Cubic critical

Percentage Accurate: 51.8% → 85.5%

Time: 14.6s

Alternatives: 15

Speedup: 16.4×

Specification

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: 51.8% 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: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+153)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 2e-54)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+153) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2e-54) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} 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 <= (-1d+153)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 2d-54) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    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 <= -1e+153) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2e-54) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+153:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 2e-54:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+153)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 2e-54)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	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 <= -1e+153)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 2e-54)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+153], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-54], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e153

   1. Initial program 38.9%

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

      1. neg-sub0 38.9%

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

      2. associate-+l- 38.9%

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

      3. sub0-neg 38.9%

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

      4. neg-mul-1 38.9%

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

      5. associate-*r/ 38.9%

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

      6. metadata-eval 38.9%

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

      7. metadata-eval 38.9%

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

      8. times-frac 38.9%

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

      9. *-commutative 38.9%

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

      10. times-frac 38.9%

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

      11. associate-*l/ 38.9%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in b around -inf 99.8%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   if -1e153 < b < 2.0000000000000001e-54

   1. Initial program 84.4%

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

   if 2.0000000000000001e-54 < b

   1. Initial program 16.0%

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

      1. /-rgt-identity 16.0%

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

      2. metadata-eval 16.0%

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

      3. associate-/l* 16.0%

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

      4. associate-*r/ 16.0%

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

      5. *-commutative 16.0%

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

      6. associate-*l/ 16.0%

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

      7. associate-*r/ 16.0%

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

      8. metadata-eval 16.0%

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

      9. metadata-eval 16.0%

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

      10. times-frac 16.0%

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

      11. neg-mul-1 16.0%

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

      12. distribute-rgt-neg-in 16.0%

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

      13. times-frac 16.0%

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

      14. metadata-eval 16.0%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 16.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 15.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 16.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around inf 93.2%

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

      1. associate-*r/ 93.2%

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

      2. *-commutative 93.2%

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

   10. Simplified 93.2%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.3333333333333333 \cdot \frac{b - \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-51}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -0.3333333333333333 (/ (- b (sqrt (* (* a c) -3.0))) a))))
   (if (<= b -3.2e-45)
     (/ (* b -2.0) (* 3.0 a))
     (if (<= b -9.5e-106)
       t_0
       (if (<= b -1.32e-129)
         (/ b (/ a -0.6666666666666666))
         (if (<= b 5.2e-51) t_0 (/ (* c -0.5) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = -0.3333333333333333 * ((b - sqrt(((a * c) * -3.0))) / a);
	double tmp;
	if (b <= -3.2e-45) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -9.5e-106) {
		tmp = t_0;
	} else if (b <= -1.32e-129) {
		tmp = b / (a / -0.6666666666666666);
	} else if (b <= 5.2e-51) {
		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 = (-0.3333333333333333d0) * ((b - sqrt(((a * c) * (-3.0d0)))) / a)
    if (b <= (-3.2d-45)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= (-9.5d-106)) then
        tmp = t_0
    else if (b <= (-1.32d-129)) then
        tmp = b / (a / (-0.6666666666666666d0))
    else if (b <= 5.2d-51) 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 = -0.3333333333333333 * ((b - Math.sqrt(((a * c) * -3.0))) / a);
	double tmp;
	if (b <= -3.2e-45) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -9.5e-106) {
		tmp = t_0;
	} else if (b <= -1.32e-129) {
		tmp = b / (a / -0.6666666666666666);
	} else if (b <= 5.2e-51) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -0.3333333333333333 * ((b - math.sqrt(((a * c) * -3.0))) / a)
	tmp = 0
	if b <= -3.2e-45:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= -9.5e-106:
		tmp = t_0
	elif b <= -1.32e-129:
		tmp = b / (a / -0.6666666666666666)
	elif b <= 5.2e-51:
		tmp = t_0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(Float64(a * c) * -3.0))) / a))
	tmp = 0.0
	if (b <= -3.2e-45)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= -9.5e-106)
		tmp = t_0;
	elseif (b <= -1.32e-129)
		tmp = Float64(b / Float64(a / -0.6666666666666666));
	elseif (b <= 5.2e-51)
		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 = -0.3333333333333333 * ((b - sqrt(((a * c) * -3.0))) / a);
	tmp = 0.0;
	if (b <= -3.2e-45)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= -9.5e-106)
		tmp = t_0;
	elseif (b <= -1.32e-129)
		tmp = b / (a / -0.6666666666666666);
	elseif (b <= 5.2e-51)
		tmp = t_0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.2e-45], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -9.5e-106], t$95$0, If[LessEqual[b, -1.32e-129], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-51], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.20000000000000007e-45

   1. Initial program 64.3%

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

      1. neg-sub0 64.3%

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

      2. associate-+l- 64.3%

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

      3. sub0-neg 64.3%

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

      4. neg-mul-1 64.3%

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

      5. associate-*r/ 64.3%

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

      6. metadata-eval 64.3%

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

      7. metadata-eval 64.3%

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

      8. times-frac 64.3%

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

      9. *-commutative 64.3%

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

      10. times-frac 64.2%

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

      11. associate-*l/ 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in b around -inf 85.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 85.3%

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

   6. Simplified 85.3%

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

   if -3.20000000000000007e-45 < b < -9.4999999999999994e-106 or -1.31999999999999992e-129 < b < 5.2e-51

   1. Initial program 80.8%

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

      1. /-rgt-identity 80.8%

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

      2. metadata-eval 80.8%

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

      3. associate-/l* 80.8%

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

      4. associate-*r/ 80.6%

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

      5. *-commutative 80.6%

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

      6. associate-*l/ 80.8%

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

      7. associate-*r/ 80.8%

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

      8. metadata-eval 80.8%

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

      9. metadata-eval 80.8%

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

      10. times-frac 80.8%

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

      11. neg-mul-1 80.8%

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

      12. distribute-rgt-neg-in 80.8%

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

      13. times-frac 80.4%

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

      14. metadata-eval 80.4%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 80.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 80.3%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around 0 76.4%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{a} \]

   if -9.4999999999999994e-106 < b < -1.31999999999999992e-129

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-*r/ 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*l/ 99.7%

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

      7. associate-*r/ 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. times-frac 99.7%

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

      11. neg-mul-1 99.7%

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

      12. distribute-rgt-neg-in 99.7%

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

      13. times-frac 99.1%

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

      14. metadata-eval 99.1%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 99.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 99.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around -inf 99.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. associate-/l* 99.7%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

      3. associate-/r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   11. Taylor expanded in a around 0 99.1%

       \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 99.4%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

       2. *-commutative 99.4%

          \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

       3. associate-/l* 99.7%

          \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   13. Simplified 99.7%

       \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   if 5.2e-51 < b

   1. Initial program 16.0%

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

      1. /-rgt-identity 16.0%

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

      2. metadata-eval 16.0%

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

      3. associate-/l* 16.0%

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

      4. associate-*r/ 16.0%

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

      5. *-commutative 16.0%

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

      6. associate-*l/ 16.0%

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

      7. associate-*r/ 16.0%

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

      8. metadata-eval 16.0%

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

      9. metadata-eval 16.0%

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

      10. times-frac 16.0%

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

      11. neg-mul-1 16.0%

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

      12. distribute-rgt-neg-in 16.0%

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

      13. times-frac 16.0%

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

      14. metadata-eval 16.0%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 16.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 15.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 16.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around inf 93.2%

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

      1. associate-*r/ 93.2%

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

      2. *-commutative 93.2%

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

   10. Simplified 93.2%

       \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{-106}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-51}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-51}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e-45)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b -6.2e-106)
     (* -0.3333333333333333 (/ (- b (sqrt (* a (* c -3.0)))) a))
     (if (<= b -1.32e-129)
       (/ b (/ a -0.6666666666666666))
       (if (<= b 8.4e-51)
         (* -0.3333333333333333 (/ (- b (sqrt (* (* a c) -3.0))) a))
         (/ (* c -0.5) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-45) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -6.2e-106) {
		tmp = -0.3333333333333333 * ((b - sqrt((a * (c * -3.0)))) / a);
	} else if (b <= -1.32e-129) {
		tmp = b / (a / -0.6666666666666666);
	} else if (b <= 8.4e-51) {
		tmp = -0.3333333333333333 * ((b - sqrt(((a * c) * -3.0))) / a);
	} 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 <= (-3.6d-45)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= (-6.2d-106)) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt((a * (c * (-3.0d0))))) / a)
    else if (b <= (-1.32d-129)) then
        tmp = b / (a / (-0.6666666666666666d0))
    else if (b <= 8.4d-51) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt(((a * c) * (-3.0d0)))) / a)
    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 <= -3.6e-45) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -6.2e-106) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt((a * (c * -3.0)))) / a);
	} else if (b <= -1.32e-129) {
		tmp = b / (a / -0.6666666666666666);
	} else if (b <= 8.4e-51) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt(((a * c) * -3.0))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.6e-45:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= -6.2e-106:
		tmp = -0.3333333333333333 * ((b - math.sqrt((a * (c * -3.0)))) / a)
	elif b <= -1.32e-129:
		tmp = b / (a / -0.6666666666666666)
	elif b <= 8.4e-51:
		tmp = -0.3333333333333333 * ((b - math.sqrt(((a * c) * -3.0))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e-45)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= -6.2e-106)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(a * Float64(c * -3.0)))) / a));
	elseif (b <= -1.32e-129)
		tmp = Float64(b / Float64(a / -0.6666666666666666));
	elseif (b <= 8.4e-51)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(Float64(a * c) * -3.0))) / a));
	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 <= -3.6e-45)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= -6.2e-106)
		tmp = -0.3333333333333333 * ((b - sqrt((a * (c * -3.0)))) / a);
	elseif (b <= -1.32e-129)
		tmp = b / (a / -0.6666666666666666);
	elseif (b <= 8.4e-51)
		tmp = -0.3333333333333333 * ((b - sqrt(((a * c) * -3.0))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e-45], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -6.2e-106], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.32e-129], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.4e-51], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -3.60000000000000001e-45

   1. Initial program 64.3%

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

      1. neg-sub0 64.3%

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

      2. associate-+l- 64.3%

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

      3. sub0-neg 64.3%

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

      4. neg-mul-1 64.3%

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

      5. associate-*r/ 64.3%

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

      6. metadata-eval 64.3%

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

      7. metadata-eval 64.3%

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

      8. times-frac 64.3%

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

      9. *-commutative 64.3%

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

      10. times-frac 64.2%

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

      11. associate-*l/ 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in b around -inf 85.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 85.3%

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

   6. Simplified 85.3%

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

   if -3.60000000000000001e-45 < b < -6.19999999999999971e-106

   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. /-rgt-identity 84.7%

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

      2. metadata-eval 84.7%

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

      3. associate-/l* 84.7%

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

      4. associate-*r/ 84.4%

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

      5. *-commutative 84.4%

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

      6. associate-*l/ 84.7%

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

      7. associate-*r/ 84.7%

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

      8. metadata-eval 84.7%

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

      9. metadata-eval 84.7%

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

      10. times-frac 84.7%

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

      11. neg-mul-1 84.7%

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

      12. distribute-rgt-neg-in 84.7%

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

      13. times-frac 84.4%

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

      14. metadata-eval 84.4%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 84.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 84.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around 0 63.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}}{a} \]

      2. *-commutative 63.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -3}}{a} \]

      3. associate-*r* 63.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{a} \]

   6. Simplified 63.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{a} \]

   if -6.19999999999999971e-106 < b < -1.31999999999999992e-129

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-*r/ 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*l/ 99.7%

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

      7. associate-*r/ 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. times-frac 99.7%

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

      11. neg-mul-1 99.7%

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

      12. distribute-rgt-neg-in 99.7%

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

      13. times-frac 99.1%

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

      14. metadata-eval 99.1%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 99.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 99.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around -inf 99.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. associate-/l* 99.7%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

      3. associate-/r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   11. Taylor expanded in a around 0 99.1%

       \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 99.4%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

       2. *-commutative 99.4%

          \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

       3. associate-/l* 99.7%

          \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   13. Simplified 99.7%

       \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   if -1.31999999999999992e-129 < b < 8.40000000000000006e-51

   1. Initial program 80.0%

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

      1. /-rgt-identity 80.0%

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

      2. metadata-eval 80.0%

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

      3. associate-/l* 80.0%

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

      4. associate-*r/ 79.8%

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

      5. *-commutative 79.8%

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

      6. associate-*l/ 80.0%

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

      7. associate-*r/ 80.0%

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

      8. metadata-eval 80.0%

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

      9. metadata-eval 80.0%

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

      10. times-frac 80.0%

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

      11. neg-mul-1 80.0%

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

      12. distribute-rgt-neg-in 80.0%

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

      13. times-frac 79.6%

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

      14. metadata-eval 79.6%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 79.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 79.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around 0 79.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{a} \]

   if 8.40000000000000006e-51 < b

   1. Initial program 16.0%

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

      1. /-rgt-identity 16.0%

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

      2. metadata-eval 16.0%

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

      3. associate-/l* 16.0%

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

      4. associate-*r/ 16.0%

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

      5. *-commutative 16.0%

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

      6. associate-*l/ 16.0%

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

      7. associate-*r/ 16.0%

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

      8. metadata-eval 16.0%

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

      9. metadata-eval 16.0%

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

      10. times-frac 16.0%

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

      11. neg-mul-1 16.0%

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

      12. distribute-rgt-neg-in 16.0%

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

      13. times-frac 16.0%

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

      14. metadata-eval 16.0%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 16.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 15.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 16.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around inf 93.2%

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

      1. associate-*r/ 93.2%

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

      2. *-commutative 93.2%

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

   10. Simplified 93.2%

       \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-51}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{elif}\;b \leq -1.25 \cdot 10^{-130}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-52}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e-45)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b -6.2e-106)
     (* -0.3333333333333333 (/ (- b (sqrt (* a (* c -3.0)))) a))
     (if (<= b -1.25e-130)
       (/ b (/ a -0.6666666666666666))
       (if (<= b 9.8e-52)
         (* -0.3333333333333333 (/ (- b (sqrt (* c (* a -3.0)))) a))
         (/ (* c -0.5) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-45) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -6.2e-106) {
		tmp = -0.3333333333333333 * ((b - sqrt((a * (c * -3.0)))) / a);
	} else if (b <= -1.25e-130) {
		tmp = b / (a / -0.6666666666666666);
	} else if (b <= 9.8e-52) {
		tmp = -0.3333333333333333 * ((b - sqrt((c * (a * -3.0)))) / a);
	} 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 <= (-5.5d-45)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= (-6.2d-106)) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt((a * (c * (-3.0d0))))) / a)
    else if (b <= (-1.25d-130)) then
        tmp = b / (a / (-0.6666666666666666d0))
    else if (b <= 9.8d-52) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt((c * (a * (-3.0d0))))) / a)
    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 <= -5.5e-45) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -6.2e-106) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt((a * (c * -3.0)))) / a);
	} else if (b <= -1.25e-130) {
		tmp = b / (a / -0.6666666666666666);
	} else if (b <= 9.8e-52) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.5e-45:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= -6.2e-106:
		tmp = -0.3333333333333333 * ((b - math.sqrt((a * (c * -3.0)))) / a)
	elif b <= -1.25e-130:
		tmp = b / (a / -0.6666666666666666)
	elif b <= 9.8e-52:
		tmp = -0.3333333333333333 * ((b - math.sqrt((c * (a * -3.0)))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.5e-45)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= -6.2e-106)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(a * Float64(c * -3.0)))) / a));
	elseif (b <= -1.25e-130)
		tmp = Float64(b / Float64(a / -0.6666666666666666));
	elseif (b <= 9.8e-52)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(c * Float64(a * -3.0)))) / a));
	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 <= -5.5e-45)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= -6.2e-106)
		tmp = -0.3333333333333333 * ((b - sqrt((a * (c * -3.0)))) / a);
	elseif (b <= -1.25e-130)
		tmp = b / (a / -0.6666666666666666);
	elseif (b <= 9.8e-52)
		tmp = -0.3333333333333333 * ((b - sqrt((c * (a * -3.0)))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.5e-45], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -6.2e-106], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.25e-130], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.8e-52], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -5.5000000000000003e-45

   1. Initial program 64.3%

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

      1. neg-sub0 64.3%

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

      2. associate-+l- 64.3%

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

      3. sub0-neg 64.3%

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

      4. neg-mul-1 64.3%

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

      5. associate-*r/ 64.3%

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

      6. metadata-eval 64.3%

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

      7. metadata-eval 64.3%

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

      8. times-frac 64.3%

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

      9. *-commutative 64.3%

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

      10. times-frac 64.2%

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

      11. associate-*l/ 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in b around -inf 85.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 85.3%

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

   6. Simplified 85.3%

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

   if -5.5000000000000003e-45 < b < -6.19999999999999971e-106

   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. /-rgt-identity 84.7%

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

      2. metadata-eval 84.7%

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

      3. associate-/l* 84.7%

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

      4. associate-*r/ 84.4%

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

      5. *-commutative 84.4%

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

      6. associate-*l/ 84.7%

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

      7. associate-*r/ 84.7%

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

      8. metadata-eval 84.7%

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

      9. metadata-eval 84.7%

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

      10. times-frac 84.7%

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

      11. neg-mul-1 84.7%

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

      12. distribute-rgt-neg-in 84.7%

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

      13. times-frac 84.4%

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

      14. metadata-eval 84.4%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 84.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 84.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around 0 63.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}}{a} \]

      2. *-commutative 63.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -3}}{a} \]

      3. associate-*r* 63.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{a} \]

   6. Simplified 63.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{a} \]

   if -6.19999999999999971e-106 < b < -1.2499999999999999e-130

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-*r/ 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*l/ 99.7%

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

      7. associate-*r/ 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. times-frac 99.7%

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

      11. neg-mul-1 99.7%

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

      12. distribute-rgt-neg-in 99.7%

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

      13. times-frac 99.1%

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

      14. metadata-eval 99.1%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 99.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 99.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around -inf 99.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. associate-/l* 99.7%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

      3. associate-/r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   11. Taylor expanded in a around 0 99.1%

       \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 99.4%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

       2. *-commutative 99.4%

          \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

       3. associate-/l* 99.7%

          \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   13. Simplified 99.7%

       \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   if -1.2499999999999999e-130 < b < 9.80000000000000037e-52

   1. Initial program 80.0%

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

      1. /-rgt-identity 80.0%

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

      2. metadata-eval 80.0%

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

      3. associate-/l* 80.0%

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

      4. associate-*r/ 79.8%

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

      5. *-commutative 79.8%

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

      6. associate-*l/ 80.0%

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

      7. associate-*r/ 80.0%

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

      8. metadata-eval 80.0%

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

      9. metadata-eval 80.0%

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

      10. times-frac 80.0%

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

      11. neg-mul-1 80.0%

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

      12. distribute-rgt-neg-in 80.0%

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

      13. times-frac 79.6%

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

      14. metadata-eval 79.6%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 79.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 79.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around 0 79.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 79.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}}{a} \]

      2. associate-*l* 79.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}}{a} \]

   6. Simplified 79.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}}{a} \]

   if 9.80000000000000037e-52 < b

   1. Initial program 16.0%

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

      1. /-rgt-identity 16.0%

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

      2. metadata-eval 16.0%

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

      3. associate-/l* 16.0%

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

      4. associate-*r/ 16.0%

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

      5. *-commutative 16.0%

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

      6. associate-*l/ 16.0%

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

      7. associate-*r/ 16.0%

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

      8. metadata-eval 16.0%

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

      9. metadata-eval 16.0%

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

      10. times-frac 16.0%

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

      11. neg-mul-1 16.0%

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

      12. distribute-rgt-neg-in 16.0%

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

      13. times-frac 16.0%

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

      14. metadata-eval 16.0%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 16.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 15.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 16.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around inf 93.2%

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

      1. associate-*r/ 93.2%

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

      2. *-commutative 93.2%

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

   10. Simplified 93.2%

       \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{elif}\;b \leq -1.25 \cdot 10^{-130}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-52}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-134}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-55}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-45)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b -6.2e-106)
     (* -0.3333333333333333 (/ (- b (sqrt (* a (* c -3.0)))) a))
     (if (<= b -9e-134)
       (/ b (/ a -0.6666666666666666))
       (if (<= b 4.4e-55)
         (* (- b (sqrt (* c (* a -3.0)))) (/ -0.3333333333333333 a))
         (/ (* c -0.5) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-45) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -6.2e-106) {
		tmp = -0.3333333333333333 * ((b - sqrt((a * (c * -3.0)))) / a);
	} else if (b <= -9e-134) {
		tmp = b / (a / -0.6666666666666666);
	} else if (b <= 4.4e-55) {
		tmp = (b - sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	} 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 <= (-2.4d-45)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= (-6.2d-106)) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt((a * (c * (-3.0d0))))) / a)
    else if (b <= (-9d-134)) then
        tmp = b / (a / (-0.6666666666666666d0))
    else if (b <= 4.4d-55) then
        tmp = (b - sqrt((c * (a * (-3.0d0))))) * ((-0.3333333333333333d0) / a)
    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 <= -2.4e-45) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -6.2e-106) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt((a * (c * -3.0)))) / a);
	} else if (b <= -9e-134) {
		tmp = b / (a / -0.6666666666666666);
	} else if (b <= 4.4e-55) {
		tmp = (b - Math.sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e-45:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= -6.2e-106:
		tmp = -0.3333333333333333 * ((b - math.sqrt((a * (c * -3.0)))) / a)
	elif b <= -9e-134:
		tmp = b / (a / -0.6666666666666666)
	elif b <= 4.4e-55:
		tmp = (b - math.sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e-45)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= -6.2e-106)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(a * Float64(c * -3.0)))) / a));
	elseif (b <= -9e-134)
		tmp = Float64(b / Float64(a / -0.6666666666666666));
	elseif (b <= 4.4e-55)
		tmp = Float64(Float64(b - sqrt(Float64(c * Float64(a * -3.0)))) * Float64(-0.3333333333333333 / a));
	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 <= -2.4e-45)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= -6.2e-106)
		tmp = -0.3333333333333333 * ((b - sqrt((a * (c * -3.0)))) / a);
	elseif (b <= -9e-134)
		tmp = b / (a / -0.6666666666666666);
	elseif (b <= 4.4e-55)
		tmp = (b - sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e-45], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -6.2e-106], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -9e-134], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-55], N[(N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -2.3999999999999999e-45

   1. Initial program 64.3%

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

      1. neg-sub0 64.3%

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

      2. associate-+l- 64.3%

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

      3. sub0-neg 64.3%

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

      4. neg-mul-1 64.3%

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

      5. associate-*r/ 64.3%

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

      6. metadata-eval 64.3%

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

      7. metadata-eval 64.3%

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

      8. times-frac 64.3%

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

      9. *-commutative 64.3%

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

      10. times-frac 64.2%

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

      11. associate-*l/ 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in b around -inf 85.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 85.3%

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

   6. Simplified 85.3%

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

   if -2.3999999999999999e-45 < b < -6.19999999999999971e-106

   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. /-rgt-identity 84.7%

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

      2. metadata-eval 84.7%

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

      3. associate-/l* 84.7%

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

      4. associate-*r/ 84.4%

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

      5. *-commutative 84.4%

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

      6. associate-*l/ 84.7%

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

      7. associate-*r/ 84.7%

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

      8. metadata-eval 84.7%

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

      9. metadata-eval 84.7%

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

      10. times-frac 84.7%

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

      11. neg-mul-1 84.7%

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

      12. distribute-rgt-neg-in 84.7%

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

      13. times-frac 84.4%

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

      14. metadata-eval 84.4%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 84.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 84.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around 0 63.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}}{a} \]

      2. *-commutative 63.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -3}}{a} \]

      3. associate-*r* 63.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{a} \]

   6. Simplified 63.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{a} \]

   if -6.19999999999999971e-106 < b < -9.000000000000001e-134

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-*r/ 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*l/ 99.7%

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

      7. associate-*r/ 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. times-frac 99.7%

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

      11. neg-mul-1 99.7%

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

      12. distribute-rgt-neg-in 99.7%

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

      13. times-frac 99.1%

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

      14. metadata-eval 99.1%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 99.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 99.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around -inf 99.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. associate-/l* 99.7%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

      3. associate-/r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   11. Taylor expanded in a around 0 99.1%

       \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 99.4%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

       2. *-commutative 99.4%

          \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

       3. associate-/l* 99.7%

          \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   13. Simplified 99.7%

       \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   if -9.000000000000001e-134 < b < 4.3999999999999999e-55

   1. Initial program 80.0%

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

      1. /-rgt-identity 80.0%

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

      2. metadata-eval 80.0%

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

      3. associate-/l* 80.0%

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

      4. associate-*r/ 79.8%

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

      5. *-commutative 79.8%

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

      6. associate-*l/ 80.0%

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

      7. associate-*r/ 80.0%

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

      8. metadata-eval 80.0%

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

      9. metadata-eval 80.0%

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

      10. times-frac 80.0%

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

      11. neg-mul-1 80.0%

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

      12. distribute-rgt-neg-in 80.0%

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

      13. times-frac 79.6%

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

      14. metadata-eval 79.6%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 79.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 79.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around 0 79.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 79.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}}{a} \]

      2. associate-*l* 79.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}}{a} \]

   6. Simplified 79.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}}{a} \]
   7. Step-by-step derivation

      1. associate-*r/ 79.4%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}} \]

   8. Applied egg-rr 79.4%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 58.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\right)\right)} \]

      2. expm1-udef 22.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\right)} - 1} \]

   10. Applied egg-rr 22.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 58.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\right)\right)} \]

       2. expm1-log1p 79.4%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}} \]

       3. associate-/l* 79.3%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{a}{b - \sqrt{c \cdot \left(a \cdot -3\right)}}}} \]

       4. associate-/r/ 79.2%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)} \]

   12. Simplified 79.2%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)} \]

   if 4.3999999999999999e-55 < b

   1. Initial program 16.0%

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

      1. /-rgt-identity 16.0%

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

      2. metadata-eval 16.0%

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

      3. associate-/l* 16.0%

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

      4. associate-*r/ 16.0%

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

      5. *-commutative 16.0%

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

      6. associate-*l/ 16.0%

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

      7. associate-*r/ 16.0%

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

      8. metadata-eval 16.0%

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

      9. metadata-eval 16.0%

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

      10. times-frac 16.0%

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

      11. neg-mul-1 16.0%

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

      12. distribute-rgt-neg-in 16.0%

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

      13. times-frac 16.0%

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

      14. metadata-eval 16.0%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 16.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 15.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 16.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around inf 93.2%

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

      1. associate-*r/ 93.2%

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

      2. *-commutative 93.2%

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

   10. Simplified 93.2%

       \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-134}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-55}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b - \sqrt{a \cdot \left(c \cdot -3\right)}\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{t_0}{a}\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{elif}\;b \leq 3.25 \cdot 10^{-50}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{a}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- b (sqrt (* a (* c -3.0))))))
   (if (<= b -1.5e-44)
     (/ (* b -2.0) (* 3.0 a))
     (if (<= b -6.2e-106)
       (* -0.3333333333333333 (/ t_0 a))
       (if (<= b -1.32e-129)
         (/ b (/ a -0.6666666666666666))
         (if (<= b 3.25e-50)
           (/ -0.3333333333333333 (/ a t_0))
           (/ (* c -0.5) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = b - sqrt((a * (c * -3.0)));
	double tmp;
	if (b <= -1.5e-44) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -6.2e-106) {
		tmp = -0.3333333333333333 * (t_0 / a);
	} else if (b <= -1.32e-129) {
		tmp = b / (a / -0.6666666666666666);
	} else if (b <= 3.25e-50) {
		tmp = -0.3333333333333333 / (a / 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 = b - sqrt((a * (c * (-3.0d0))))
    if (b <= (-1.5d-44)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= (-6.2d-106)) then
        tmp = (-0.3333333333333333d0) * (t_0 / a)
    else if (b <= (-1.32d-129)) then
        tmp = b / (a / (-0.6666666666666666d0))
    else if (b <= 3.25d-50) then
        tmp = (-0.3333333333333333d0) / (a / 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 = b - Math.sqrt((a * (c * -3.0)));
	double tmp;
	if (b <= -1.5e-44) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -6.2e-106) {
		tmp = -0.3333333333333333 * (t_0 / a);
	} else if (b <= -1.32e-129) {
		tmp = b / (a / -0.6666666666666666);
	} else if (b <= 3.25e-50) {
		tmp = -0.3333333333333333 / (a / t_0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b - math.sqrt((a * (c * -3.0)))
	tmp = 0
	if b <= -1.5e-44:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= -6.2e-106:
		tmp = -0.3333333333333333 * (t_0 / a)
	elif b <= -1.32e-129:
		tmp = b / (a / -0.6666666666666666)
	elif b <= 3.25e-50:
		tmp = -0.3333333333333333 / (a / t_0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b - sqrt(Float64(a * Float64(c * -3.0))))
	tmp = 0.0
	if (b <= -1.5e-44)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= -6.2e-106)
		tmp = Float64(-0.3333333333333333 * Float64(t_0 / a));
	elseif (b <= -1.32e-129)
		tmp = Float64(b / Float64(a / -0.6666666666666666));
	elseif (b <= 3.25e-50)
		tmp = Float64(-0.3333333333333333 / Float64(a / t_0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b - sqrt((a * (c * -3.0)));
	tmp = 0.0;
	if (b <= -1.5e-44)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= -6.2e-106)
		tmp = -0.3333333333333333 * (t_0 / a);
	elseif (b <= -1.32e-129)
		tmp = b / (a / -0.6666666666666666);
	elseif (b <= 3.25e-50)
		tmp = -0.3333333333333333 / (a / t_0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.5e-44], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -6.2e-106], N[(-0.3333333333333333 * N[(t$95$0 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.32e-129], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.25e-50], N[(-0.3333333333333333 / N[(a / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.5000000000000001e-44

   1. Initial program 64.3%

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

      1. neg-sub0 64.3%

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

      2. associate-+l- 64.3%

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

      3. sub0-neg 64.3%

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

      4. neg-mul-1 64.3%

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

      5. associate-*r/ 64.3%

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

      6. metadata-eval 64.3%

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

      7. metadata-eval 64.3%

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

      8. times-frac 64.3%

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

      9. *-commutative 64.3%

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

      10. times-frac 64.2%

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

      11. associate-*l/ 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in b around -inf 85.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 85.3%

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

   6. Simplified 85.3%

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

   if -1.5000000000000001e-44 < b < -6.19999999999999971e-106

   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. /-rgt-identity 84.7%

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

      2. metadata-eval 84.7%

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

      3. associate-/l* 84.7%

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

      4. associate-*r/ 84.4%

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

      5. *-commutative 84.4%

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

      6. associate-*l/ 84.7%

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

      7. associate-*r/ 84.7%

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

      8. metadata-eval 84.7%

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

      9. metadata-eval 84.7%

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

      10. times-frac 84.7%

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

      11. neg-mul-1 84.7%

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

      12. distribute-rgt-neg-in 84.7%

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

      13. times-frac 84.4%

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

      14. metadata-eval 84.4%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 84.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 84.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around 0 63.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}}{a} \]

      2. *-commutative 63.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -3}}{a} \]

      3. associate-*r* 63.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{a} \]

   6. Simplified 63.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{a} \]

   if -6.19999999999999971e-106 < b < -1.31999999999999992e-129

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-*r/ 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*l/ 99.7%

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

      7. associate-*r/ 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. times-frac 99.7%

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

      11. neg-mul-1 99.7%

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

      12. distribute-rgt-neg-in 99.7%

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

      13. times-frac 99.1%

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

      14. metadata-eval 99.1%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 99.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 99.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around -inf 99.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. associate-/l* 99.7%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

      3. associate-/r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   11. Taylor expanded in a around 0 99.1%

       \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 99.4%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

       2. *-commutative 99.4%

          \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

       3. associate-/l* 99.7%

          \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   13. Simplified 99.7%

       \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   if -1.31999999999999992e-129 < b < 3.24999999999999994e-50

   1. Initial program 80.0%

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

      1. /-rgt-identity 80.0%

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

      2. metadata-eval 80.0%

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

      3. associate-/l* 80.0%

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

      4. associate-*r/ 79.8%

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

      5. *-commutative 79.8%

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

      6. associate-*l/ 80.0%

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

      7. associate-*r/ 80.0%

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

      8. metadata-eval 80.0%

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

      9. metadata-eval 80.0%

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

      10. times-frac 80.0%

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

      11. neg-mul-1 80.0%

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

      12. distribute-rgt-neg-in 80.0%

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

      13. times-frac 79.6%

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

      14. metadata-eval 79.6%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 79.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 79.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around 0 79.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 79.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}}{a} \]

      2. *-commutative 79.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -3}}{a} \]

      3. associate-*r* 79.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{a} \]

   6. Simplified 79.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{a} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 58.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\right)\right)} \]

      2. expm1-udef 22.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\right)} - 1} \]

   8. Applied egg-rr 22.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 58.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\right)\right)} \]

      2. expm1-log1p 79.0%

         \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}} \]

      3. associate-*r/ 79.2%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{a \cdot \left(c \cdot -3\right)}\right)}{a}} \]

      4. associate-/l* 79.2%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{a}{b - \sqrt{a \cdot \left(c \cdot -3\right)}}}} \]

   10. Simplified 79.2%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{a}{b - \sqrt{a \cdot \left(c \cdot -3\right)}}}} \]

   if 3.24999999999999994e-50 < b

   1. Initial program 16.0%

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

      1. /-rgt-identity 16.0%

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

      2. metadata-eval 16.0%

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

      3. associate-/l* 16.0%

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

      4. associate-*r/ 16.0%

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

      5. *-commutative 16.0%

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

      6. associate-*l/ 16.0%

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

      7. associate-*r/ 16.0%

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

      8. metadata-eval 16.0%

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

      9. metadata-eval 16.0%

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

      10. times-frac 16.0%

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

      11. neg-mul-1 16.0%

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

      12. distribute-rgt-neg-in 16.0%

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

      13. times-frac 16.0%

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

      14. metadata-eval 16.0%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 16.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 15.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 16.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around inf 93.2%

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

      1. associate-*r/ 93.2%

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

      2. *-commutative 93.2%

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

   10. Simplified 93.2%

       \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{elif}\;b \leq 3.25 \cdot 10^{-50}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{a}{b - \sqrt{a \cdot \left(c \cdot -3\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -3.5 \cdot 10^{-105}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e-43)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b -3.5e-105)
     (* -0.3333333333333333 (/ (- b (sqrt (* a (* c -3.0)))) a))
     (if (<= b -1.32e-129)
       (/ b (/ a -0.6666666666666666))
       (if (<= b 2.5e-54)
         (/ (* -0.3333333333333333 (- b (sqrt (* c (* a -3.0))))) a)
         (/ (* c -0.5) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e-43) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -3.5e-105) {
		tmp = -0.3333333333333333 * ((b - sqrt((a * (c * -3.0)))) / a);
	} else if (b <= -1.32e-129) {
		tmp = b / (a / -0.6666666666666666);
	} else if (b <= 2.5e-54) {
		tmp = (-0.3333333333333333 * (b - sqrt((c * (a * -3.0))))) / a;
	} 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 <= (-1.2d-43)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= (-3.5d-105)) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt((a * (c * (-3.0d0))))) / a)
    else if (b <= (-1.32d-129)) then
        tmp = b / (a / (-0.6666666666666666d0))
    else if (b <= 2.5d-54) then
        tmp = ((-0.3333333333333333d0) * (b - sqrt((c * (a * (-3.0d0)))))) / a
    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 <= -1.2e-43) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -3.5e-105) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt((a * (c * -3.0)))) / a);
	} else if (b <= -1.32e-129) {
		tmp = b / (a / -0.6666666666666666);
	} else if (b <= 2.5e-54) {
		tmp = (-0.3333333333333333 * (b - Math.sqrt((c * (a * -3.0))))) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.2e-43:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= -3.5e-105:
		tmp = -0.3333333333333333 * ((b - math.sqrt((a * (c * -3.0)))) / a)
	elif b <= -1.32e-129:
		tmp = b / (a / -0.6666666666666666)
	elif b <= 2.5e-54:
		tmp = (-0.3333333333333333 * (b - math.sqrt((c * (a * -3.0))))) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e-43)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= -3.5e-105)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(a * Float64(c * -3.0)))) / a));
	elseif (b <= -1.32e-129)
		tmp = Float64(b / Float64(a / -0.6666666666666666));
	elseif (b <= 2.5e-54)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(b - sqrt(Float64(c * Float64(a * -3.0))))) / a);
	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 <= -1.2e-43)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= -3.5e-105)
		tmp = -0.3333333333333333 * ((b - sqrt((a * (c * -3.0)))) / a);
	elseif (b <= -1.32e-129)
		tmp = b / (a / -0.6666666666666666);
	elseif (b <= 2.5e-54)
		tmp = (-0.3333333333333333 * (b - sqrt((c * (a * -3.0))))) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e-43], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -3.5e-105], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.32e-129], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5e-54], N[(N[(-0.3333333333333333 * N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.2000000000000001e-43

   1. Initial program 64.3%

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

      1. neg-sub0 64.3%

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

      2. associate-+l- 64.3%

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

      3. sub0-neg 64.3%

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

      4. neg-mul-1 64.3%

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

      5. associate-*r/ 64.3%

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

      6. metadata-eval 64.3%

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

      7. metadata-eval 64.3%

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

      8. times-frac 64.3%

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

      9. *-commutative 64.3%

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

      10. times-frac 64.2%

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

      11. associate-*l/ 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in b around -inf 85.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 85.3%

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

   6. Simplified 85.3%

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

   if -1.2000000000000001e-43 < b < -3.5e-105

   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. /-rgt-identity 84.7%

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

      2. metadata-eval 84.7%

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

      3. associate-/l* 84.7%

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

      4. associate-*r/ 84.4%

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

      5. *-commutative 84.4%

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

      6. associate-*l/ 84.7%

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

      7. associate-*r/ 84.7%

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

      8. metadata-eval 84.7%

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

      9. metadata-eval 84.7%

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

      10. times-frac 84.7%

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

      11. neg-mul-1 84.7%

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

      12. distribute-rgt-neg-in 84.7%

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

      13. times-frac 84.4%

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

      14. metadata-eval 84.4%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 84.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 84.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around 0 63.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}}{a} \]

      2. *-commutative 63.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -3}}{a} \]

      3. associate-*r* 63.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{a} \]

   6. Simplified 63.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{a} \]

   if -3.5e-105 < b < -1.31999999999999992e-129

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-*r/ 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*l/ 99.7%

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

      7. associate-*r/ 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. times-frac 99.7%

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

      11. neg-mul-1 99.7%

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

      12. distribute-rgt-neg-in 99.7%

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

      13. times-frac 99.1%

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

      14. metadata-eval 99.1%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 99.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 99.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around -inf 99.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. associate-/l* 99.7%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

      3. associate-/r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   11. Taylor expanded in a around 0 99.1%

       \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 99.4%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

       2. *-commutative 99.4%

          \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

       3. associate-/l* 99.7%

          \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   13. Simplified 99.7%

       \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   if -1.31999999999999992e-129 < b < 2.50000000000000008e-54

   1. Initial program 80.0%

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

      1. /-rgt-identity 80.0%

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

      2. metadata-eval 80.0%

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

      3. associate-/l* 80.0%

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

      4. associate-*r/ 79.8%

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

      5. *-commutative 79.8%

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

      6. associate-*l/ 80.0%

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

      7. associate-*r/ 80.0%

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

      8. metadata-eval 80.0%

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

      9. metadata-eval 80.0%

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

      10. times-frac 80.0%

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

      11. neg-mul-1 80.0%

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

      12. distribute-rgt-neg-in 80.0%

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

      13. times-frac 79.6%

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

      14. metadata-eval 79.6%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 79.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 79.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around 0 79.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 79.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}}{a} \]

      2. associate-*l* 79.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}}{a} \]

   6. Simplified 79.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}}{a} \]
   7. Step-by-step derivation

      1. associate-*r/ 79.4%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}} \]

   8. Applied egg-rr 79.4%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}} \]

   if 2.50000000000000008e-54 < b

   1. Initial program 16.0%

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

      1. /-rgt-identity 16.0%

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

      2. metadata-eval 16.0%

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

      3. associate-/l* 16.0%

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

      4. associate-*r/ 16.0%

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

      5. *-commutative 16.0%

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

      6. associate-*l/ 16.0%

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

      7. associate-*r/ 16.0%

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

      8. metadata-eval 16.0%

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

      9. metadata-eval 16.0%

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

      10. times-frac 16.0%

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

      11. neg-mul-1 16.0%

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

      12. distribute-rgt-neg-in 16.0%

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

      13. times-frac 16.0%

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

      14. metadata-eval 16.0%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 16.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 15.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 16.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around inf 93.2%

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

      1. associate-*r/ 93.2%

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

      2. *-commutative 93.2%

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

   10. Simplified 93.2%

       \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -3.5 \cdot 10^{-105}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 8: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-43}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-106}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{b}{a} - \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\right)\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-51}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-43)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b -7e-106)
     (* -0.3333333333333333 (- (/ b a) (/ (sqrt (* a (* c -3.0))) a)))
     (if (<= b -1.32e-129)
       (/ b (/ a -0.6666666666666666))
       (if (<= b 4.3e-51)
         (/ (* -0.3333333333333333 (- b (sqrt (* c (* a -3.0))))) a)
         (/ (* c -0.5) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-43) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -7e-106) {
		tmp = -0.3333333333333333 * ((b / a) - (sqrt((a * (c * -3.0))) / a));
	} else if (b <= -1.32e-129) {
		tmp = b / (a / -0.6666666666666666);
	} else if (b <= 4.3e-51) {
		tmp = (-0.3333333333333333 * (b - sqrt((c * (a * -3.0))))) / a;
	} 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 <= (-2d-43)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= (-7d-106)) then
        tmp = (-0.3333333333333333d0) * ((b / a) - (sqrt((a * (c * (-3.0d0)))) / a))
    else if (b <= (-1.32d-129)) then
        tmp = b / (a / (-0.6666666666666666d0))
    else if (b <= 4.3d-51) then
        tmp = ((-0.3333333333333333d0) * (b - sqrt((c * (a * (-3.0d0)))))) / a
    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 <= -2e-43) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= -7e-106) {
		tmp = -0.3333333333333333 * ((b / a) - (Math.sqrt((a * (c * -3.0))) / a));
	} else if (b <= -1.32e-129) {
		tmp = b / (a / -0.6666666666666666);
	} else if (b <= 4.3e-51) {
		tmp = (-0.3333333333333333 * (b - Math.sqrt((c * (a * -3.0))))) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-43:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= -7e-106:
		tmp = -0.3333333333333333 * ((b / a) - (math.sqrt((a * (c * -3.0))) / a))
	elif b <= -1.32e-129:
		tmp = b / (a / -0.6666666666666666)
	elif b <= 4.3e-51:
		tmp = (-0.3333333333333333 * (b - math.sqrt((c * (a * -3.0))))) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-43)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= -7e-106)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b / a) - Float64(sqrt(Float64(a * Float64(c * -3.0))) / a)));
	elseif (b <= -1.32e-129)
		tmp = Float64(b / Float64(a / -0.6666666666666666));
	elseif (b <= 4.3e-51)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(b - sqrt(Float64(c * Float64(a * -3.0))))) / a);
	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 <= -2e-43)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= -7e-106)
		tmp = -0.3333333333333333 * ((b / a) - (sqrt((a * (c * -3.0))) / a));
	elseif (b <= -1.32e-129)
		tmp = b / (a / -0.6666666666666666);
	elseif (b <= 4.3e-51)
		tmp = (-0.3333333333333333 * (b - sqrt((c * (a * -3.0))))) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-43], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -7e-106], N[(-0.3333333333333333 * N[(N[(b / a), $MachinePrecision] - N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.32e-129], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.3e-51], N[(N[(-0.3333333333333333 * N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -2.00000000000000015e-43

   1. Initial program 64.3%

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

      1. neg-sub0 64.3%

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

      2. associate-+l- 64.3%

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

      3. sub0-neg 64.3%

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

      4. neg-mul-1 64.3%

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

      5. associate-*r/ 64.3%

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

      6. metadata-eval 64.3%

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

      7. metadata-eval 64.3%

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

      8. times-frac 64.3%

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

      9. *-commutative 64.3%

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

      10. times-frac 64.2%

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

      11. associate-*l/ 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in b around -inf 85.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 85.3%

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

   6. Simplified 85.3%

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

   if -2.00000000000000015e-43 < b < -7e-106

   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. /-rgt-identity 84.7%

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

      2. metadata-eval 84.7%

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

      3. associate-/l* 84.7%

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

      4. associate-*r/ 84.4%

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

      5. *-commutative 84.4%

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

      6. associate-*l/ 84.7%

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

      7. associate-*r/ 84.7%

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

      8. metadata-eval 84.7%

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

      9. metadata-eval 84.7%

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

      10. times-frac 84.7%

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

      11. neg-mul-1 84.7%

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

      12. distribute-rgt-neg-in 84.7%

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

      13. times-frac 84.4%

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

      14. metadata-eval 84.4%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 84.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 84.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around 0 63.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}}{a} \]

      2. *-commutative 63.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -3}}{a} \]

      3. associate-*r* 63.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{a} \]

   6. Simplified 63.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{a} \]
   7. Step-by-step derivation

      1. div-sub 63.5%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{b}{a} - \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\right)} \]

   8. Applied egg-rr 63.5%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{b}{a} - \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\right)} \]

   if -7e-106 < b < -1.31999999999999992e-129

   1. Initial program 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-*r/ 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*l/ 99.7%

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

      7. associate-*r/ 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. times-frac 99.7%

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

      11. neg-mul-1 99.7%

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

      12. distribute-rgt-neg-in 99.7%

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

      13. times-frac 99.1%

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

      14. metadata-eval 99.1%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 99.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 99.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 99.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 99.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around -inf 99.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. associate-/l* 99.7%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

      3. associate-/r/ 99.4%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   11. Taylor expanded in a around 0 99.1%

       \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 99.4%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

       2. *-commutative 99.4%

          \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

       3. associate-/l* 99.7%

          \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   13. Simplified 99.7%

       \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   if -1.31999999999999992e-129 < b < 4.2999999999999997e-51

   1. Initial program 80.0%

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

      1. /-rgt-identity 80.0%

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

      2. metadata-eval 80.0%

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

      3. associate-/l* 80.0%

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

      4. associate-*r/ 79.8%

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

      5. *-commutative 79.8%

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

      6. associate-*l/ 80.0%

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

      7. associate-*r/ 80.0%

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

      8. metadata-eval 80.0%

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

      9. metadata-eval 80.0%

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

      10. times-frac 80.0%

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

      11. neg-mul-1 80.0%

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

      12. distribute-rgt-neg-in 80.0%

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

      13. times-frac 79.6%

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

      14. metadata-eval 79.6%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 79.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 79.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around 0 79.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right)}}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 79.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}}{a} \]

      2. associate-*l* 79.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}}{a} \]

   6. Simplified 79.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}}{a} \]
   7. Step-by-step derivation

      1. associate-*r/ 79.4%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}} \]

   8. Applied egg-rr 79.4%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}} \]

   if 4.2999999999999997e-51 < b

   1. Initial program 16.0%

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

      1. /-rgt-identity 16.0%

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

      2. metadata-eval 16.0%

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

      3. associate-/l* 16.0%

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

      4. associate-*r/ 16.0%

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

      5. *-commutative 16.0%

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

      6. associate-*l/ 16.0%

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

      7. associate-*r/ 16.0%

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

      8. metadata-eval 16.0%

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

      9. metadata-eval 16.0%

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

      10. times-frac 16.0%

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

      11. neg-mul-1 16.0%

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

      12. distribute-rgt-neg-in 16.0%

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

      13. times-frac 16.0%

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

      14. metadata-eval 16.0%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 16.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 15.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 16.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around inf 93.2%

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

      1. associate-*r/ 93.2%

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

      2. *-commutative 93.2%

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

   10. Simplified 93.2%

       \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-43}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-106}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{b}{a} - \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\right)\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-51}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 9: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.06 \cdot 10^{+108}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 3.95 \cdot 10^{-54}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.06e+108)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 3.95e-54)
     (* -0.3333333333333333 (/ (- b (sqrt (- (* b b) (* 3.0 (* a c))))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.06e+108) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 3.95e-54) {
		tmp = -0.3333333333333333 * ((b - sqrt(((b * b) - (3.0 * (a * c))))) / a);
	} 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 <= (-1.06d+108)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 3.95d-54) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt(((b * b) - (3.0d0 * (a * c))))) / a)
    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 <= -1.06e+108) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 3.95e-54) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt(((b * b) - (3.0 * (a * c))))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.06e+108:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 3.95e-54:
		tmp = -0.3333333333333333 * ((b - math.sqrt(((b * b) - (3.0 * (a * c))))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.06e+108)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 3.95e-54)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c))))) / a));
	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 <= -1.06e+108)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 3.95e-54)
		tmp = -0.3333333333333333 * ((b - sqrt(((b * b) - (3.0 * (a * c))))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.06e+108], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.95e-54], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.06e108

   1. Initial program 50.9%

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

      1. neg-sub0 50.9%

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

      2. associate-+l- 50.9%

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

      3. sub0-neg 50.9%

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

      4. neg-mul-1 50.9%

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

      5. associate-*r/ 50.9%

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

      6. metadata-eval 50.9%

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

      7. metadata-eval 50.9%

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

      8. times-frac 50.9%

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

      9. *-commutative 50.9%

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

      10. times-frac 50.8%

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

      11. associate-*l/ 50.9%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in b around -inf 99.7%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   if -1.06e108 < b < 3.9500000000000002e-54

   1. Initial program 83.0%

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

      1. /-rgt-identity 83.0%

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

      2. metadata-eval 83.0%

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

      3. associate-/l* 83.0%

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

      4. associate-*r/ 82.8%

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

      5. *-commutative 82.8%

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

      6. associate-*l/ 83.0%

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

      7. associate-*r/ 83.0%

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

      8. metadata-eval 83.0%

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

      9. metadata-eval 83.0%

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

      10. times-frac 83.0%

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

      11. neg-mul-1 83.0%

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

      12. distribute-rgt-neg-in 83.0%

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

      13. times-frac 82.6%

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

      14. metadata-eval 82.6%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 82.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 82.6%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 82.6%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 82.6%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 82.6%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 82.6%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 82.6%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 82.6%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 82.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 82.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 82.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 82.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 82.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 82.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 82.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 82.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 82.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 82.6%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 82.6%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 82.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 82.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 82.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 82.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 82.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 82.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 82.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 82.6%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   if 3.9500000000000002e-54 < b

   1. Initial program 16.0%

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

      1. /-rgt-identity 16.0%

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

      2. metadata-eval 16.0%

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

      3. associate-/l* 16.0%

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

      4. associate-*r/ 16.0%

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

      5. *-commutative 16.0%

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

      6. associate-*l/ 16.0%

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

      7. associate-*r/ 16.0%

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

      8. metadata-eval 16.0%

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

      9. metadata-eval 16.0%

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

      10. times-frac 16.0%

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

      11. neg-mul-1 16.0%

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

      12. distribute-rgt-neg-in 16.0%

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

      13. times-frac 16.0%

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

      14. metadata-eval 16.0%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 16.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 15.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 16.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around inf 93.2%

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

      1. associate-*r/ 93.2%

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

      2. *-commutative 93.2%

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

   10. Simplified 93.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.06 \cdot 10^{+108}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 3.95 \cdot 10^{-54}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 10: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+152}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+152)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 1.04e-53)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+152) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.04e-53) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} 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 <= (-1d+152)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 1.04d-53) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    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 <= -1e+152) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.04e-53) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+152:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 1.04e-53:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+152)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 1.04e-53)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	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 <= -1e+152)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 1.04e-53)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+152], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.04e-53], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e152

   1. Initial program 38.9%

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

      1. neg-sub0 38.9%

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

      2. associate-+l- 38.9%

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

      3. sub0-neg 38.9%

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

      4. neg-mul-1 38.9%

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

      5. associate-*r/ 38.9%

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

      6. metadata-eval 38.9%

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

      7. metadata-eval 38.9%

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

      8. times-frac 38.9%

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

      9. *-commutative 38.9%

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

      10. times-frac 38.9%

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

      11. associate-*l/ 38.9%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in b around -inf 99.8%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   if -1e152 < b < 1.04000000000000001e-53

   1. Initial program 84.4%

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

      1. neg-sub0 84.4%

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

      2. associate-+l- 84.4%

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

      3. sub0-neg 84.4%

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

      4. neg-mul-1 84.4%

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

      5. associate-*r/ 84.4%

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

      6. metadata-eval 84.4%

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

      7. metadata-eval 84.4%

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

      8. times-frac 84.4%

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

      9. *-commutative 84.4%

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

      10. times-frac 84.2%

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

      11. associate-*l/ 84.4%

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

   3. Simplified 84.3%

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

   if 1.04000000000000001e-53 < b

   1. Initial program 16.0%

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

      1. /-rgt-identity 16.0%

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

      2. metadata-eval 16.0%

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

      3. associate-/l* 16.0%

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

      4. associate-*r/ 16.0%

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

      5. *-commutative 16.0%

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

      6. associate-*l/ 16.0%

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

      7. associate-*r/ 16.0%

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

      8. metadata-eval 16.0%

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

      9. metadata-eval 16.0%

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

      10. times-frac 16.0%

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

      11. neg-mul-1 16.0%

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

      12. distribute-rgt-neg-in 16.0%

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

      13. times-frac 16.0%

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

      14. metadata-eval 16.0%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 16.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 15.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 15.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 16.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 16.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 16.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around inf 93.2%

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

      1. associate-*r/ 93.2%

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

      2. *-commutative 93.2%

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

   10. Simplified 93.2%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+152}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 11: 67.5% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.6e-300) (/ (* b -2.0) (* 3.0 a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.6e-300) {
		tmp = (b * -2.0) / (3.0 * a);
	} 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 <= 7.6d-300) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    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 <= 7.6e-300) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.6e-300:
		tmp = (b * -2.0) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.6e-300)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	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 <= 7.6e-300)
		tmp = (b * -2.0) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.6e-300], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.60000000000000026e-300

   1. Initial program 72.4%

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

      1. neg-sub0 72.4%

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

      2. associate-+l- 72.4%

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

      3. sub0-neg 72.4%

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

      4. neg-mul-1 72.4%

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

      5. associate-*r/ 72.4%

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

      6. metadata-eval 72.4%

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

      7. metadata-eval 72.4%

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

      8. times-frac 72.4%

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

      9. *-commutative 72.4%

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

      10. times-frac 72.3%

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

      11. associate-*l/ 72.4%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in b around -inf 64.8%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 64.8%

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

   6. Simplified 64.8%

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

   if 7.60000000000000026e-300 < b

   1. Initial program 36.2%

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

      1. /-rgt-identity 36.2%

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

      2. metadata-eval 36.2%

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

      3. associate-/l* 36.2%

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

      4. associate-*r/ 36.1%

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

      5. *-commutative 36.1%

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

      6. associate-*l/ 36.2%

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

      7. associate-*r/ 36.2%

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

      8. metadata-eval 36.2%

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

      9. metadata-eval 36.2%

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

      10. times-frac 36.2%

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

      11. neg-mul-1 36.2%

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

      12. distribute-rgt-neg-in 36.2%

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

      13. times-frac 36.1%

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

      14. metadata-eval 36.1%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 36.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 35.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 36.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around inf 68.0%

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

      1. associate-*r/ 68.0%

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

      2. *-commutative 68.0%

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

   10. Simplified 68.0%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 12: 67.5% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{-300}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.6e-300) (* b (/ -0.6666666666666666 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.6e-300) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / 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 <= 7.6d-300) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.6e-300) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.6e-300:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.6e-300)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7.6e-300)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.6e-300], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.60000000000000026e-300

   1. Initial program 72.4%

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

      1. /-rgt-identity 72.4%

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

      2. metadata-eval 72.4%

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

      3. associate-/l* 72.4%

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

      4. associate-*r/ 72.3%

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

      5. *-commutative 72.3%

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

      6. associate-*l/ 72.4%

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

      7. associate-*r/ 72.4%

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

      8. metadata-eval 72.4%

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

      9. metadata-eval 72.4%

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

      10. times-frac 72.4%

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

      11. neg-mul-1 72.4%

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

      12. distribute-rgt-neg-in 72.4%

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

      13. times-frac 72.1%

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

      14. metadata-eval 72.1%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 72.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 71.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 72.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around -inf 64.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 64.6%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. associate-/l* 64.8%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

      3. associate-/r/ 64.7%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   10. Simplified 64.7%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   if 7.60000000000000026e-300 < b

   1. Initial program 36.2%

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

      1. /-rgt-identity 36.2%

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

      2. metadata-eval 36.2%

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

      3. associate-/r/ 36.2%

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

      4. metadata-eval 36.2%

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

      5. metadata-eval 36.2%

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

      6. times-frac 36.2%

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

      7. *-commutative 36.2%

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

      8. times-frac 36.1%

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

      9. *-commutative 36.1%

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

      10. associate-/r* 36.1%

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

      11. associate-*l/ 36.2%

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

   3. Simplified 36.1%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]

   4. Taylor expanded in b around inf 68.0%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{-300}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 13: 67.5% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-299}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.35e-299) (/ b (/ a -0.6666666666666666)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.35e-299) {
		tmp = b / (a / -0.6666666666666666);
	} else {
		tmp = -0.5 * (c / 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 <= 1.35d-299) then
        tmp = b / (a / (-0.6666666666666666d0))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.35e-299) {
		tmp = b / (a / -0.6666666666666666);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.35e-299:
		tmp = b / (a / -0.6666666666666666)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.35e-299)
		tmp = Float64(b / Float64(a / -0.6666666666666666));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.35e-299)
		tmp = b / (a / -0.6666666666666666);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.35e-299], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35000000000000001e-299

   1. Initial program 72.4%

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

      1. /-rgt-identity 72.4%

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

      2. metadata-eval 72.4%

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

      3. associate-/l* 72.4%

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

      4. associate-*r/ 72.3%

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

      5. *-commutative 72.3%

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

      6. associate-*l/ 72.4%

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

      7. associate-*r/ 72.4%

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

      8. metadata-eval 72.4%

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

      9. metadata-eval 72.4%

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

      10. times-frac 72.4%

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

      11. neg-mul-1 72.4%

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

      12. distribute-rgt-neg-in 72.4%

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

      13. times-frac 72.1%

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

      14. metadata-eval 72.1%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 72.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 71.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 72.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around -inf 64.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 64.6%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. associate-/l* 64.8%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

      3. associate-/r/ 64.7%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   10. Simplified 64.7%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   11. Taylor expanded in a around 0 64.6%

       \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 64.6%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

       2. *-commutative 64.6%

          \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

       3. associate-/l* 64.8%

          \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   13. Simplified 64.8%

       \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   if 1.35000000000000001e-299 < b

   1. Initial program 36.2%

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

      1. /-rgt-identity 36.2%

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

      2. metadata-eval 36.2%

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

      3. associate-/r/ 36.2%

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

      4. metadata-eval 36.2%

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

      5. metadata-eval 36.2%

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

      6. times-frac 36.2%

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

      7. *-commutative 36.2%

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

      8. times-frac 36.1%

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

      9. *-commutative 36.1%

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

      10. associate-/r* 36.1%

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

      11. associate-*l/ 36.2%

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

   3. Simplified 36.1%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]

   4. Taylor expanded in b around inf 68.0%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-299}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 14: 67.5% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.6e-300) (/ b (/ a -0.6666666666666666)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.6e-300) {
		tmp = b / (a / -0.6666666666666666);
	} 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 <= 7.6d-300) then
        tmp = b / (a / (-0.6666666666666666d0))
    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 <= 7.6e-300) {
		tmp = b / (a / -0.6666666666666666);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.6e-300:
		tmp = b / (a / -0.6666666666666666)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.6e-300)
		tmp = Float64(b / Float64(a / -0.6666666666666666));
	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 <= 7.6e-300)
		tmp = b / (a / -0.6666666666666666);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.6e-300], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.60000000000000026e-300

   1. Initial program 72.4%

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

      1. /-rgt-identity 72.4%

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

      2. metadata-eval 72.4%

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

      3. associate-/l* 72.4%

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

      4. associate-*r/ 72.3%

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

      5. *-commutative 72.3%

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

      6. associate-*l/ 72.4%

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

      7. associate-*r/ 72.4%

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

      8. metadata-eval 72.4%

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

      9. metadata-eval 72.4%

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

      10. times-frac 72.4%

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

      11. neg-mul-1 72.4%

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

      12. distribute-rgt-neg-in 72.4%

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

      13. times-frac 72.1%

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

      14. metadata-eval 72.1%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 72.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 71.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 71.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 72.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 72.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 72.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around -inf 64.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 64.6%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. associate-/l* 64.8%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

      3. associate-/r/ 64.7%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   10. Simplified 64.7%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   11. Taylor expanded in a around 0 64.6%

       \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 64.6%

          \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

       2. *-commutative 64.6%

          \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

       3. associate-/l* 64.8%

          \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   13. Simplified 64.8%

       \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

   if 7.60000000000000026e-300 < b

   1. Initial program 36.2%

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

      1. /-rgt-identity 36.2%

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

      2. metadata-eval 36.2%

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

      3. associate-/l* 36.2%

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

      4. associate-*r/ 36.1%

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

      5. *-commutative 36.1%

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

      6. associate-*l/ 36.2%

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

      7. associate-*r/ 36.2%

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

      8. metadata-eval 36.2%

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

      9. metadata-eval 36.2%

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

      10. times-frac 36.2%

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

      11. neg-mul-1 36.2%

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

      12. distribute-rgt-neg-in 36.2%

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

      13. times-frac 36.1%

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

      14. metadata-eval 36.1%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 36.1%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
   4. Step-by-step derivation

      1. associate-*r* 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      2. *-commutative 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      3. metadata-eval 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-lft-neg-in 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}{a} \]

      5. fma-neg 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}{a} \]

      6. *-un-lft-identity 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{1 \cdot \left(3 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      7. prod-diff 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right) + \mathsf{fma}\left(-3 \cdot \left(a \cdot c\right), 1, \left(3 \cdot \left(a \cdot c\right)\right) \cdot 1\right)}}}{a} \]

   5. Applied egg-rr 35.9%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]
   6. Step-by-step derivation

      1. *-rgt-identity 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -3\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      2. fma-neg 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -3\right), 1, \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}{a} \]

      3. fma-udef 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -3\right)\right) \cdot 1\right)}}}{a} \]

      4. distribute-rgt-in 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{1 \cdot \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      5. *-lft-identity 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\left(b \cdot b - a \cdot \left(c \cdot -3\right)\right) + \color{blue}{\left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      6. associate--r- 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - \left(a \cdot \left(c \cdot -3\right) - \left(a \cdot \left(c \cdot -3\right) + a \cdot \left(c \cdot -3\right)\right)\right)}}}{a} \]

      7. associate--r+ 35.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(\left(a \cdot \left(c \cdot -3\right) - a \cdot \left(c \cdot -3\right)\right) - a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      8. +-inverses 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -3\right)\right)}}{a} \]

      9. neg-sub0 36.1%

         \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}}}{a} \]

      10. associate-*r* 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{a} \]

      11. *-commutative 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}}{a} \]

      12. *-commutative 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(-\color{blue}{-3 \cdot \left(c \cdot a\right)}\right)}}{a} \]

      13. *-commutative 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \left(--3 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} \]

      14. distribute-lft-neg-in 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{\left(--3\right) \cdot \left(a \cdot c\right)}}}{a} \]

      15. metadata-eval 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - \color{blue}{3} \cdot \left(a \cdot c\right)}}{a} \]

      16. *-commutative 36.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \color{blue}{\left(c \cdot a\right)}}}{a} \]

   7. Simplified 36.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{b - \sqrt{\color{blue}{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a} \]

   8. Taylor expanded in b around inf 68.0%

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

      1. associate-*r/ 68.0%

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

      2. *-commutative 68.0%

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

   10. Simplified 68.0%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 15: 35.4% 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 55.3%

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

   1. /-rgt-identity 55.3%

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

   2. metadata-eval 55.3%

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

   3. associate-/r/ 55.3%

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

   4. metadata-eval 55.3%

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

   5. metadata-eval 55.3%

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

   6. times-frac 55.3%

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

   7. *-commutative 55.3%

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

   8. times-frac 55.2%

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

   9. *-commutative 55.2%

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

   10. associate-/r* 55.1%

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

   11. associate-*l/ 55.2%

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

3. Simplified 55.1%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]

4. Taylor expanded in b around inf 33.3%

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

5. Final simplification 33.3%

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

Reproduce

herbie shell --seed 2023230 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))