Cubic critical, medium range

Percentage Accurate: 31.4% → 99.4%

Time: 19.5s

Alternatives: 7

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.4% 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{1}{c} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ -1.0 (* (/ 1.0 c) (+ b (sqrt (fma b b (* (* c a) -3.0)))))))

C

double code(double a, double b, double c) {
	return -1.0 / ((1.0 / c) * (b + sqrt(fma(b, b, ((c * a) * -3.0)))));
}

Julia

function code(a, b, c)
	return Float64(-1.0 / Float64(Float64(1.0 / c) * Float64(b + sqrt(fma(b, b, Float64(Float64(c * a) * -3.0))))))
end

Wolfram

code[a_, b_, c_] := N[(-1.0 / N[(N[(1.0 / c), $MachinePrecision] * N[(b + N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.7%

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

   1. add-cube-cbrt 31.7%

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

   2. pow3 31.7%

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

4. Applied egg-rr 31.7%

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

   1. flip-+ 31.8%

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

   2. pow2 31.8%

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

   3. add-sqr-sqrt 32.6%

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

   4. pow2 32.6%

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

   5. *-commutative 32.6%

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

   6. *-commutative 32.6%

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

   7. pow2 32.6%

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

   8. *-commutative 32.6%

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

   9. *-commutative 32.6%

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

6. Applied egg-rr 32.6%

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

   1. associate--r- 97.8%

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

   2. associate-*r* 97.7%

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

   3. *-commutative 97.7%

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

   4. associate-*r* 97.8%

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

   5. associate-*r* 97.8%

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

   6. *-commutative 97.8%

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

   7. associate-*r* 97.8%

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

8. Simplified 97.8%

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

   1. rem-cube-cbrt 99.1%

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

   2. clear-num 99.1%

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

   3. inv-pow 99.1%

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

10. Applied egg-rr 99.1%

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

    1. unpow-1 99.1%

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

    2. associate-/r/ 99.1%

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

    3. *-commutative 99.1%

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

    4. fma-undefine 99.1%

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

    5. +-inverses 99.1%

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

    6. +-rgt-identity 99.1%

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

    7. associate-*r* 99.1%

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

    8. *-commutative 99.1%

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

    9. associate-*r* 99.1%

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

    10. *-commutative 99.1%

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

    11. associate-*r* 99.1%

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

    12. unpow2 99.1%

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

    13. associate-*r* 99.1%

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

    14. *-commutative 99.1%

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

    15. associate-*r* 99.1%

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

12. Simplified 99.1%

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

13. Taylor expanded in a around 0 99.4%

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

14. Final simplification 99.4%

    \[\leadsto \frac{-1}{\frac{1}{c} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}\right)} \]
15. Add Preprocessing

Alternative 2: 94.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -10000000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{c \cdot a}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -10000000.0)
   (/ (- (sqrt (- (* b b) (* (* c a) 3.0))) b) (* a 3.0))
   (/
    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 tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -10000000.0) {
		tmp = (sqrt(((b * b) - ((c * a) * 3.0))) - b) / (a * 3.0);
	} 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) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-10000000.0d0)) then
        tmp = (sqrt(((b * b) - ((c * a) * 3.0d0))) - b) / (a * 3.0d0)
    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 tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -10000000.0) {
		tmp = (Math.sqrt(((b * b) - ((c * a) * 3.0))) - b) / (a * 3.0);
	} 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):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -10000000.0:
		tmp = (math.sqrt(((b * b) - ((c * a) * 3.0))) - b) / (a * 3.0)
	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)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -10000000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(c * a) * 3.0))) - b) / Float64(a * 3.0));
	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)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -10000000.0)
		tmp = (sqrt(((b * b) - ((c * a) * 3.0))) - b) / (a * 3.0);
	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_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -10000000.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], 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)) < -1e7

   1. Initial program 95.9%

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

      1. sqr-neg 95.9%

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

      2. sqr-neg 95.9%

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

      3. associate-*l* 95.9%

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

   3. Simplified 95.9%

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

   if -1e7 < (/.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 29.4%

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

      1. add-cube-cbrt 29.3%

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

      2. pow3 29.3%

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

   4. Applied egg-rr 29.3%

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

      1. flip-+ 29.5%

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

      2. pow2 29.5%

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

      3. add-sqr-sqrt 30.3%

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

      4. pow2 30.3%

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

      5. *-commutative 30.3%

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

      6. *-commutative 30.3%

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

      7. pow2 30.3%

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

      8. *-commutative 30.3%

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

      9. *-commutative 30.3%

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

   6. Applied egg-rr 30.3%

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

      1. associate--r- 97.8%

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

      2. associate-*r* 97.7%

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

      3. *-commutative 97.7%

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

      4. associate-*r* 97.8%

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

      5. associate-*r* 97.8%

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

      6. *-commutative 97.8%

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

      7. associate-*r* 97.8%

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

   8. Simplified 97.8%

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

      1. rem-cube-cbrt 99.2%

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

      2. clear-num 99.1%

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

      3. inv-pow 99.1%

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

   10. Applied egg-rr 99.1%

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

       1. unpow-1 99.1%

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

       2. associate-/r/ 99.1%

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

       3. *-commutative 99.1%

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

       4. fma-undefine 99.1%

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

       5. +-inverses 99.1%

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

       6. +-rgt-identity 99.1%

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

       7. associate-*r* 99.1%

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

       8. *-commutative 99.1%

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

       9. associate-*r* 99.1%

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

       10. *-commutative 99.1%

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

       11. associate-*r* 99.1%

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

       12. unpow2 99.1%

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

       13. associate-*r* 99.1%

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

       14. *-commutative 99.1%

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

       15. associate-*r* 99.1%

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

   12. Simplified 99.1%

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

   13. Taylor expanded in a around 0 94.4%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -10000000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + a \cdot \left(1.125 \cdot \frac{c \cdot a}{{b}^{3}} + 1.5 \cdot \frac{1}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -10000000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -10000000.0)
   (/ (- (sqrt (- (* b b) (* (* c a) 3.0))) b) (* a 3.0))
   (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -10000000.0) {
		tmp = (sqrt(((b * b) - ((c * a) * 3.0))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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 (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-10000000.0d0)) then
        tmp = (sqrt(((b * b) - ((c * a) * 3.0d0))) - b) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -10000000.0) {
		tmp = (Math.sqrt(((b * b) - ((c * a) * 3.0))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -10000000.0:
		tmp = (math.sqrt(((b * b) - ((c * a) * 3.0))) - b) / (a * 3.0)
	else:
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -10000000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(c * a) * 3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -10000000.0)
		tmp = (sqrt(((b * b) - ((c * a) * 3.0))) - b) / (a * 3.0);
	else
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -10000000.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $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)) < -1e7

   1. Initial program 95.9%

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

      1. sqr-neg 95.9%

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

      2. sqr-neg 95.9%

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

      3. associate-*l* 95.9%

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

   3. Simplified 95.9%

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

   if -1e7 < (/.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 29.4%

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

      1. add-cube-cbrt 29.3%

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

      2. pow3 29.3%

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

   4. Applied egg-rr 29.3%

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

      1. flip-+ 29.5%

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

      2. pow2 29.5%

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

      3. add-sqr-sqrt 30.3%

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

      4. pow2 30.3%

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

      5. *-commutative 30.3%

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

      6. *-commutative 30.3%

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

      7. pow2 30.3%

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

      8. *-commutative 30.3%

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

      9. *-commutative 30.3%

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

   6. Applied egg-rr 30.3%

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

      1. associate--r- 97.8%

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

      2. associate-*r* 97.7%

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

      3. *-commutative 97.7%

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

      4. associate-*r* 97.8%

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

      5. associate-*r* 97.8%

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

      6. *-commutative 97.8%

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

      7. associate-*r* 97.8%

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

   8. Simplified 97.8%

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

      1. rem-cube-cbrt 99.2%

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

      2. clear-num 99.1%

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

      3. inv-pow 99.1%

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

   10. Applied egg-rr 99.1%

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

       1. unpow-1 99.1%

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

       2. associate-/r/ 99.1%

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

       3. *-commutative 99.1%

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

       4. fma-undefine 99.1%

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

       5. +-inverses 99.1%

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

       6. +-rgt-identity 99.1%

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

       7. associate-*r* 99.1%

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

       8. *-commutative 99.1%

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

       9. associate-*r* 99.1%

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

       10. *-commutative 99.1%

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

       11. associate-*r* 99.1%

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

       12. unpow2 99.1%

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

       13. associate-*r* 99.1%

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

       14. *-commutative 99.1%

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

       15. associate-*r* 99.1%

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

   12. Simplified 99.1%

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

   13. Taylor expanded in a around 0 91.8%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -10000000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% 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 31.7%

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

   1. add-cube-cbrt 31.7%

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

   2. pow3 31.7%

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

4. Applied egg-rr 31.7%

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

   1. flip-+ 31.8%

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

   2. pow2 31.8%

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

   3. add-sqr-sqrt 32.6%

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

   4. pow2 32.6%

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

   5. *-commutative 32.6%

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

   6. *-commutative 32.6%

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

   7. pow2 32.6%

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

   8. *-commutative 32.6%

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

   9. *-commutative 32.6%

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

6. Applied egg-rr 32.6%

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

   1. associate--r- 97.8%

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

   2. associate-*r* 97.7%

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

   3. *-commutative 97.7%

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

   4. associate-*r* 97.8%

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

   5. associate-*r* 97.8%

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

   6. *-commutative 97.8%

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

   7. associate-*r* 97.8%

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

8. Simplified 97.8%

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

   1. rem-cube-cbrt 99.1%

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

   2. clear-num 99.1%

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

   3. inv-pow 99.1%

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

10. Applied egg-rr 99.1%

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

    1. unpow-1 99.1%

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

    2. associate-/r/ 99.1%

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

    3. *-commutative 99.1%

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

    4. fma-undefine 99.1%

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

    5. +-inverses 99.1%

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

    6. +-rgt-identity 99.1%

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

    7. associate-*r* 99.1%

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

    8. *-commutative 99.1%

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

    9. associate-*r* 99.1%

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

    10. *-commutative 99.1%

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

    11. associate-*r* 99.1%

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

    12. unpow2 99.1%

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

    13. associate-*r* 99.1%

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

    14. *-commutative 99.1%

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

    15. associate-*r* 99.1%

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

12. Simplified 99.1%

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

13. Taylor expanded in a around 0 89.8%

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

Alternative 5: 81.2% 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 31.7%

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

   1. sqr-neg 31.7%

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

   2. sqr-neg 31.7%

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

   3. associate-*l* 31.7%

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

3. Simplified 31.7%

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

5. Taylor expanded in b around inf 80.9%

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

   1. associate-*r/ 80.9%

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

   2. *-commutative 80.9%

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

7. Simplified 80.9%

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

Alternative 6: 81.0% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

   1. sqr-neg 31.7%

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

   2. sqr-neg 31.7%

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

   3. associate-*l* 31.7%

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

3. Simplified 31.7%

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

5. Taylor expanded in b around inf 80.9%

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

   1. associate-*r/ 80.9%

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

   2. *-commutative 80.9%

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

7. Simplified 80.9%

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

   1. clear-num 80.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{b}{c \cdot -0.5}}} \]

   2. inv-pow 80.7%

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

9. Applied egg-rr 80.7%

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

    1. unpow-1 80.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{b}{c \cdot -0.5}}} \]

    2. associate-/r* 80.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{b}{c}}{-0.5}}} \]

11. Simplified 80.7%

    \[\leadsto \color{blue}{\frac{1}{\frac{\frac{b}{c}}{-0.5}}} \]
12. Step-by-step derivation

    1. *-un-lft-identity 80.7%

       \[\leadsto \color{blue}{1 \cdot \frac{1}{\frac{\frac{b}{c}}{-0.5}}} \]

    2. associate-/r/ 80.7%

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

13. Applied egg-rr 80.7%

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

    1. *-lft-identity 80.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{b}{c}} \cdot -0.5} \]

    2. associate-*l/ 80.7%

       \[\leadsto \color{blue}{\frac{1 \cdot -0.5}{\frac{b}{c}}} \]

    3. metadata-eval 80.7%

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

15. Simplified 80.7%

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

Alternative 7: 81.0% 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 31.7%

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

   1. sqr-neg 31.7%

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

   2. sqr-neg 31.7%

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

   3. associate-*l* 31.7%

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

3. Simplified 31.7%

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

5. Taylor expanded in b around inf 80.9%

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

   1. associate-*r/ 80.9%

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

   2. *-commutative 80.9%

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

7. Simplified 80.9%

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

8. Taylor expanded in c around 0 80.9%

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

   1. associate-*r/ 80.9%

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

   2. *-commutative 80.9%

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

   3. associate-*r/ 80.7%

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

10. Simplified 80.7%

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

Reproduce

herbie shell --seed 2024114 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))