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

      11. pow2 51.8%

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

      12. associate-*l* 51.8%

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

   10. Applied egg-rr 51.9%

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

       1. associate--r- 99.4%

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

       2. unpow2 99.4%

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

       3. unpow2 99.4%

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

       4. difference-of-squares 99.4%

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

       5. +-commutative 99.4%

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

       6. neg-mul-1 99.4%

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

       7. distribute-rgt1-in 99.4%

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

       8. metadata-eval 99.4%

          \[\leadsto \frac{\frac{\left(\color{blue}{0} \cdot b\right) \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

       9. mul0-lft 99.4%

          \[\leadsto \frac{\frac{\color{blue}{0} \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

       10. unpow2 99.4%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{b \cdot b} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

       11. fmm-def 99.3%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

       12. *-commutative 99.3%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)}}}{a \cdot 2} \]

       13. distribute-rgt-neg-in 99.3%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)}}}{a \cdot 2} \]

       14. metadata-eval 99.3%

           \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)}}}{a \cdot 2} \]

   12. Simplified 99.3%

       \[\leadsto \frac{\color{blue}{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}}}{a \cdot 2} \]

   13. Taylor expanded in b around 0 99.3%

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

   14. Taylor expanded in a around 0 87.1%

       \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{a \cdot c}{b}\right)}}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.094:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \left(b + -2 \cdot \frac{a \cdot c}{b}\right)}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.1% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

   1. *-commutative 58.2%

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

3. Simplified 58.2%

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

   1. neg-sub0 58.2%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. flip-- 58.0%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - b \cdot b}{0 + b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 58.0%

      \[\leadsto \frac{\frac{\color{blue}{0} - b \cdot b}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 58.0%

      \[\leadsto \frac{\frac{0 - \color{blue}{{b}^{2}}}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. add-sqr-sqrt 56.9%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. sqrt-prod 58.0%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   7. sqr-neg 58.0%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. sqrt-unprod 0.0%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   9. add-sqr-sqrt 1.6%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\left(-b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   10. sub-neg 1.6%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{0 - b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   11. neg-sub0 1.6%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{-b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   12. add-sqr-sqrt 0.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   13. sqrt-unprod 58.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   14. sqr-neg 58.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\sqrt{\color{blue}{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   15. sqrt-prod 56.9%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   16. add-sqr-sqrt 58.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 58.0%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{2}}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. neg-sub0 58.0%

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

8. Simplified 58.0%

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

   1. flip-+ 58.0%

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

   2. pow2 58.0%

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

   3. pow2 58.0%

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

   4. distribute-frac-neg 58.0%

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

   5. pow2 58.0%

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

   6. pow1 58.0%

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

   7. pow-div 58.1%

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

   8. metadata-eval 58.1%

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

   9. pow1 58.1%

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

   10. add-sqr-sqrt 59.7%

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

   11. pow2 59.7%

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

   12. associate-*l* 59.7%

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

10. Applied egg-rr 59.7%

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

    1. associate--r- 99.3%

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

    2. unpow2 99.3%

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

    3. unpow2 99.3%

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

    4. difference-of-squares 99.3%

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

    5. +-commutative 99.3%

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

    6. neg-mul-1 99.3%

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

    7. distribute-rgt1-in 99.3%

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

    8. metadata-eval 99.3%

       \[\leadsto \frac{\frac{\left(\color{blue}{0} \cdot b\right) \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

    9. mul0-lft 99.3%

       \[\leadsto \frac{\frac{\color{blue}{0} \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

    10. unpow2 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{b \cdot b} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

    11. fmm-def 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

    12. *-commutative 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)}}}{a \cdot 2} \]

    13. distribute-rgt-neg-in 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)}}}{a \cdot 2} \]

    14. metadata-eval 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)}}}{a \cdot 2} \]

12. Simplified 99.3%

    \[\leadsto \frac{\color{blue}{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}}}{a \cdot 2} \]

13. Taylor expanded in b around 0 99.3%

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

14. Taylor expanded in a around 0 81.1%

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

Alternative 6: 82.1% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, b, c):
	return ((4.0 * (a * c)) / (2.0 * ((a * (c / b)) - b))) / (a * 2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

   1. *-commutative 58.2%

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

3. Simplified 58.2%

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

   1. neg-sub0 58.2%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. flip-- 58.0%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - b \cdot b}{0 + b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 58.0%

      \[\leadsto \frac{\frac{\color{blue}{0} - b \cdot b}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 58.0%

      \[\leadsto \frac{\frac{0 - \color{blue}{{b}^{2}}}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. add-sqr-sqrt 56.9%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. sqrt-prod 58.0%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   7. sqr-neg 58.0%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. sqrt-unprod 0.0%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   9. add-sqr-sqrt 1.6%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\left(-b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   10. sub-neg 1.6%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{0 - b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   11. neg-sub0 1.6%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{-b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   12. add-sqr-sqrt 0.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   13. sqrt-unprod 58.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   14. sqr-neg 58.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\sqrt{\color{blue}{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   15. sqrt-prod 56.9%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   16. add-sqr-sqrt 58.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 58.0%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{2}}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. neg-sub0 58.0%

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

8. Simplified 58.0%

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

   1. flip-+ 58.0%

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

   2. pow2 58.0%

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

   3. pow2 58.0%

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

   4. distribute-frac-neg 58.0%

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

   5. pow2 58.0%

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

   6. pow1 58.0%

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

   7. pow-div 58.1%

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

   8. metadata-eval 58.1%

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

   9. pow1 58.1%

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

   10. add-sqr-sqrt 59.7%

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

   11. pow2 59.7%

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

   12. associate-*l* 59.7%

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

10. Applied egg-rr 59.7%

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

    1. associate--r- 99.3%

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

    2. unpow2 99.3%

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

    3. unpow2 99.3%

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

    4. difference-of-squares 99.3%

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

    5. +-commutative 99.3%

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

    6. neg-mul-1 99.3%

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

    7. distribute-rgt1-in 99.3%

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

    8. metadata-eval 99.3%

       \[\leadsto \frac{\frac{\left(\color{blue}{0} \cdot b\right) \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

    9. mul0-lft 99.3%

       \[\leadsto \frac{\frac{\color{blue}{0} \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

    10. unpow2 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{b \cdot b} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

    11. fmm-def 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

    12. *-commutative 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)}}}{a \cdot 2} \]

    13. distribute-rgt-neg-in 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)}}}{a \cdot 2} \]

    14. metadata-eval 99.3%

        \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)}}}{a \cdot 2} \]

12. Simplified 99.3%

    \[\leadsto \frac{\color{blue}{\frac{0 \cdot \left(\left(-b\right) - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}}}{a \cdot 2} \]

13. Taylor expanded in b around 0 99.3%

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

14. Taylor expanded in a around 0 81.0%

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

    1. distribute-lft-out-- 81.0%

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

    2. associate-/l* 81.0%

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

16. Simplified 81.0%

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

Alternative 7: 64.5% 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(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 58.2%

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

   1. *-commutative 58.2%

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

3. Simplified 58.2%

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

5. Taylor expanded in b around inf 62.6%

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

   1. associate-*r/ 62.6%

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

   2. mul-1-neg 62.6%

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

7. Simplified 62.6%

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

8. Final simplification 62.6%

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

Alternative 8: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 58.2%

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

   1. *-commutative 58.2%

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

3. Simplified 58.2%

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

   1. neg-sub0 58.2%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. flip-- 58.0%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - b \cdot b}{0 + b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 58.0%

      \[\leadsto \frac{\frac{\color{blue}{0} - b \cdot b}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 58.0%

      \[\leadsto \frac{\frac{0 - \color{blue}{{b}^{2}}}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. add-sqr-sqrt 56.9%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. sqrt-prod 58.0%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   7. sqr-neg 58.0%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. sqrt-unprod 0.0%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   9. add-sqr-sqrt 1.6%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\left(-b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   10. sub-neg 1.6%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{0 - b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   11. neg-sub0 1.6%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{-b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   12. add-sqr-sqrt 0.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   13. sqrt-unprod 58.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   14. sqr-neg 58.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\sqrt{\color{blue}{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   15. sqrt-prod 56.9%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   16. add-sqr-sqrt 58.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 58.0%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{2}}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. neg-sub0 58.0%

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

8. Simplified 58.0%

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

9. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 3.2%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left(b + -1 \cdot b\right)}{a}} \]

    2. distribute-rgt1-in 3.2%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

    3. metadata-eval 3.2%

       \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 3.2%

       \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.2%

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

11. Simplified 3.2%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024186 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))