Cubic critical, narrow range

Percentage Accurate: 55.2% → 99.5%

Time: 19.1s

Alternatives: 10

Speedup: 23.2×

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(c \cdot 3\right)}\\ \frac{\frac{c \cdot a}{a}}{\left(-b\right) - \sqrt{\left(b + t\_0\right) \cdot \left(b - t\_0\right)}} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a (* c 3.0)))))
   (/ (/ (* c a) a) (- (- b) (sqrt (* (+ b t_0) (- b t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (c * 3.0)));
	return ((c * a) / a) / (-b - sqrt(((b + t_0) * (b - t_0))));
}

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
    t_0 = sqrt((a * (c * 3.0d0)))
    code = ((c * a) / a) / (-b - sqrt(((b + t_0) * (b - t_0))))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((a * (c * 3.0)));
	return ((c * a) / a) / (-b - Math.sqrt(((b + t_0) * (b - t_0))));
}

Python

def code(a, b, c):
	t_0 = math.sqrt((a * (c * 3.0)))
	return ((c * a) / a) / (-b - math.sqrt(((b + t_0) * (b - t_0))))

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * Float64(c * 3.0)))
	return Float64(Float64(Float64(c * a) / a) / Float64(Float64(-b) - sqrt(Float64(Float64(b + t_0) * Float64(b - t_0)))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = sqrt((a * (c * 3.0)));
	tmp = ((c * a) / a) / (-b - sqrt(((b + t_0) * (b - t_0))));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(c * a), $MachinePrecision] / a), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b + t$95$0), $MachinePrecision] * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 57.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 57.3%

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

   2. difference-of-squares 57.4%

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

   3. associate-*l* 57.4%

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

   4. associate-*l* 57.4%

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

4. Applied egg-rr 57.4%

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

   1. associate-*r* 57.4%

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

   2. *-commutative 57.4%

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

   3. associate-*l* 57.4%

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

   4. associate-*r* 57.4%

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

   5. *-commutative 57.4%

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

   6. associate-*l* 57.4%

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

6. Simplified 57.4%

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

   1. flip-+ 57.1%

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

   2. pow2 57.1%

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

   3. add-sqr-sqrt 58.5%

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

   4. associate-*r* 58.5%

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

   5. associate-*r* 58.5%

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

8. Applied egg-rr 58.5%

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

   1. unpow2 58.5%

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

   2. sqr-neg 58.5%

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

   3. unpow2 58.5%

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

   4. associate-*l* 58.5%

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

   5. *-commutative 58.5%

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

   6. associate-*l* 58.5%

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

   7. *-commutative 58.5%

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

10. Simplified 58.5%

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

11. Taylor expanded in b around 0 98.5%

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

    1. unpow2 98.5%

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

    2. rem-square-sqrt 99.1%

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

13. Simplified 99.1%

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

    1. expm1-log1p-u 85.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{a \cdot \left(c \cdot 3\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a}\right)\right)} \]

    2. expm1-udef 62.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{a \cdot \left(c \cdot 3\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a}\right)} - 1} \]

    3. associate-/l/ 62.4%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a \cdot \left(c \cdot 3\right)}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)}}\right)} - 1 \]

    4. *-commutative 62.4%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{a \cdot \left(c \cdot 3\right)}{\color{blue}{\left(a \cdot 3\right)} \cdot \left(\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)}\right)} - 1 \]

15. Applied egg-rr 62.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot \left(c \cdot 3\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)}\right)} - 1} \]
16. Step-by-step derivation

    1. expm1-def 85.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot \left(c \cdot 3\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)}\right)\right)} \]

    2. expm1-log1p 99.1%

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

    3. associate-/r* 99.2%

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

    4. associate-*r* 99.1%

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

    5. times-frac 99.5%

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

    6. *-commutative 99.5%

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

    7. metadata-eval 99.5%

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

17. Simplified 99.5%

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

18. Final simplification 99.5%

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

Alternative 2: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -4e-7)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -4e-7) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -4e-7)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -4e-7], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.9999999999999998e-7

   1. Initial program 74.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. +-commutative 74.0%

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

      2. sqr-neg 74.0%

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

      3. unsub-neg 74.0%

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

      4. div-sub 73.1%

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

      5. --rgt-identity 73.1%

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

      6. div-sub 74.0%

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

   3. Simplified 74.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}} \]
   4. Add Preprocessing

   if -3.9999999999999998e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.9%

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

   3. Taylor expanded in b around inf 84.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 84.6%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   5. Simplified 84.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) + \mathsf{fma}\left(c \cdot \left(-a\right), 3, 3 \cdot \left(c \cdot a\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot 3\right)}{\mathsf{fma}\left(b, -2, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(-a, c \cdot 3, 0\right)}{b}, 0\right)\right)}}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.8)
   (/
    (-
     (sqrt
      (+ (fma b b (* a (* c -3.0))) (fma (* c (- a)) 3.0 (* 3.0 (* c a)))))
     b)
    (* a 3.0))
   (/
    (/
     (* a (* c 3.0))
     (fma b -2.0 (fma -0.5 (/ (fma (- a) (* c 3.0) 0.0) b) 0.0)))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.8) {
		tmp = (sqrt((fma(b, b, (a * (c * -3.0))) + fma((c * -a), 3.0, (3.0 * (c * a))))) - b) / (a * 3.0);
	} else {
		tmp = ((a * (c * 3.0)) / fma(b, -2.0, fma(-0.5, (fma(-a, (c * 3.0), 0.0) / b), 0.0))) / (a * 3.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.8)
		tmp = Float64(Float64(sqrt(Float64(fma(b, b, Float64(a * Float64(c * -3.0))) + fma(Float64(c * Float64(-a)), 3.0, Float64(3.0 * Float64(c * a))))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(a * Float64(c * 3.0)) / fma(b, -2.0, fma(-0.5, Float64(fma(Float64(-a), Float64(c * 3.0), 0.0) / b), 0.0))) / Float64(a * 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.8], N[(N[(N[Sqrt[N[(N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * (-a)), $MachinePrecision] * 3.0 + N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision] / N[(b * -2.0 + N[(-0.5 * N[(N[((-a) * N[(c * 3.0), $MachinePrecision] + 0.0), $MachinePrecision] / b), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000004

   1. Initial program 84.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 84.7%

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

      2. associate-*l* 84.7%

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

      3. prod-diff 84.8%

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

      4. *-commutative 84.8%

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

      5. associate-*l* 84.7%

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

      6. fma-def 84.7%

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

      7. *-un-lft-identity 84.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(\color{blue}{b \cdot b} + \left(-\left(3 \cdot a\right) \cdot c\right)\right) + \mathsf{fma}\left(-a \cdot c, 3, \left(a \cdot c\right) \cdot 3\right)}}{3 \cdot a} \]

      8. fma-def 85.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)} + \mathsf{fma}\left(-a \cdot c, 3, \left(a \cdot c\right) \cdot 3\right)}}{3 \cdot a} \]

      9. distribute-lft-neg-in 85.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3 \cdot a\right) \cdot c}\right) + \mathsf{fma}\left(-a \cdot c, 3, \left(a \cdot c\right) \cdot 3\right)}}{3 \cdot a} \]

      10. *-commutative 85.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-\color{blue}{a \cdot 3}\right) \cdot c\right) + \mathsf{fma}\left(-a \cdot c, 3, \left(a \cdot c\right) \cdot 3\right)}}{3 \cdot a} \]

      11. distribute-rgt-neg-in 85.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot \left(-3\right)\right)} \cdot c\right) + \mathsf{fma}\left(-a \cdot c, 3, \left(a \cdot c\right) \cdot 3\right)}}{3 \cdot a} \]

      12. metadata-eval 85.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot \color{blue}{-3}\right) \cdot c\right) + \mathsf{fma}\left(-a \cdot c, 3, \left(a \cdot c\right) \cdot 3\right)}}{3 \cdot a} \]

      13. associate-*r* 85.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-3 \cdot c\right)}\right) + \mathsf{fma}\left(-a \cdot c, 3, \left(a \cdot c\right) \cdot 3\right)}}{3 \cdot a} \]

      14. *-commutative 85.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(c \cdot -3\right)}\right) + \mathsf{fma}\left(-a \cdot c, 3, \left(a \cdot c\right) \cdot 3\right)}}{3 \cdot a} \]

      15. *-commutative 85.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) + \mathsf{fma}\left(-a \cdot c, 3, \color{blue}{3 \cdot \left(a \cdot c\right)}\right)}}{3 \cdot a} \]

   4. Applied egg-rr 85.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) + \mathsf{fma}\left(-a \cdot c, 3, 3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]

   if 1.80000000000000004 < b

   1. Initial program 49.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 49.8%

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

      2. difference-of-squares 49.9%

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

      3. associate-*l* 49.9%

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

      4. associate-*l* 49.9%

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

   4. Applied egg-rr 49.9%

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

      1. associate-*r* 49.9%

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

      2. *-commutative 49.9%

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

      3. associate-*l* 49.9%

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

      4. associate-*r* 49.9%

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

      5. *-commutative 49.9%

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

      6. associate-*l* 49.9%

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

   6. Simplified 49.9%

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

      1. flip-+ 49.6%

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

      2. pow2 49.6%

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

      3. add-sqr-sqrt 51.1%

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

      4. associate-*r* 51.1%

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

      5. associate-*r* 51.1%

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

   8. Applied egg-rr 51.1%

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

      1. unpow2 51.1%

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

      2. sqr-neg 51.1%

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

      3. unpow2 51.1%

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

      4. associate-*l* 51.1%

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

      5. *-commutative 51.1%

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

      6. associate-*l* 51.1%

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

      7. *-commutative 51.1%

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

   10. Simplified 51.1%

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

   11. Taylor expanded in b around 0 98.6%

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

       1. unpow2 98.6%

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

       2. rem-square-sqrt 99.1%

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

   13. Simplified 99.1%

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

   14. Taylor expanded in b around inf 86.6%

       \[\leadsto \frac{\frac{a \cdot \left(c \cdot 3\right)}{\color{blue}{-2 \cdot b + \left(-0.5 \cdot \left(-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}\right) + -0.5 \cdot \frac{-1 \cdot \left(a \cdot \left(c \cdot {\left(\sqrt{3}\right)}^{2}\right)\right) - {\left(0.5 \cdot \left(-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}^{2}}{b}\right)}}}{3 \cdot a} \]

   16. Simplified 86.6%

       \[\leadsto \frac{\frac{a \cdot \left(c \cdot 3\right)}{\color{blue}{\mathsf{fma}\left(b, -2, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(-a, c \cdot 3, 0\right)}{b}, 0\right)\right)}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) + \mathsf{fma}\left(c \cdot \left(-a\right), 3, 3 \cdot \left(c \cdot a\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot 3\right)}{\mathsf{fma}\left(b, -2, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(-a, c \cdot 3, 0\right)}{b}, 0\right)\right)}}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot 3\right)}{\mathsf{fma}\left(b, -2, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(-a, c \cdot 3, 0\right)}{b}, 0\right)\right)}}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.1)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/
    (/
     (* a (* c 3.0))
     (fma b -2.0 (fma -0.5 (/ (fma (- a) (* c 3.0) 0.0) b) 0.0)))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = ((a * (c * 3.0)) / fma(b, -2.0, fma(-0.5, (fma(-a, (c * 3.0), 0.0) / b), 0.0))) / (a * 3.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.1)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(a * Float64(c * 3.0)) / fma(b, -2.0, fma(-0.5, Float64(fma(Float64(-a), Float64(c * 3.0), 0.0) / b), 0.0))) / Float64(a * 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.1], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision] / N[(b * -2.0 + N[(-0.5 * N[(N[((-a) * N[(c * 3.0), $MachinePrecision] + 0.0), $MachinePrecision] / b), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.10000000000000009

   1. Initial program 84.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. +-commutative 84.7%

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

      2. sqr-neg 84.7%

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

      3. unsub-neg 84.7%

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

      4. div-sub 83.7%

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

      5. --rgt-identity 83.7%

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

      6. div-sub 84.7%

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

   3. Simplified 84.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}} \]
   4. Add Preprocessing

   if 2.10000000000000009 < b

   1. Initial program 49.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 49.8%

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

      2. difference-of-squares 49.9%

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

      3. associate-*l* 49.9%

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

      4. associate-*l* 49.9%

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

   4. Applied egg-rr 49.9%

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

      1. associate-*r* 49.9%

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

      2. *-commutative 49.9%

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

      3. associate-*l* 49.9%

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

      4. associate-*r* 49.9%

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

      5. *-commutative 49.9%

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

      6. associate-*l* 49.9%

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

   6. Simplified 49.9%

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

      1. flip-+ 49.6%

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

      2. pow2 49.6%

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

      3. add-sqr-sqrt 51.1%

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

      4. associate-*r* 51.1%

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

      5. associate-*r* 51.1%

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

   8. Applied egg-rr 51.1%

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

      1. unpow2 51.1%

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

      2. sqr-neg 51.1%

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

      3. unpow2 51.1%

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

      4. associate-*l* 51.1%

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

      5. *-commutative 51.1%

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

      6. associate-*l* 51.1%

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

      7. *-commutative 51.1%

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

   10. Simplified 51.1%

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

   11. Taylor expanded in b around 0 98.6%

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

       1. unpow2 98.6%

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

       2. rem-square-sqrt 99.1%

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

   13. Simplified 99.1%

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

   14. Taylor expanded in b around inf 86.6%

       \[\leadsto \frac{\frac{a \cdot \left(c \cdot 3\right)}{\color{blue}{-2 \cdot b + \left(-0.5 \cdot \left(-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}\right) + -0.5 \cdot \frac{-1 \cdot \left(a \cdot \left(c \cdot {\left(\sqrt{3}\right)}^{2}\right)\right) - {\left(0.5 \cdot \left(-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}^{2}}{b}\right)}}}{3 \cdot a} \]

   16. Simplified 86.6%

       \[\leadsto \frac{\frac{a \cdot \left(c \cdot 3\right)}{\color{blue}{\mathsf{fma}\left(b, -2, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(-a, c \cdot 3, 0\right)}{b}, 0\right)\right)}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot 3\right)}{\mathsf{fma}\left(b, -2, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(-a, c \cdot 3, 0\right)}{b}, 0\right)\right)}}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -4e-7) t_0 (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -4e-7) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	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 * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-4d-7)) then
        tmp = t_0
    else
        tmp = (c * (-0.5d0)) / b
    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 * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -4e-7) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -4e-7:
		tmp = t_0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -4e-7)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -4e-7)
		tmp = t_0;
	else
		tmp = (c * -0.5) / b;
	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[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-7], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.9999999999999998e-7

   1. Initial program 74.0%

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

   if -3.9999999999999998e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.9%

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

   3. Taylor expanded in b around inf 84.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 84.6%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   5. Simplified 84.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.8)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.8) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.8], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000004

   1. Initial program 84.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. +-commutative 84.7%

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

      2. sqr-neg 84.7%

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

      3. unsub-neg 84.7%

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

      4. div-sub 83.7%

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

      5. --rgt-identity 83.7%

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

      6. div-sub 84.7%

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

   3. Simplified 84.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}} \]
   4. Add Preprocessing

   if 1.80000000000000004 < b

   1. Initial program 49.8%

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

   3. Taylor expanded in b around inf 86.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 820:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 820.0)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 820.0) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	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) :: tmp
    if (b <= 820.0d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 820.0) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 820.0:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 820.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 820.0)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 820.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 820

   1. Initial program 77.3%

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

   3. Taylor expanded in a around 0 77.3%

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

   if 820 < b

   1. Initial program 41.5%

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

   3. Taylor expanded in b around inf 75.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 75.9%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   5. Simplified 75.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 820:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.5% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.5}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* c (/ -0.5 b)))

C

double code(double a, double b, double c) {
	return c * (-0.5 / 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 * ((-0.5d0) / b)
end function

Java

public static double code(double a, double b, double c) {
	return c * (-0.5 / b);
}

Python

def code(a, b, c):
	return c * (-0.5 / b)

Julia

function code(a, b, c)
	return Float64(c * Float64(-0.5 / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * (-0.5 / b);
end

Wolfram

code[a_, b_, c_] := N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in b around inf 62.4%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ 62.4%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   2. associate-/l* 62.3%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{b}{c}}} \]

5. Simplified 62.3%

   \[\leadsto \color{blue}{\frac{-0.5}{\frac{b}{c}}} \]
6. Step-by-step derivation

   1. associate-/r/ 62.3%

      \[\leadsto \color{blue}{\frac{-0.5}{b} \cdot c} \]

7. Applied egg-rr 62.3%

   \[\leadsto \color{blue}{\frac{-0.5}{b} \cdot c} \]

8. Final simplification 62.3%

   \[\leadsto c \cdot \frac{-0.5}{b} \]
9. Add Preprocessing

Alternative 9: 64.6% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (* c -0.5) b))

C

double code(double a, double b, double c) {
	return (c * -0.5) / 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 * (-0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * -0.5) / b;
}

Python

def code(a, b, c):
	return (c * -0.5) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * -0.5) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * -0.5) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in b around inf 62.4%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ 62.4%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

5. Simplified 62.4%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

6. Final simplification 62.4%

   \[\leadsto \frac{c \cdot -0.5}{b} \]
7. Add Preprocessing

Alternative 10: 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 57.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 57.3%

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

   2. difference-of-squares 57.4%

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

   3. associate-*l* 57.4%

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

   4. associate-*l* 57.4%

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

4. Applied egg-rr 57.4%

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

   1. associate-*r* 57.4%

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

   2. *-commutative 57.4%

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

   3. associate-*l* 57.4%

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

   4. associate-*r* 57.4%

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

   5. *-commutative 57.4%

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

   6. associate-*l* 57.4%

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

6. Simplified 57.4%

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

7. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}\right)}{a}} \]

   2. distribute-lft1-in 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \left(\color{blue}{0} \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

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

9. Simplified 3.2%

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

10. Final simplification 3.2%

    \[\leadsto \frac{0}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))