Cubic critical, narrow range

Percentage Accurate: 55.3% → 99.6%

Time: 20.1s

Alternatives: 9

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ c (- (- b) (sqrt (fma -3.0 (* c a) (pow b 2.0))))))

C

double code(double a, double b, double c) {
	return c / (-b - sqrt(fma(-3.0, (c * a), pow(b, 2.0))));
}

Julia

function code(a, b, c)
	return Float64(c / Float64(Float64(-b) - sqrt(fma(-3.0, Float64(c * a), (b ^ 2.0)))))
end

Wolfram

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

Derivation

1. Initial program 57.5%

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

   1. neg-sub0 57.5%

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

   2. flip-- 57.3%

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

   3. metadata-eval 57.3%

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

   4. pow2 57.3%

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

   5. add-sqr-sqrt 56.0%

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

   6. sqrt-prod 57.3%

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

   7. sqr-neg 57.3%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. neg-sub0 1.6%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 57.3%

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

   14. sqr-neg 57.3%

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

   15. sqrt-prod 56.0%

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

   16. add-sqr-sqrt 57.3%

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

4. Applied egg-rr 57.3%

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

   1. neg-sub0 57.3%

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

6. Simplified 57.3%

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

   1. flip-+ 57.5%

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

8. Applied egg-rr 59.0%

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

   1. associate--r- 99.3%

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

10. Simplified 99.3%

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

    1. div-inv 99.3%

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

    2. +-commutative 99.3%

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

    3. *-commutative 99.3%

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

    4. fma-define 99.3%

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

    5. *-commutative 99.3%

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

    6. neg-mul-1 99.3%

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

    7. unpow-prod-down 99.3%

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

    8. metadata-eval 99.3%

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

    9. *-un-lft-identity 99.3%

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

    10. *-commutative 99.3%

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

12. Applied egg-rr 99.3%

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

    1. *-commutative 99.3%

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

    2. times-frac 99.3%

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

    3. *-lft-identity 99.3%

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

    4. associate-/r* 99.5%

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

    5. fma-undefine 99.5%

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

    6. +-inverses 99.5%

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

    7. +-rgt-identity 99.5%

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

    8. associate-*r* 99.5%

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

    9. *-commutative 99.5%

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

    10. *-commutative 99.5%

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

    11. cancel-sign-sub-inv 99.5%

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

    12. metadata-eval 99.5%

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

    13. +-commutative 99.5%

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

14. Simplified 99.5%

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

15. Taylor expanded in c around 0 99.6%

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

Alternative 2: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{c \cdot a}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -1.0)
     t_0
     (/
      1.0
      (+
       (* -2.0 (/ b c))
       (* a (+ (* 1.125 (/ (* c a) (pow b 3.0))) (* 1.5 (/ 1.0 b)))))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (a * ((1.125 * ((c * a) / pow(b, 3.0))) + (1.5 * (1.0 / b)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-1.0d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / (((-2.0d0) * (b / c)) + (a * ((1.125d0 * ((c * a) / (b ** 3.0d0))) + (1.5d0 * (1.0d0 / b)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (a * ((1.125 * ((c * a) / Math.pow(b, 3.0))) + (1.5 * (1.0 / b)))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -1.0:
		tmp = t_0
	else:
		tmp = 1.0 / ((-2.0 * (b / c)) + (a * ((1.125 * ((c * a) / math.pow(b, 3.0))) + (1.5 * (1.0 / b)))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(a * Float64(Float64(1.125 * Float64(Float64(c * a) / (b ^ 3.0))) + Float64(1.5 * Float64(1.0 / b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -1.0)
		tmp = t_0;
	else
		tmp = 1.0 / ((-2.0 * (b / c)) + (a * ((1.125 * ((c * a) / (b ^ 3.0))) + (1.5 * (1.0 / b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], t$95$0, N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(1.125 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1

   1. Initial program 85.8%

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

   if -1 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 52.4%

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

      1. neg-sub0 52.4%

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

      2. flip-- 52.2%

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

      3. metadata-eval 52.2%

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

      4. pow2 52.3%

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

      5. add-sqr-sqrt 51.0%

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

      6. sqrt-prod 52.3%

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

      7. sqr-neg 52.3%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 1.6%

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

      10. sub-neg 1.6%

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

      11. neg-sub0 1.6%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 52.3%

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

      14. sqr-neg 52.3%

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

      15. sqrt-prod 51.0%

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

      16. add-sqr-sqrt 52.3%

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

   4. Applied egg-rr 52.3%

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

      1. neg-sub0 52.3%

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

   6. Simplified 52.3%

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

      1. flip-+ 52.5%

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

   8. Applied egg-rr 53.8%

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

      1. associate--r- 99.3%

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

   10. Simplified 99.3%

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

       1. clear-num 99.2%

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

       2. inv-pow 99.2%

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

   12. Applied egg-rr 99.2%

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

       1. unpow-1 99.2%

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

       2. associate-/r/ 99.2%

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

       3. *-commutative 99.2%

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

       4. *-lft-identity 99.2%

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

       5. times-frac 99.0%

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

       6. metadata-eval 99.0%

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

       7. fma-undefine 99.0%

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

       8. +-inverses 99.0%

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

       9. +-rgt-identity 99.0%

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

       10. associate-*r* 99.0%

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

       11. *-commutative 99.0%

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

       12. *-commutative 99.0%

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

       13. cancel-sign-sub-inv 99.0%

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

   14. Simplified 99.0%

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

   15. Taylor expanded in a around 0 91.4%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{a \cdot c}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{c \cdot a}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 3\right)\\ t_1 := \frac{\sqrt{b \cdot b - t\_0} - b}{a \cdot 3}\\ \mathbf{if}\;t\_1 \leq -0.003112:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{a \cdot 3}}{1.5 \cdot \frac{c \cdot a}{b} - b \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 3.0))) (t_1 (/ (- (sqrt (- (* b b) t_0)) b) (* a 3.0))))
   (if (<= t_1 -0.003112)
     t_1
     (/ (/ t_0 (* a 3.0)) (- (* 1.5 (/ (* c a) b)) (* b 2.0))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	double t_1 = (sqrt(((b * b) - t_0)) - b) / (a * 3.0);
	double tmp;
	if (t_1 <= -0.003112) {
		tmp = t_1;
	} else {
		tmp = (t_0 / (a * 3.0)) / ((1.5 * ((c * a) / b)) - (b * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (a * 3.0d0)
    t_1 = (sqrt(((b * b) - t_0)) - b) / (a * 3.0d0)
    if (t_1 <= (-0.003112d0)) then
        tmp = t_1
    else
        tmp = (t_0 / (a * 3.0d0)) / ((1.5d0 * ((c * a) / b)) - (b * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	double t_1 = (Math.sqrt(((b * b) - t_0)) - b) / (a * 3.0);
	double tmp;
	if (t_1 <= -0.003112) {
		tmp = t_1;
	} else {
		tmp = (t_0 / (a * 3.0)) / ((1.5 * ((c * a) / b)) - (b * 2.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (a * 3.0)
	t_1 = (math.sqrt(((b * b) - t_0)) - b) / (a * 3.0)
	tmp = 0
	if t_1 <= -0.003112:
		tmp = t_1
	else:
		tmp = (t_0 / (a * 3.0)) / ((1.5 * ((c * a) / b)) - (b * 2.0))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 3.0))
	t_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_1 <= -0.003112)
		tmp = t_1;
	else
		tmp = Float64(Float64(t_0 / Float64(a * 3.0)) / Float64(Float64(1.5 * Float64(Float64(c * a) / b)) - Float64(b * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a * 3.0);
	t_1 = (sqrt(((b * b) - t_0)) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_1 <= -0.003112)
		tmp = t_1;
	else
		tmp = (t_0 / (a * 3.0)) / ((1.5 * ((c * a) / b)) - (b * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.003112], t$95$1, N[(N[(t$95$0 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0031120000000000002

   1. Initial program 78.8%

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

   if -0.0031120000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 47.5%

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

      1. neg-sub0 47.5%

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

      2. flip-- 47.4%

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

      3. metadata-eval 47.4%

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

      4. pow2 47.5%

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

      5. add-sqr-sqrt 46.1%

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

      6. sqrt-prod 47.5%

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

      7. sqr-neg 47.5%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 1.6%

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

      10. sub-neg 1.6%

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

      11. neg-sub0 1.6%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 47.5%

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

      14. sqr-neg 47.5%

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

      15. sqrt-prod 46.1%

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

      16. add-sqr-sqrt 47.5%

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

   4. Applied egg-rr 47.5%

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

      1. neg-sub0 47.5%

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

   6. Simplified 47.5%

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

      1. flip-+ 47.6%

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

   8. Applied egg-rr 48.9%

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

      1. associate--r- 99.3%

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

   10. Simplified 99.3%

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

       1. div-inv 99.3%

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

       2. +-commutative 99.3%

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

       3. *-commutative 99.3%

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

       4. fma-define 99.3%

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

       5. *-commutative 99.3%

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

       6. neg-mul-1 99.3%

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

       7. unpow-prod-down 99.3%

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

       8. metadata-eval 99.3%

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

       9. *-un-lft-identity 99.3%

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

       10. *-commutative 99.3%

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

   12. Applied egg-rr 99.3%

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

       1. *-commutative 99.3%

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

       2. times-frac 99.3%

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

       3. *-lft-identity 99.3%

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

       4. associate-/r* 99.5%

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

       5. fma-undefine 99.5%

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

       6. +-inverses 99.5%

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

       7. +-rgt-identity 99.5%

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

       8. associate-*r* 99.5%

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

       9. *-commutative 99.5%

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

       10. *-commutative 99.5%

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

       11. cancel-sign-sub-inv 99.5%

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

       12. metadata-eval 99.5%

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

       13. +-commutative 99.5%

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

   14. Simplified 99.5%

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

   15. Taylor expanded in c around 0 90.2%

       \[\leadsto \frac{\frac{c \cdot \left(a \cdot 3\right)}{a \cdot 3}}{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.003112:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot 3\right)}{a \cdot 3}}{1.5 \cdot \frac{c \cdot a}{b} - b \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{c \cdot \left(a \cdot 3\right)}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} + -3 \cdot \left(c \cdot a\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (* c (* a 3.0)) (* a 3.0))
  (- (- b) (sqrt (+ (pow b 2.0) (* -3.0 (* c a)))))))

C

double code(double a, double b, double c) {
	return ((c * (a * 3.0)) / (a * 3.0)) / (-b - sqrt((pow(b, 2.0) + (-3.0 * (c * 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 = ((c * (a * 3.0d0)) / (a * 3.0d0)) / (-b - sqrt(((b ** 2.0d0) + ((-3.0d0) * (c * a)))))
end function

Java

public static double code(double a, double b, double c) {
	return ((c * (a * 3.0)) / (a * 3.0)) / (-b - Math.sqrt((Math.pow(b, 2.0) + (-3.0 * (c * a)))));
}

Python

def code(a, b, c):
	return ((c * (a * 3.0)) / (a * 3.0)) / (-b - math.sqrt((math.pow(b, 2.0) + (-3.0 * (c * a)))))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * Float64(a * 3.0)) / Float64(a * 3.0)) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) + Float64(-3.0 * Float64(c * a))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * (a * 3.0)) / (a * 3.0)) / (-b - sqrt(((b ^ 2.0) + (-3.0 * (c * a)))));
end

Wolfram

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

Derivation

1. Initial program 57.5%

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

   1. neg-sub0 57.5%

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

   2. flip-- 57.3%

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

   3. metadata-eval 57.3%

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

   4. pow2 57.3%

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

   5. add-sqr-sqrt 56.0%

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

   6. sqrt-prod 57.3%

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

   7. sqr-neg 57.3%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. neg-sub0 1.6%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 57.3%

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

   14. sqr-neg 57.3%

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

   15. sqrt-prod 56.0%

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

   16. add-sqr-sqrt 57.3%

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

4. Applied egg-rr 57.3%

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

   1. neg-sub0 57.3%

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

6. Simplified 57.3%

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

   1. flip-+ 57.5%

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

8. Applied egg-rr 59.0%

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

   1. associate--r- 99.3%

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

10. Simplified 99.3%

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

    1. div-inv 99.3%

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

    2. +-commutative 99.3%

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

    3. *-commutative 99.3%

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

    4. fma-define 99.3%

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

    5. *-commutative 99.3%

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

    6. neg-mul-1 99.3%

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

    7. unpow-prod-down 99.3%

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

    8. metadata-eval 99.3%

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

    9. *-un-lft-identity 99.3%

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

    10. *-commutative 99.3%

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

12. Applied egg-rr 99.3%

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

    1. *-commutative 99.3%

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

    2. times-frac 99.3%

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

    3. *-lft-identity 99.3%

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

    4. associate-/r* 99.5%

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

    5. fma-undefine 99.5%

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

    6. +-inverses 99.5%

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

    7. +-rgt-identity 99.5%

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

    8. associate-*r* 99.5%

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

    9. *-commutative 99.5%

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

    10. *-commutative 99.5%

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

    11. cancel-sign-sub-inv 99.5%

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

    12. metadata-eval 99.5%

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

    13. +-commutative 99.5%

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

14. Simplified 99.5%

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

    1. fma-undefine 99.5%

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

16. Applied egg-rr 99.5%

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

17. Final simplification 99.5%

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

Alternative 5: 82.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{c \cdot \left(a \cdot 3\right)}{a \cdot 3}}{1.5 \cdot \frac{c \cdot a}{b} - b \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (/ (* c (* a 3.0)) (* a 3.0)) (- (* 1.5 (/ (* c a) b)) (* b 2.0))))

C

double code(double a, double b, double c) {
	return ((c * (a * 3.0)) / (a * 3.0)) / ((1.5 * ((c * a) / b)) - (b * 2.0));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((c * (a * 3.0d0)) / (a * 3.0d0)) / ((1.5d0 * ((c * a) / b)) - (b * 2.0d0))
end function

Java

public static double code(double a, double b, double c) {
	return ((c * (a * 3.0)) / (a * 3.0)) / ((1.5 * ((c * a) / b)) - (b * 2.0));
}

Python

def code(a, b, c):
	return ((c * (a * 3.0)) / (a * 3.0)) / ((1.5 * ((c * a) / b)) - (b * 2.0))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * Float64(a * 3.0)) / Float64(a * 3.0)) / Float64(Float64(1.5 * Float64(Float64(c * a) / b)) - Float64(b * 2.0)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * (a * 3.0)) / (a * 3.0)) / ((1.5 * ((c * a) / b)) - (b * 2.0));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.5%

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

   1. neg-sub0 57.5%

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

   2. flip-- 57.3%

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

   3. metadata-eval 57.3%

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

   4. pow2 57.3%

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

   5. add-sqr-sqrt 56.0%

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

   6. sqrt-prod 57.3%

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

   7. sqr-neg 57.3%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. neg-sub0 1.6%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 57.3%

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

   14. sqr-neg 57.3%

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

   15. sqrt-prod 56.0%

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

   16. add-sqr-sqrt 57.3%

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

4. Applied egg-rr 57.3%

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

   1. neg-sub0 57.3%

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

6. Simplified 57.3%

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

   1. flip-+ 57.5%

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

8. Applied egg-rr 59.0%

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

   1. associate--r- 99.3%

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

10. Simplified 99.3%

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

    1. div-inv 99.3%

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

    2. +-commutative 99.3%

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

    3. *-commutative 99.3%

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

    4. fma-define 99.3%

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

    5. *-commutative 99.3%

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

    6. neg-mul-1 99.3%

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

    7. unpow-prod-down 99.3%

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

    8. metadata-eval 99.3%

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

    9. *-un-lft-identity 99.3%

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

    10. *-commutative 99.3%

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

12. Applied egg-rr 99.3%

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

    1. *-commutative 99.3%

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

    2. times-frac 99.3%

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

    3. *-lft-identity 99.3%

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

    4. associate-/r* 99.5%

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

    5. fma-undefine 99.5%

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

    6. +-inverses 99.5%

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

    7. +-rgt-identity 99.5%

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

    8. associate-*r* 99.5%

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

    9. *-commutative 99.5%

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

    10. *-commutative 99.5%

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

    11. cancel-sign-sub-inv 99.5%

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

    12. metadata-eval 99.5%

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

    13. +-commutative 99.5%

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

14. Simplified 99.5%

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

15. Taylor expanded in c around 0 81.6%

    \[\leadsto \frac{\frac{c \cdot \left(a \cdot 3\right)}{a \cdot 3}}{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}} \]

16. Final simplification 81.6%

    \[\leadsto \frac{\frac{c \cdot \left(a \cdot 3\right)}{a \cdot 3}}{1.5 \cdot \frac{c \cdot a}{b} - b \cdot 2} \]
17. Add Preprocessing

Alternative 6: 82.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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 = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

   1. neg-sub0 57.5%

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

   2. flip-- 57.3%

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

   3. metadata-eval 57.3%

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

   4. pow2 57.3%

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

   5. add-sqr-sqrt 56.0%

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

   6. sqrt-prod 57.3%

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

   7. sqr-neg 57.3%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. neg-sub0 1.6%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 57.3%

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

   14. sqr-neg 57.3%

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

   15. sqrt-prod 56.0%

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

   16. add-sqr-sqrt 57.3%

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

4. Applied egg-rr 57.3%

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

   1. neg-sub0 57.3%

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

6. Simplified 57.3%

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

   1. flip-+ 57.5%

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

8. Applied egg-rr 59.0%

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

   1. associate--r- 99.3%

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

10. Simplified 99.3%

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

    1. clear-num 99.2%

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

    2. inv-pow 99.2%

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

12. Applied egg-rr 99.2%

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

    1. unpow-1 99.2%

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

    2. associate-/r/ 99.2%

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

    3. *-commutative 99.2%

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

    4. *-lft-identity 99.2%

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

    5. times-frac 99.0%

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

    6. metadata-eval 99.0%

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

    7. fma-undefine 99.0%

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

    8. +-inverses 99.0%

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

    9. +-rgt-identity 99.0%

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

    10. associate-*r* 99.0%

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

    11. *-commutative 99.0%

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

    12. *-commutative 99.0%

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

    13. cancel-sign-sub-inv 99.0%

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

14. Simplified 99.0%

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

15. Taylor expanded in a around 0 81.6%

    \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
16. Add Preprocessing

Alternative 7: 64.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c * (-0.5d0)) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in b around inf 62.7%

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

   1. associate-*r/ 62.7%

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

   2. *-commutative 62.7%

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

5. Simplified 62.7%

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

Alternative 8: 64.4% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * ((-0.5d0) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in b around inf 62.7%

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

   1. associate-*r/ 62.7%

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

   2. *-commutative 62.7%

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

5. Simplified 62.7%

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

6. Taylor expanded in c around 0 62.7%

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

   1. *-commutative 62.7%

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

   2. associate-*l/ 62.7%

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

   3. associate-*r/ 62.6%

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

8. Simplified 62.6%

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

Alternative 9: 3.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 57.5%

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

   1. neg-sub0 57.5%

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

   2. flip-- 57.3%

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

   3. metadata-eval 57.3%

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

   4. pow2 57.3%

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

   5. add-sqr-sqrt 56.0%

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

   6. sqrt-prod 57.3%

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

   7. sqr-neg 57.3%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. neg-sub0 1.6%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 57.3%

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

   14. sqr-neg 57.3%

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

   15. sqrt-prod 56.0%

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

   16. add-sqr-sqrt 57.3%

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

4. Applied egg-rr 57.3%

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

   1. neg-sub0 57.3%

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

6. Simplified 57.3%

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

7. Taylor expanded in a around 0 3.2%

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

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

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

   4. mul0-lft 3.2%

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

   5. metadata-eval 3.2%

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

9. Simplified 3.2%

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

10. Taylor expanded in a around 0 3.2%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))