Cubic critical, medium range

Percentage Accurate: 30.7% → 99.7%

Time: 14.9s

Alternatives: 6

Speedup: 23.2×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 30.7% 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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.5%

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

   1. neg-sub0 28.5%

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

   2. flip-- 28.7%

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

   3. metadata-eval 28.7%

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

   4. pow2 28.7%

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

   5. add-sqr-sqrt 29.0%

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

   6. sqrt-prod 28.7%

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

   7. sqr-neg 28.7%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. neg-sub0 1.6%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 28.7%

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

   14. sqr-neg 28.7%

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

   15. sqrt-prod 29.0%

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

   16. add-sqr-sqrt 28.7%

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

4. Applied egg-rr 28.7%

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

   1. neg-sub0 28.7%

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

6. Simplified 28.7%

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

   1. flip-+ 28.8%

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

8. Applied egg-rr 29.6%

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

   1. associate--r- 99.3%

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

   2. associate-*r* 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*r* 99.2%

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

   5. associate-*r* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*r* 99.2%

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

10. Simplified 99.2%

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

11. Taylor expanded in b around 0 99.1%

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

    1. associate-*r* 99.3%

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

13. Simplified 99.3%

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

    1. *-un-lft-identity 99.3%

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

    2. associate-/l/ 99.3%

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

    3. *-commutative 99.3%

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

    4. *-commutative 99.3%

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

    5. *-commutative 99.3%

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

15. Applied egg-rr 99.3%

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

    1. *-lft-identity 99.3%

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

    2. *-commutative 99.3%

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

    3. times-frac 99.7%

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

    4. *-inverses 99.7%

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

    5. sub-neg 99.7%

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

    6. unpow2 99.7%

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

    7. fma-define 99.8%

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

    8. associate-*r* 99.8%

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

    9. *-commutative 99.8%

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

    10. distribute-rgt-neg-in 99.8%

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

    11. metadata-eval 99.8%

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

17. Simplified 99.8%

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

18. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((c * (a * 3.0d0)) / (-b - sqrt(((b * b) - (a * (c * 3.0d0)))))) / (a * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.5%

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

   1. neg-sub0 28.5%

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

   2. flip-- 28.7%

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

   3. metadata-eval 28.7%

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

   4. pow2 28.7%

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

   5. add-sqr-sqrt 29.0%

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

   6. sqrt-prod 28.7%

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

   7. sqr-neg 28.7%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. neg-sub0 1.6%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 28.7%

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

   14. sqr-neg 28.7%

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

   15. sqrt-prod 29.0%

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

   16. add-sqr-sqrt 28.7%

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

4. Applied egg-rr 28.7%

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

   1. neg-sub0 28.7%

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

6. Simplified 28.7%

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

   1. flip-+ 28.8%

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

8. Applied egg-rr 29.6%

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

   1. associate--r- 99.3%

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

   2. associate-*r* 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*r* 99.2%

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

   5. associate-*r* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*r* 99.2%

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

10. Simplified 99.2%

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

11. Taylor expanded in b around 0 99.1%

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

    1. associate-*r* 99.3%

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

13. Simplified 99.3%

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

    1. pow2 99.3%

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

15. Applied egg-rr 99.3%

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

16. Final simplification 99.3%

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

Alternative 3: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - \frac{2}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (* b (- (* 1.5 (/ a (pow b 2.0))) (/ 2.0 c)))))

C

double code(double a, double b, double c) {
	return 1.0 / (b * ((1.5 * (a / pow(b, 2.0))) - (2.0 / c)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (b * ((1.5d0 * (a / (b ** 2.0d0))) - (2.0d0 / c)))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / (b * ((1.5 * (a / Math.pow(b, 2.0))) - (2.0 / c)));
}

Python

def code(a, b, c):
	return 1.0 / (b * ((1.5 * (a / math.pow(b, 2.0))) - (2.0 / c)))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(b * Float64(Float64(1.5 * Float64(a / (b ^ 2.0))) - Float64(2.0 / c))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / (b * ((1.5 * (a / (b ^ 2.0))) - (2.0 / c)));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(b * N[(N[(1.5 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.5%

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

   1. add-cube-cbrt 28.5%

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

   2. pow3 28.5%

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

4. Applied egg-rr 28.5%

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

   1. pow1 28.5%

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

   2. metadata-eval 28.5%

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

   3. pow-div 28.7%

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

   4. pow2 28.7%

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

   5. pow1 28.7%

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

   6. distribute-frac-neg 28.7%

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

   7. pow2 28.7%

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

   8. rem-cube-cbrt 28.7%

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

   9. clear-num 28.7%

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

   10. inv-pow 28.7%

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

6. Applied egg-rr 28.5%

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

   1. unpow-1 28.5%

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

   2. associate-/l* 28.5%

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

   3. associate-*r* 28.5%

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

   4. *-commutative 28.5%

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

   5. associate-*r* 28.5%

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

8. Simplified 28.5%

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

9. Taylor expanded in b around inf 91.8%

   \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)}} \]
10. Step-by-step derivation

    1. associate-*r/ 91.8%

       \[\leadsto \frac{1}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - \color{blue}{\frac{2 \cdot 1}{c}}\right)} \]

    2. metadata-eval 91.8%

       \[\leadsto \frac{1}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - \frac{\color{blue}{2}}{c}\right)} \]

11. Simplified 91.8%

    \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - \frac{2}{c}\right)}} \]
12. Add Preprocessing

Alternative 4: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{3}} - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (* -0.375 (/ (* c a) (pow b 3.0))) (/ 0.5 b))))

C

double code(double a, double b, double c) {
	return c * ((-0.375 * ((c * a) / pow(b, 3.0))) - (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 * (((-0.375d0) * ((c * a) / (b ** 3.0d0))) - (0.5d0 / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-0.375 * ((c * a) / Math.pow(b, 3.0))) - (0.5 / b));
}

Python

def code(a, b, c):
	return c * ((-0.375 * ((c * a) / math.pow(b, 3.0))) - (0.5 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-0.375 * Float64(Float64(c * a) / (b ^ 3.0))) - Float64(0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-0.375 * ((c * a) / (b ^ 3.0))) - (0.5 / b));
end

Wolfram

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

Derivation

1. Initial program 28.5%

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

   1. /-rgt-identity 28.5%

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

   2. metadata-eval 28.5%

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

3. Simplified 28.6%

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

5. Taylor expanded in c around 0 91.7%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 91.7%

      \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \]

   2. metadata-eval 91.7%

      \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{\color{blue}{0.5}}{b}\right) \]

7. Simplified 91.7%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{0.5}{b}\right)} \]

8. Final simplification 91.7%

   \[\leadsto c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{3}} - \frac{0.5}{b}\right) \]
9. Add Preprocessing

Alternative 5: 81.9% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.5%

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

   1. /-rgt-identity 28.5%

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

   2. metadata-eval 28.5%

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

3. Simplified 28.6%

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

5. Taylor expanded in b around inf 83.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Alternative 6: 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 28.5%

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

   1. neg-sub0 28.5%

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

   2. flip-- 28.7%

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

   3. metadata-eval 28.7%

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

   4. pow2 28.7%

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

   5. add-sqr-sqrt 29.0%

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

   6. sqrt-prod 28.7%

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

   7. sqr-neg 28.7%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. neg-sub0 1.6%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 28.7%

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

   14. sqr-neg 28.7%

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

   15. sqrt-prod 29.0%

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

   16. add-sqr-sqrt 28.7%

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

4. Applied egg-rr 28.7%

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

   1. neg-sub0 28.7%

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

6. Simplified 28.7%

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

7. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(b + -1 \cdot b\right)}{a}} \]

   2. distribute-rgt1-in 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

   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 2024139 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))