Quadratic roots, narrow range

Percentage Accurate: 55.4% → 99.3%

Time: 13.3s

Alternatives: 7

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: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

   1. add-sqr-sqrt 55.1%

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

   2. distribute-rgt-neg-in 55.1%

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

6. Applied egg-rr 55.1%

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

   1. flip-+ 55.0%

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

   2. distribute-rgt-neg-out 55.0%

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

   3. add-sqr-sqrt 55.8%

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

   4. distribute-rgt-neg-out 55.8%

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

   5. add-sqr-sqrt 56.0%

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

   6. pow2 56.0%

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

   7. add-sqr-sqrt 57.5%

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

   8. pow2 57.5%

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

   9. associate-*l* 57.5%

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

8. Applied egg-rr 57.5%

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

   1. associate--r- 99.2%

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

   2. unpow2 99.2%

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

   3. unpow2 99.2%

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

   4. difference-of-squares 99.2%

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

   5. neg-mul-1 99.2%

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

   6. distribute-lft1-in 99.2%

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

   7. metadata-eval 99.2%

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

   8. mul0-lft 99.2%

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

   9. unpow2 99.2%

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

   10. fmm-def 99.3%

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

   11. *-commutative 99.3%

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

   12. distribute-rgt-neg-in 99.3%

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

   13. metadata-eval 99.3%

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

10. Simplified 99.3%

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

11. Taylor expanded in b around 0 99.3%

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

    1. *-commutative 99.3%

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

13. Simplified 99.3%

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

    1. neg-sub0 99.3%

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

    2. flip-- 99.3%

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

    3. metadata-eval 99.3%

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

    4. pow2 99.3%

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

    5. add-sqr-sqrt 99.2%

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

    6. sqrt-prod 99.3%

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

    7. sqr-neg 99.3%

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

    8. sqrt-unprod 0.0%

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

    9. add-sqr-sqrt 1.4%

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

    10. sub-neg 1.4%

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

    11. neg-sub0 1.4%

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

    12. add-sqr-sqrt 0.0%

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

    13. sqrt-unprod 99.3%

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

    14. sqr-neg 99.3%

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

    15. sqrt-prod 99.2%

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

    16. add-sqr-sqrt 99.3%

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

15. Applied egg-rr 99.3%

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

    1. neg-sub0 99.3%

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

17. Simplified 99.3%

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

18. Final simplification 99.3%

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

Alternative 2: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(c \cdot a\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -0.2)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ (/ (* 4.0 (* c a)) (* 2.0 (- (* a (/ c 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)) <= -0.2) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((4.0 * (c * a)) / (2.0 * ((a * (c / 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)) <= -0.2)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(c * a)) / Float64(2.0 * Float64(Float64(a * Float64(c / 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], -0.2], 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[(4.0 * N[(c * a), $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. 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)) < -0.20000000000000001

   1. Initial program 84.4%

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

      1. *-commutative 84.4%

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

   3. Simplified 84.5%

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

   if -0.20000000000000001 < (/.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 47.5%

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

      1. *-commutative 47.5%

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

   3. Simplified 47.5%

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

      1. add-sqr-sqrt 46.6%

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

      2. distribute-rgt-neg-in 46.6%

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

   6. Applied egg-rr 46.6%

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

      1. flip-+ 46.5%

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

      2. distribute-rgt-neg-out 46.5%

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

      3. add-sqr-sqrt 47.2%

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

      4. distribute-rgt-neg-out 47.2%

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

      5. add-sqr-sqrt 47.3%

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

      6. pow2 47.3%

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

      7. add-sqr-sqrt 48.8%

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

      8. pow2 48.8%

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

      9. associate-*l* 48.8%

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

   8. Applied egg-rr 48.8%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]
   9. 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. neg-mul-1 99.3%

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

      6. distribute-lft1-in 99.3%

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

      7. metadata-eval 99.3%

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

      8. mul0-lft 99.3%

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

      9. unpow2 99.3%

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

      10. fmm-def 99.3%

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

      11. *-commutative 99.3%

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

      12. distribute-rgt-neg-in 99.3%

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

      13. metadata-eval 99.3%

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

   10. Simplified 99.3%

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

   11. Taylor expanded in b around 0 99.3%

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

       1. *-commutative 99.3%

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

   13. Simplified 99.3%

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

   14. Taylor expanded in a around 0 88.4%

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

       1. distribute-lft-out-- 88.4%

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

       2. associate-*r/ 88.4%

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

   16. Simplified 88.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(c \cdot a\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(c \cdot a\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))))
   (if (<= t_0 -0.2)
     t_0
     (/ (/ (* 4.0 (* c a)) (* 2.0 (- (* a (/ c 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 <= -0.2) {
		tmp = t_0;
	} else {
		tmp = ((4.0 * (c * a)) / (2.0 * ((a * (c / 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 <= (-0.2d0)) then
        tmp = t_0
    else
        tmp = ((4.0d0 * (c * a)) / (2.0d0 * ((a * (c / 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 <= -0.2) {
		tmp = t_0;
	} else {
		tmp = ((4.0 * (c * a)) / (2.0 * ((a * (c / 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 <= -0.2:
		tmp = t_0
	else:
		tmp = ((4.0 * (c * a)) / (2.0 * ((a * (c / 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 <= -0.2)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(c * a)) / Float64(2.0 * Float64(Float64(a * Float64(c / 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 <= -0.2)
		tmp = t_0;
	else
		tmp = ((4.0 * (c * a)) / (2.0 * ((a * (c / 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, -0.2], t$95$0, N[(N[(N[(4.0 * N[(c * a), $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. 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)) < -0.20000000000000001

   1. Initial program 84.4%

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

   if -0.20000000000000001 < (/.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 47.5%

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

      1. *-commutative 47.5%

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

   3. Simplified 47.5%

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

      1. add-sqr-sqrt 46.6%

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

      2. distribute-rgt-neg-in 46.6%

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

   6. Applied egg-rr 46.6%

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

      1. flip-+ 46.5%

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

      2. distribute-rgt-neg-out 46.5%

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

      3. add-sqr-sqrt 47.2%

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

      4. distribute-rgt-neg-out 47.2%

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

      5. add-sqr-sqrt 47.3%

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

      6. pow2 47.3%

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

      7. add-sqr-sqrt 48.8%

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

      8. pow2 48.8%

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

      9. associate-*l* 48.8%

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

   8. Applied egg-rr 48.8%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]
   9. 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. neg-mul-1 99.3%

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

      6. distribute-lft1-in 99.3%

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

      7. metadata-eval 99.3%

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

      8. mul0-lft 99.3%

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

      9. unpow2 99.3%

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

      10. fmm-def 99.3%

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

      11. *-commutative 99.3%

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

      12. distribute-rgt-neg-in 99.3%

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

      13. metadata-eval 99.3%

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

   10. Simplified 99.3%

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

   11. Taylor expanded in b around 0 99.3%

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

       1. *-commutative 99.3%

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

   13. Simplified 99.3%

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

   14. Taylor expanded in a around 0 88.4%

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

       1. distribute-lft-out-- 88.4%

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

       2. associate-*r/ 88.4%

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

   16. Simplified 88.4%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(c \cdot a\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

   1. add-sqr-sqrt 55.1%

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

   2. distribute-rgt-neg-in 55.1%

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

6. Applied egg-rr 55.1%

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

   1. flip-+ 55.0%

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

   2. distribute-rgt-neg-out 55.0%

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

   3. add-sqr-sqrt 55.8%

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

   4. distribute-rgt-neg-out 55.8%

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

   5. add-sqr-sqrt 56.0%

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

   6. pow2 56.0%

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

   7. add-sqr-sqrt 57.5%

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

   8. pow2 57.5%

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

   9. associate-*l* 57.5%

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

8. Applied egg-rr 57.5%

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

   1. associate--r- 99.2%

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

   2. unpow2 99.2%

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

   3. unpow2 99.2%

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

   4. difference-of-squares 99.2%

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

   5. neg-mul-1 99.2%

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

   6. distribute-lft1-in 99.2%

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

   7. metadata-eval 99.2%

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

   8. mul0-lft 99.2%

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

   9. unpow2 99.2%

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

   10. fmm-def 99.3%

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

   11. *-commutative 99.3%

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

   12. distribute-rgt-neg-in 99.3%

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

   13. metadata-eval 99.3%

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

10. Simplified 99.3%

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

11. Taylor expanded in b around 0 99.3%

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

    1. *-commutative 99.3%

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

13. Simplified 99.3%

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

14. Final simplification 99.3%

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

Alternative 5: 82.0% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(N[(N[(4.0 * N[(c * a), $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 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

   1. add-sqr-sqrt 55.1%

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

   2. distribute-rgt-neg-in 55.1%

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

6. Applied egg-rr 55.1%

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

   1. flip-+ 55.0%

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

   2. distribute-rgt-neg-out 55.0%

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

   3. add-sqr-sqrt 55.8%

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

   4. distribute-rgt-neg-out 55.8%

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

   5. add-sqr-sqrt 56.0%

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

   6. pow2 56.0%

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

   7. add-sqr-sqrt 57.5%

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

   8. pow2 57.5%

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

   9. associate-*l* 57.5%

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

8. Applied egg-rr 57.5%

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

   1. associate--r- 99.2%

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

   2. unpow2 99.2%

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

   3. unpow2 99.2%

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

   4. difference-of-squares 99.2%

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

   5. neg-mul-1 99.2%

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

   6. distribute-lft1-in 99.2%

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

   7. metadata-eval 99.2%

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

   8. mul0-lft 99.2%

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

   9. unpow2 99.2%

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

   10. fmm-def 99.3%

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

   11. *-commutative 99.3%

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

   12. distribute-rgt-neg-in 99.3%

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

   13. metadata-eval 99.3%

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

10. Simplified 99.3%

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

11. Taylor expanded in b around 0 99.3%

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

    1. *-commutative 99.3%

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

13. Simplified 99.3%

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

14. Taylor expanded in a around 0 80.2%

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

    1. distribute-lft-out-- 80.2%

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

    2. associate-*r/ 80.2%

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

16. Simplified 80.2%

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

Alternative 6: 64.4% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-b} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / -b
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(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 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.2%

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

5. Taylor expanded in b around inf 63.7%

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

   1. associate-*r/ 63.7%

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

   2. mul-1-neg 63.7%

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

7. Simplified 63.7%

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

8. Final simplification 63.7%

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

Alternative 7: 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 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

   1. add-sqr-sqrt 55.1%

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

   2. distribute-rgt-neg-in 55.1%

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

6. Applied egg-rr 55.1%

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

7. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
8. 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%

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

9. Simplified 3.2%

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

Reproduce

herbie shell --seed 2024188 
(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)))