Quadratic roots, narrow range

Percentage Accurate: 55.4% → 90.2%

Time: 16.5s

Alternatives: 11

Speedup: 29.0×

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

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 80:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(a \cdot \left(-2 \cdot \frac{a \cdot \left(\left({c}^{2} + -2.5 \cdot {c}^{2}\right) - {c}^{2} \cdot -0.5\right)}{{b}^{5}} + \frac{c}{{b}^{3}}\right) + \frac{1}{b}\right) - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* c (* a 4.0)))))
   (if (<= b 80.0)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 2.0 a))
     (/
      1.0
      (-
       (*
        a
        (+
         (*
          a
          (+
           (*
            -2.0
            (/
             (*
              a
              (- (+ (pow c 2.0) (* -2.5 (pow c 2.0))) (* (pow c 2.0) -0.5)))
             (pow b 5.0)))
           (/ c (pow b 3.0))))
         (/ 1.0 b)))
       (/ b c))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 80.0) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = 1.0 / ((a * ((a * ((-2.0 * ((a * ((pow(c, 2.0) + (-2.5 * pow(c, 2.0))) - (pow(c, 2.0) * -0.5))) / pow(b, 5.0))) + (c / pow(b, 3.0)))) + (1.0 / b))) - (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b ** 2.0d0) - (c * (a * 4.0d0))
    if (b <= 80.0d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (2.0d0 * a)
    else
        tmp = 1.0d0 / ((a * ((a * (((-2.0d0) * ((a * (((c ** 2.0d0) + ((-2.5d0) * (c ** 2.0d0))) - ((c ** 2.0d0) * (-0.5d0)))) / (b ** 5.0d0))) + (c / (b ** 3.0d0)))) + (1.0d0 / b))) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 80.0) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = 1.0 / ((a * ((a * ((-2.0 * ((a * ((Math.pow(c, 2.0) + (-2.5 * Math.pow(c, 2.0))) - (Math.pow(c, 2.0) * -0.5))) / Math.pow(b, 5.0))) + (c / Math.pow(b, 3.0)))) + (1.0 / b))) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.pow(b, 2.0) - (c * (a * 4.0))
	tmp = 0
	if b <= 80.0:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (2.0 * a)
	else:
		tmp = 1.0 / ((a * ((a * ((-2.0 * ((a * ((math.pow(c, 2.0) + (-2.5 * math.pow(c, 2.0))) - (math.pow(c, 2.0) * -0.5))) / math.pow(b, 5.0))) + (c / math.pow(b, 3.0)))) + (1.0 / b))) - (b / c))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64((b ^ 2.0) - Float64(c * Float64(a * 4.0)))
	tmp = 0.0
	if (b <= 80.0)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(1.0 / Float64(Float64(a * Float64(Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * Float64(Float64((c ^ 2.0) + Float64(-2.5 * (c ^ 2.0))) - Float64((c ^ 2.0) * -0.5))) / (b ^ 5.0))) + Float64(c / (b ^ 3.0)))) + Float64(1.0 / b))) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b ^ 2.0) - (c * (a * 4.0));
	tmp = 0.0;
	if (b <= 80.0)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	else
		tmp = 1.0 / ((a * ((a * ((-2.0 * ((a * (((c ^ 2.0) + (-2.5 * (c ^ 2.0))) - ((c ^ 2.0) * -0.5))) / (b ^ 5.0))) + (c / (b ^ 3.0)))) + (1.0 / b))) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 80.0], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[(a * N[(N[(N[Power[c, 2.0], $MachinePrecision] + N[(-2.5 * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 80

   1. Initial program 81.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 81.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 80.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 79.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 79.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 79.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 79.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 79.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 80.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 80.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. flip-+ 80.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow2 80.5%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      3. add-sqr-sqrt 80.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      4. pow1/3 79.3%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      5. pow-pow 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      6. metadata-eval 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      7. *-commutative 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      8. *-commutative 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      9. pow1/3 82.7%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      10. pow-pow 82.9%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      11. metadata-eval 82.9%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      12. *-commutative 82.9%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}}}{a \cdot 2} \]

      13. *-commutative 82.9%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}}}{a \cdot 2} \]

   10. Applied egg-rr 82.9%

       \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}}{a \cdot 2} \]

   if 80 < b

   1. Initial program 48.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 48.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 48.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 45.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 45.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 45.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 45.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 45.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 45.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 48.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 48.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. clear-num 48.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}} \]

      2. inv-pow 48.4%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1}} \]

      3. neg-mul-1 48.4%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1} \]

      4. fma-define 48.4%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}}\right)}^{-1} \]

      5. pow1/3 45.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      6. pow-pow 48.6%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      7. metadata-eval 48.6%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      8. *-commutative 48.6%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}\right)}^{-1} \]

      9. *-commutative 48.6%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}\right)}^{-1} \]

   10. Applied egg-rr 48.6%

       \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 48.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

       2. associate-/l* 48.6%

          \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

       3. *-commutative 48.6%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(a \cdot 4\right) \cdot c}}\right)}} \]

       4. *-commutative 48.6%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(4 \cdot a\right)} \cdot c}\right)}} \]

       5. associate-*r* 48.6%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}} \]

       6. cancel-sign-sub-inv 48.6%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)}}\right)}} \]

       7. metadata-eval 48.6%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} + \color{blue}{-4} \cdot \left(a \cdot c\right)}\right)}} \]

       8. +-commutative 48.6%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}\right)}} \]

       9. fma-define 48.6%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}\right)}} \]

       10. *-commutative 48.6%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, \color{blue}{c \cdot a}, {b}^{2}\right)}\right)}} \]

   12. Simplified 48.6%

       \[\leadsto \color{blue}{\frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right)}}} \]

   13. Taylor expanded in b around inf 94.7%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-2 \cdot \frac{-1 \cdot \left(a \cdot \left(c \cdot \left(-1 \cdot \left({a}^{2} \cdot c\right) + 0.5 \cdot \left({a}^{2} \cdot c\right)\right)\right)\right) + \left(-0.125 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {c}^{2}} + {a}^{3} \cdot {c}^{2}\right)}{{b}^{6}} + \left(-2 \cdot \frac{-1 \cdot \left({a}^{2} \cdot c\right) + 0.5 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + \frac{a}{{b}^{2}}\right)\right) - \frac{1}{c}\right)}} \]

   15. Simplified 94.7%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-0.125, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{c}^{2}}, {c}^{2} \cdot {a}^{3}\right) - \left(c \cdot a\right) \cdot \left(\left(c \cdot {a}^{2}\right) \cdot -0.5\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.5}{{b}^{4}}, \frac{a}{{b}^{2}}\right)\right) - \frac{1}{c}\right)}} \]

   16. Taylor expanded in a around 0 94.9%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-2 \cdot \frac{a \cdot \left(\left(-2.5 \cdot {c}^{2} + {c}^{2}\right) - -0.5 \cdot {c}^{2}\right)}{{b}^{5}} + \frac{c}{{b}^{3}}\right) + \frac{1}{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 80:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(a \cdot \left(-2 \cdot \frac{a \cdot \left(\left({c}^{2} + -2.5 \cdot {c}^{2}\right) - {c}^{2} \cdot -0.5\right)}{{b}^{5}} + \frac{c}{{b}^{3}}\right) + \frac{1}{b}\right) - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 80:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - {\left(\frac{c}{-b}\right)}^{2}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* c (* a 4.0)))))
   (if (<= b 80.0)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 2.0 a))
     (/
      (-
       (*
        a
        (-
         (*
          a
          (+
           (* -5.0 (/ (* a (pow c 4.0)) (pow b 6.0)))
           (* -2.0 (/ (pow c 3.0) (pow b 4.0)))))
         (pow (/ c (- b)) 2.0)))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 80.0) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = ((a * ((a * ((-5.0 * ((a * pow(c, 4.0)) / pow(b, 6.0))) + (-2.0 * (pow(c, 3.0) / pow(b, 4.0))))) - pow((c / -b), 2.0))) - c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b ** 2.0d0) - (c * (a * 4.0d0))
    if (b <= 80.0d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (2.0d0 * a)
    else
        tmp = ((a * ((a * (((-5.0d0) * ((a * (c ** 4.0d0)) / (b ** 6.0d0))) + ((-2.0d0) * ((c ** 3.0d0) / (b ** 4.0d0))))) - ((c / -b) ** 2.0d0))) - c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 80.0) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = ((a * ((a * ((-5.0 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 6.0))) + (-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 4.0))))) - Math.pow((c / -b), 2.0))) - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.pow(b, 2.0) - (c * (a * 4.0))
	tmp = 0
	if b <= 80.0:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (2.0 * a)
	else:
		tmp = ((a * ((a * ((-5.0 * ((a * math.pow(c, 4.0)) / math.pow(b, 6.0))) + (-2.0 * (math.pow(c, 3.0) / math.pow(b, 4.0))))) - math.pow((c / -b), 2.0))) - c) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64((b ^ 2.0) - Float64(c * Float64(a * 4.0)))
	tmp = 0.0
	if (b <= 80.0)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(a * Float64(Float64(a * Float64(Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 6.0))) + Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 4.0))))) - (Float64(c / Float64(-b)) ^ 2.0))) - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b ^ 2.0) - (c * (a * 4.0));
	tmp = 0.0;
	if (b <= 80.0)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	else
		tmp = ((a * ((a * ((-5.0 * ((a * (c ^ 4.0)) / (b ^ 6.0))) + (-2.0 * ((c ^ 3.0) / (b ^ 4.0))))) - ((c / -b) ^ 2.0))) - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 80.0], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(N[(a * N[(N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 80

   1. Initial program 81.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 81.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 80.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 79.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 79.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 79.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 79.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 79.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 80.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 80.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. flip-+ 80.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow2 80.5%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      3. add-sqr-sqrt 80.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      4. pow1/3 79.3%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      5. pow-pow 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      6. metadata-eval 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      7. *-commutative 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      8. *-commutative 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      9. pow1/3 82.7%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      10. pow-pow 82.9%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      11. metadata-eval 82.9%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      12. *-commutative 82.9%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}}}{a \cdot 2} \]

      13. *-commutative 82.9%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}}}{a \cdot 2} \]

   10. Applied egg-rr 82.9%

       \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}}{a \cdot 2} \]

   if 80 < b

   1. Initial program 48.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

   3. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 94.8%

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.25 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   6. Step-by-step derivation

   7. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \left(-0.25 \cdot \left(\frac{{c}^{4} \cdot {a}^{4}}{a} \cdot \frac{20}{{b}^{6}}\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}\right) - c\right)}{b}} \]

   8. Taylor expanded in a around 0 94.8%

      \[\leadsto \frac{\color{blue}{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{{c}^{2}}{{b}^{2}}\right) - c}}{b} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 94.8%

         \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \color{blue}{1 \cdot \frac{{c}^{2}}{{b}^{2}}}\right) - c}{b} \]

   10. Applied egg-rr 94.8%

       \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \color{blue}{1 \cdot \frac{{c}^{2}}{{b}^{2}}}\right) - c}{b} \]
   11. Step-by-step derivation

       1. *-lft-identity 94.8%

          \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \color{blue}{\frac{{c}^{2}}{{b}^{2}}}\right) - c}{b} \]

       2. unpow2 94.8%

          \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right) - c}{b} \]

       3. unpow2 94.8%

          \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{c \cdot c}{\color{blue}{b \cdot b}}\right) - c}{b} \]

       4. times-frac 94.8%

          \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \color{blue}{\frac{c}{b} \cdot \frac{c}{b}}\right) - c}{b} \]

       5. sqr-neg 94.8%

          \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \color{blue}{\left(-\frac{c}{b}\right) \cdot \left(-\frac{c}{b}\right)}\right) - c}{b} \]

       6. distribute-frac-neg 94.8%

          \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \color{blue}{\frac{-c}{b}} \cdot \left(-\frac{c}{b}\right)\right) - c}{b} \]

       7. distribute-frac-neg 94.8%

          \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{-c}{b} \cdot \color{blue}{\frac{-c}{b}}\right) - c}{b} \]

       8. unpow1 94.8%

          \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \color{blue}{{\left(\frac{-c}{b}\right)}^{1}} \cdot \frac{-c}{b}\right) - c}{b} \]

       9. pow-plus 94.8%

          \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \color{blue}{{\left(\frac{-c}{b}\right)}^{\left(1 + 1\right)}}\right) - c}{b} \]

       10. distribute-frac-neg 94.8%

           \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - {\color{blue}{\left(-\frac{c}{b}\right)}}^{\left(1 + 1\right)}\right) - c}{b} \]

       11. distribute-neg-frac2 94.8%

           \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - {\color{blue}{\left(\frac{c}{-b}\right)}}^{\left(1 + 1\right)}\right) - c}{b} \]

       12. metadata-eval 94.8%

           \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - {\left(\frac{c}{-b}\right)}^{\color{blue}{2}}\right) - c}{b} \]

   12. Simplified 94.8%

       \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \color{blue}{{\left(\frac{c}{-b}\right)}^{2}}\right) - c}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 80:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - {\left(\frac{c}{-b}\right)}^{2}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 80:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(c \cdot \left({a}^{3} \cdot \left(\frac{c \cdot -5}{{b}^{7}} - \frac{2}{a \cdot {b}^{5}}\right)\right) - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* c (* a 4.0)))))
   (if (<= b 80.0)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 2.0 a))
     (*
      c
      (+
       (*
        c
        (-
         (*
          c
          (*
           (pow a 3.0)
           (- (/ (* c -5.0) (pow b 7.0)) (/ 2.0 (* a (pow b 5.0))))))
         (/ a (pow b 3.0))))
       (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 80.0) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = c * ((c * ((c * (pow(a, 3.0) * (((c * -5.0) / pow(b, 7.0)) - (2.0 / (a * pow(b, 5.0)))))) - (a / pow(b, 3.0)))) + (-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 = (b ** 2.0d0) - (c * (a * 4.0d0))
    if (b <= 80.0d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (2.0d0 * a)
    else
        tmp = c * ((c * ((c * ((a ** 3.0d0) * (((c * (-5.0d0)) / (b ** 7.0d0)) - (2.0d0 / (a * (b ** 5.0d0)))))) - (a / (b ** 3.0d0)))) + ((-1.0d0) / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 80.0) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = c * ((c * ((c * (Math.pow(a, 3.0) * (((c * -5.0) / Math.pow(b, 7.0)) - (2.0 / (a * Math.pow(b, 5.0)))))) - (a / Math.pow(b, 3.0)))) + (-1.0 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.pow(b, 2.0) - (c * (a * 4.0))
	tmp = 0
	if b <= 80.0:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (2.0 * a)
	else:
		tmp = c * ((c * ((c * (math.pow(a, 3.0) * (((c * -5.0) / math.pow(b, 7.0)) - (2.0 / (a * math.pow(b, 5.0)))))) - (a / math.pow(b, 3.0)))) + (-1.0 / b))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64((b ^ 2.0) - Float64(c * Float64(a * 4.0)))
	tmp = 0.0
	if (b <= 80.0)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(c * Float64((a ^ 3.0) * Float64(Float64(Float64(c * -5.0) / (b ^ 7.0)) - Float64(2.0 / Float64(a * (b ^ 5.0)))))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b ^ 2.0) - (c * (a * 4.0));
	tmp = 0.0;
	if (b <= 80.0)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	else
		tmp = c * ((c * ((c * ((a ^ 3.0) * (((c * -5.0) / (b ^ 7.0)) - (2.0 / (a * (b ^ 5.0)))))) - (a / (b ^ 3.0)))) + (-1.0 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 80.0], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(c * N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(N[(c * -5.0), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 80

   1. Initial program 81.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 81.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 80.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 79.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 79.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 79.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 79.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 79.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 79.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 80.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 80.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. flip-+ 80.5%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow2 80.5%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      3. add-sqr-sqrt 80.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      4. pow1/3 79.3%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      5. pow-pow 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      6. metadata-eval 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      7. *-commutative 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      8. *-commutative 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      9. pow1/3 82.7%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      10. pow-pow 82.9%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      11. metadata-eval 82.9%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      12. *-commutative 82.9%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}}}{a \cdot 2} \]

      13. *-commutative 82.9%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}}}{a \cdot 2} \]

   10. Applied egg-rr 82.9%

       \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}}{a \cdot 2} \]

   if 80 < b

   1. Initial program 48.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

   3. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 94.8%

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.25 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   6. Step-by-step derivation

   7. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \left(-0.25 \cdot \left(\frac{{c}^{4} \cdot {a}^{4}}{a} \cdot \frac{20}{{b}^{6}}\right) - a \cdot \frac{{c}^{2}}{{b}^{2}}\right) - c\right)}{b}} \]

   8. Taylor expanded in c around 0 94.5%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + -2 \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{b}\right)} \]

   9. Taylor expanded in a around inf 94.5%

      \[\leadsto c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \color{blue}{\left({a}^{3} \cdot \left(-5 \cdot \frac{c}{{b}^{7}} - 2 \cdot \frac{1}{a \cdot {b}^{5}}\right)\right)}\right) - \frac{1}{b}\right) \]
   10. Step-by-step derivation

       1. associate-*r/ 94.5%

          \[\leadsto c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left({a}^{3} \cdot \left(\color{blue}{\frac{-5 \cdot c}{{b}^{7}}} - 2 \cdot \frac{1}{a \cdot {b}^{5}}\right)\right)\right) - \frac{1}{b}\right) \]

       2. associate-*r/ 94.5%

          \[\leadsto c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left({a}^{3} \cdot \left(\frac{-5 \cdot c}{{b}^{7}} - \color{blue}{\frac{2 \cdot 1}{a \cdot {b}^{5}}}\right)\right)\right) - \frac{1}{b}\right) \]

       3. metadata-eval 94.5%

          \[\leadsto c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left({a}^{3} \cdot \left(\frac{-5 \cdot c}{{b}^{7}} - \frac{\color{blue}{2}}{a \cdot {b}^{5}}\right)\right)\right) - \frac{1}{b}\right) \]

   11. Simplified 94.5%

       \[\leadsto c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \color{blue}{\left({a}^{3} \cdot \left(\frac{-5 \cdot c}{{b}^{7}} - \frac{2}{a \cdot {b}^{5}}\right)\right)}\right) - \frac{1}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 80:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(c \cdot \left({a}^{3} \cdot \left(\frac{c \cdot -5}{{b}^{7}} - \frac{2}{a \cdot {b}^{5}}\right)\right) - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 160:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{1}{b} + \frac{c \cdot a}{{b}^{3}}\right) - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* c (* a 4.0)))))
   (if (<= b 160.0)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 2.0 a))
     (/ 1.0 (- (* a (+ (/ 1.0 b) (/ (* c a) (pow b 3.0)))) (/ b c))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 160.0) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = 1.0 / ((a * ((1.0 / b) + ((c * a) / pow(b, 3.0)))) - (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b ** 2.0d0) - (c * (a * 4.0d0))
    if (b <= 160.0d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (2.0d0 * a)
    else
        tmp = 1.0d0 / ((a * ((1.0d0 / b) + ((c * a) / (b ** 3.0d0)))) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 160.0) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = 1.0 / ((a * ((1.0 / b) + ((c * a) / Math.pow(b, 3.0)))) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.pow(b, 2.0) - (c * (a * 4.0))
	tmp = 0
	if b <= 160.0:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (2.0 * a)
	else:
		tmp = 1.0 / ((a * ((1.0 / b) + ((c * a) / math.pow(b, 3.0)))) - (b / c))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64((b ^ 2.0) - Float64(c * Float64(a * 4.0)))
	tmp = 0.0
	if (b <= 160.0)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(1.0 / Float64(Float64(a * Float64(Float64(1.0 / b) + Float64(Float64(c * a) / (b ^ 3.0)))) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b ^ 2.0) - (c * (a * 4.0));
	tmp = 0.0;
	if (b <= 160.0)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (2.0 * a);
	else
		tmp = 1.0 / ((a * ((1.0 / b) + ((c * a) / (b ^ 3.0)))) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 160.0], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a * N[(N[(1.0 / b), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 160

   1. Initial program 80.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 80.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 79.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 78.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 78.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 78.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 77.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 77.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 77.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 80.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 80.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. flip-+ 79.8%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow2 79.8%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      3. add-sqr-sqrt 80.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      4. pow1/3 78.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      5. pow-pow 82.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      6. metadata-eval 82.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      7. *-commutative 82.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      8. *-commutative 82.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      9. pow1/3 82.0%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      10. pow-pow 82.1%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      11. metadata-eval 82.1%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      12. *-commutative 82.1%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}}}{a \cdot 2} \]

      13. *-commutative 82.1%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}}}{a \cdot 2} \]

   10. Applied egg-rr 82.1%

       \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}}{a \cdot 2} \]

   if 160 < b

   1. Initial program 46.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 46.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 46.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 43.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 46.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 46.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. clear-num 46.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}} \]

      2. inv-pow 46.5%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1}} \]

      3. neg-mul-1 46.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1} \]

      4. fma-define 46.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}}\right)}^{-1} \]

      5. pow1/3 43.9%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      6. pow-pow 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      7. metadata-eval 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      8. *-commutative 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}\right)}^{-1} \]

      9. *-commutative 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}\right)}^{-1} \]

   10. Applied egg-rr 46.7%

       \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 46.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

       2. associate-/l* 46.7%

          \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

       3. *-commutative 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(a \cdot 4\right) \cdot c}}\right)}} \]

       4. *-commutative 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(4 \cdot a\right)} \cdot c}\right)}} \]

       5. associate-*r* 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}} \]

       6. cancel-sign-sub-inv 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)}}\right)}} \]

       7. metadata-eval 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} + \color{blue}{-4} \cdot \left(a \cdot c\right)}\right)}} \]

       8. +-commutative 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}\right)}} \]

       9. fma-define 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}\right)}} \]

       10. *-commutative 46.7%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, \color{blue}{c \cdot a}, {b}^{2}\right)}\right)}} \]

   12. Simplified 46.7%

       \[\leadsto \color{blue}{\frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right)}}} \]

   13. Taylor expanded in b around inf 95.8%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-2 \cdot \frac{-1 \cdot \left(a \cdot \left(c \cdot \left(-1 \cdot \left({a}^{2} \cdot c\right) + 0.5 \cdot \left({a}^{2} \cdot c\right)\right)\right)\right) + \left(-0.125 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {c}^{2}} + {a}^{3} \cdot {c}^{2}\right)}{{b}^{6}} + \left(-2 \cdot \frac{-1 \cdot \left({a}^{2} \cdot c\right) + 0.5 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + \frac{a}{{b}^{2}}\right)\right) - \frac{1}{c}\right)}} \]

   15. Simplified 95.8%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-0.125, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{c}^{2}}, {c}^{2} \cdot {a}^{3}\right) - \left(c \cdot a\right) \cdot \left(\left(c \cdot {a}^{2}\right) \cdot -0.5\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.5}{{b}^{4}}, \frac{a}{{b}^{2}}\right)\right) - \frac{1}{c}\right)}} \]

   16. Taylor expanded in a around 0 94.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 160:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{1}{b} + \frac{c \cdot a}{{b}^{3}}\right) - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 160:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{1}{b} + \frac{c \cdot a}{{b}^{3}}\right) - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 160.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (/ 1.0 (- (* a (+ (/ 1.0 b) (/ (* c a) (pow b 3.0)))) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 160.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = 1.0 / ((a * ((1.0 / b) + ((c * a) / pow(b, 3.0)))) - (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 160.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(1.0 / Float64(Float64(a * Float64(Float64(1.0 / b) + Float64(Float64(c * a) / (b ^ 3.0)))) - Float64(b / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 160.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a * N[(N[(1.0 / b), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 160

   1. Initial program 80.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 80.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 80.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   if 160 < b

   1. Initial program 46.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 46.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 46.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 43.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 46.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 46.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. clear-num 46.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}} \]

      2. inv-pow 46.5%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1}} \]

      3. neg-mul-1 46.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1} \]

      4. fma-define 46.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}}\right)}^{-1} \]

      5. pow1/3 43.9%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      6. pow-pow 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      7. metadata-eval 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      8. *-commutative 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}\right)}^{-1} \]

      9. *-commutative 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}\right)}^{-1} \]

   10. Applied egg-rr 46.7%

       \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 46.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

       2. associate-/l* 46.7%

          \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

       3. *-commutative 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(a \cdot 4\right) \cdot c}}\right)}} \]

       4. *-commutative 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(4 \cdot a\right)} \cdot c}\right)}} \]

       5. associate-*r* 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}} \]

       6. cancel-sign-sub-inv 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)}}\right)}} \]

       7. metadata-eval 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} + \color{blue}{-4} \cdot \left(a \cdot c\right)}\right)}} \]

       8. +-commutative 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}\right)}} \]

       9. fma-define 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}\right)}} \]

       10. *-commutative 46.7%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, \color{blue}{c \cdot a}, {b}^{2}\right)}\right)}} \]

   12. Simplified 46.7%

       \[\leadsto \color{blue}{\frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right)}}} \]

   13. Taylor expanded in b around inf 95.8%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-2 \cdot \frac{-1 \cdot \left(a \cdot \left(c \cdot \left(-1 \cdot \left({a}^{2} \cdot c\right) + 0.5 \cdot \left({a}^{2} \cdot c\right)\right)\right)\right) + \left(-0.125 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {c}^{2}} + {a}^{3} \cdot {c}^{2}\right)}{{b}^{6}} + \left(-2 \cdot \frac{-1 \cdot \left({a}^{2} \cdot c\right) + 0.5 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + \frac{a}{{b}^{2}}\right)\right) - \frac{1}{c}\right)}} \]

   15. Simplified 95.8%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-0.125, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{c}^{2}}, {c}^{2} \cdot {a}^{3}\right) - \left(c \cdot a\right) \cdot \left(\left(c \cdot {a}^{2}\right) \cdot -0.5\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.5}{{b}^{4}}, \frac{a}{{b}^{2}}\right)\right) - \frac{1}{c}\right)}} \]

   16. Taylor expanded in a around 0 94.2%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 160:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{1}{b} + \frac{c \cdot a}{{b}^{3}}\right) - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 160:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{1}{b} + \frac{c \cdot a}{{b}^{3}}\right) - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 160.0)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))
   (/ 1.0 (- (* a (+ (/ 1.0 b) (/ (* c a) (pow b 3.0)))) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 160.0) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = 1.0 / ((a * ((1.0 / b) + ((c * a) / pow(b, 3.0)))) - (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 160.0d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    else
        tmp = 1.0d0 / ((a * ((1.0d0 / b) + ((c * a) / (b ** 3.0d0)))) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 160.0) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = 1.0 / ((a * ((1.0 / b) + ((c * a) / Math.pow(b, 3.0)))) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 160.0:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	else:
		tmp = 1.0 / ((a * ((1.0 / b) + ((c * a) / math.pow(b, 3.0)))) - (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 160.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(1.0 / Float64(Float64(a * Float64(Float64(1.0 / b) + Float64(Float64(c * a) / (b ^ 3.0)))) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 160.0)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	else
		tmp = 1.0 / ((a * ((1.0 / b) + ((c * a) / (b ^ 3.0)))) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 160.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a * N[(N[(1.0 / b), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 160

   1. Initial program 80.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if 160 < b

   1. Initial program 46.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 46.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 46.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 43.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 46.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 46.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. clear-num 46.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}} \]

      2. inv-pow 46.5%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1}} \]

      3. neg-mul-1 46.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1} \]

      4. fma-define 46.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}}\right)}^{-1} \]

      5. pow1/3 43.9%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      6. pow-pow 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      7. metadata-eval 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      8. *-commutative 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}\right)}^{-1} \]

      9. *-commutative 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}\right)}^{-1} \]

   10. Applied egg-rr 46.7%

       \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 46.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

       2. associate-/l* 46.7%

          \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

       3. *-commutative 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(a \cdot 4\right) \cdot c}}\right)}} \]

       4. *-commutative 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(4 \cdot a\right)} \cdot c}\right)}} \]

       5. associate-*r* 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}} \]

       6. cancel-sign-sub-inv 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)}}\right)}} \]

       7. metadata-eval 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} + \color{blue}{-4} \cdot \left(a \cdot c\right)}\right)}} \]

       8. +-commutative 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}\right)}} \]

       9. fma-define 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}\right)}} \]

       10. *-commutative 46.7%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, \color{blue}{c \cdot a}, {b}^{2}\right)}\right)}} \]

   12. Simplified 46.7%

       \[\leadsto \color{blue}{\frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right)}}} \]

   13. Taylor expanded in b around inf 95.8%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-2 \cdot \frac{-1 \cdot \left(a \cdot \left(c \cdot \left(-1 \cdot \left({a}^{2} \cdot c\right) + 0.5 \cdot \left({a}^{2} \cdot c\right)\right)\right)\right) + \left(-0.125 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {c}^{2}} + {a}^{3} \cdot {c}^{2}\right)}{{b}^{6}} + \left(-2 \cdot \frac{-1 \cdot \left({a}^{2} \cdot c\right) + 0.5 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + \frac{a}{{b}^{2}}\right)\right) - \frac{1}{c}\right)}} \]

   15. Simplified 95.8%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-0.125, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{c}^{2}}, {c}^{2} \cdot {a}^{3}\right) - \left(c \cdot a\right) \cdot \left(\left(c \cdot {a}^{2}\right) \cdot -0.5\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.5}{{b}^{4}}, \frac{a}{{b}^{2}}\right)\right) - \frac{1}{c}\right)}} \]

   16. Taylor expanded in a around 0 94.2%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 160:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{1}{b} + \frac{c \cdot a}{{b}^{3}}\right) - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 160:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 160.0)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))
   (/ 1.0 (/ (- (/ (* c a) b) b) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 160.0) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = 1.0 / ((((c * a) / b) - b) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 160.0d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    else
        tmp = 1.0d0 / ((((c * a) / b) - b) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 160.0) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = 1.0 / ((((c * a) / b) - b) / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 160.0:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	else:
		tmp = 1.0 / ((((c * a) / b) - b) / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 160.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c * a) / b) - b) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 160.0)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	else
		tmp = 1.0 / ((((c * a) / b) - b) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 160.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 160

   1. Initial program 80.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if 160 < b

   1. Initial program 46.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 46.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 46.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 43.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 43.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 46.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 46.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. clear-num 46.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}} \]

      2. inv-pow 46.5%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1}} \]

      3. neg-mul-1 46.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1} \]

      4. fma-define 46.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}}\right)}^{-1} \]

      5. pow1/3 43.9%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      6. pow-pow 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      7. metadata-eval 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

      8. *-commutative 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}\right)}^{-1} \]

      9. *-commutative 46.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}\right)}^{-1} \]

   10. Applied egg-rr 46.7%

       \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 46.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

       2. associate-/l* 46.7%

          \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

       3. *-commutative 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(a \cdot 4\right) \cdot c}}\right)}} \]

       4. *-commutative 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(4 \cdot a\right)} \cdot c}\right)}} \]

       5. associate-*r* 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}} \]

       6. cancel-sign-sub-inv 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)}}\right)}} \]

       7. metadata-eval 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} + \color{blue}{-4} \cdot \left(a \cdot c\right)}\right)}} \]

       8. +-commutative 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}\right)}} \]

       9. fma-define 46.7%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}\right)}} \]

       10. *-commutative 46.7%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, \color{blue}{c \cdot a}, {b}^{2}\right)}\right)}} \]

   12. Simplified 46.7%

       \[\leadsto \color{blue}{\frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right)}}} \]

   13. Taylor expanded in b around inf 95.8%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-2 \cdot \frac{-1 \cdot \left(a \cdot \left(c \cdot \left(-1 \cdot \left({a}^{2} \cdot c\right) + 0.5 \cdot \left({a}^{2} \cdot c\right)\right)\right)\right) + \left(-0.125 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {c}^{2}} + {a}^{3} \cdot {c}^{2}\right)}{{b}^{6}} + \left(-2 \cdot \frac{-1 \cdot \left({a}^{2} \cdot c\right) + 0.5 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + \frac{a}{{b}^{2}}\right)\right) - \frac{1}{c}\right)}} \]

   15. Simplified 95.8%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-0.125, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{c}^{2}}, {c}^{2} \cdot {a}^{3}\right) - \left(c \cdot a\right) \cdot \left(\left(c \cdot {a}^{2}\right) \cdot -0.5\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.5}{{b}^{4}}, \frac{a}{{b}^{2}}\right)\right) - \frac{1}{c}\right)}} \]

   16. Taylor expanded in c around 0 88.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 160:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.1% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 58.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 58.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 57.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. pow1/3 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 55.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 57.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

8. Simplified 57.8%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
9. Step-by-step derivation

   1. clear-num 57.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}} \]

   2. inv-pow 57.8%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1}} \]

   3. neg-mul-1 57.8%

      \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1} \]

   4. fma-define 57.8%

      \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}}\right)}^{-1} \]

   5. pow1/3 55.3%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

   6. pow-pow 58.1%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

   7. metadata-eval 58.1%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

   8. *-commutative 58.1%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}\right)}^{-1} \]

   9. *-commutative 58.1%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}\right)}^{-1} \]

10. Applied egg-rr 58.1%

    \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}\right)}^{-1}} \]
11. Step-by-step derivation

    1. unpow-1 58.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

    2. associate-/l* 58.1%

       \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

    3. *-commutative 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(a \cdot 4\right) \cdot c}}\right)}} \]

    4. *-commutative 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(4 \cdot a\right)} \cdot c}\right)}} \]

    5. associate-*r* 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}} \]

    6. cancel-sign-sub-inv 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)}}\right)}} \]

    7. metadata-eval 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} + \color{blue}{-4} \cdot \left(a \cdot c\right)}\right)}} \]

    8. +-commutative 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}\right)}} \]

    9. fma-define 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}\right)}} \]

    10. *-commutative 58.1%

        \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, \color{blue}{c \cdot a}, {b}^{2}\right)}\right)}} \]

12. Simplified 58.1%

    \[\leadsto \color{blue}{\frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right)}}} \]

13. Taylor expanded in b around inf 89.3%

    \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-2 \cdot \frac{-1 \cdot \left(a \cdot \left(c \cdot \left(-1 \cdot \left({a}^{2} \cdot c\right) + 0.5 \cdot \left({a}^{2} \cdot c\right)\right)\right)\right) + \left(-0.125 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {c}^{2}} + {a}^{3} \cdot {c}^{2}\right)}{{b}^{6}} + \left(-2 \cdot \frac{-1 \cdot \left({a}^{2} \cdot c\right) + 0.5 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + \frac{a}{{b}^{2}}\right)\right) - \frac{1}{c}\right)}} \]

15. Simplified 89.3%

    \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-0.125, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{c}^{2}}, {c}^{2} \cdot {a}^{3}\right) - \left(c \cdot a\right) \cdot \left(\left(c \cdot {a}^{2}\right) \cdot -0.5\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.5}{{b}^{4}}, \frac{a}{{b}^{2}}\right)\right) - \frac{1}{c}\right)}} \]

16. Taylor expanded in c around 0 79.3%

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

17. Final simplification 79.3%

    \[\leadsto \frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}} \]
18. Add Preprocessing

Alternative 9: 82.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{b} - \frac{b}{c}} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (- (/ a b) (/ b c))))

C

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

Java

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

Python

def code(a, b, c):
	return 1.0 / ((a / b) - (b / c))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((a / b) - (b / c));
end

Wolfram

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

Derivation

1. Initial program 58.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 58.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 58.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 57.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. pow1/3 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 55.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 57.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

8. Simplified 57.8%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
9. Step-by-step derivation

   1. clear-num 57.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}} \]

   2. inv-pow 57.8%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1}} \]

   3. neg-mul-1 57.8%

      \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1} \]

   4. fma-define 57.8%

      \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}}\right)}^{-1} \]

   5. pow1/3 55.3%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

   6. pow-pow 58.1%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

   7. metadata-eval 58.1%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

   8. *-commutative 58.1%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}\right)}^{-1} \]

   9. *-commutative 58.1%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}\right)}^{-1} \]

10. Applied egg-rr 58.1%

    \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}\right)}^{-1}} \]
11. Step-by-step derivation

    1. unpow-1 58.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

    2. associate-/l* 58.1%

       \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

    3. *-commutative 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(a \cdot 4\right) \cdot c}}\right)}} \]

    4. *-commutative 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(4 \cdot a\right)} \cdot c}\right)}} \]

    5. associate-*r* 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}} \]

    6. cancel-sign-sub-inv 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)}}\right)}} \]

    7. metadata-eval 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} + \color{blue}{-4} \cdot \left(a \cdot c\right)}\right)}} \]

    8. +-commutative 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}\right)}} \]

    9. fma-define 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}\right)}} \]

    10. *-commutative 58.1%

        \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, \color{blue}{c \cdot a}, {b}^{2}\right)}\right)}} \]

12. Simplified 58.1%

    \[\leadsto \color{blue}{\frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right)}}} \]

13. Taylor expanded in b around inf 89.3%

    \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-2 \cdot \frac{-1 \cdot \left(a \cdot \left(c \cdot \left(-1 \cdot \left({a}^{2} \cdot c\right) + 0.5 \cdot \left({a}^{2} \cdot c\right)\right)\right)\right) + \left(-0.125 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {c}^{2}} + {a}^{3} \cdot {c}^{2}\right)}{{b}^{6}} + \left(-2 \cdot \frac{-1 \cdot \left({a}^{2} \cdot c\right) + 0.5 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + \frac{a}{{b}^{2}}\right)\right) - \frac{1}{c}\right)}} \]

15. Simplified 89.3%

    \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-0.125, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{c}^{2}}, {c}^{2} \cdot {a}^{3}\right) - \left(c \cdot a\right) \cdot \left(\left(c \cdot {a}^{2}\right) \cdot -0.5\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.5}{{b}^{4}}, \frac{a}{{b}^{2}}\right)\right) - \frac{1}{c}\right)}} \]

16. Taylor expanded in a around 0 79.3%

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

17. Final simplification 79.3%

    \[\leadsto \frac{1}{\frac{a}{b} - \frac{b}{c}} \]
18. Add Preprocessing

Alternative 10: 64.4% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

3. Simplified 58.1%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 61.9%

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

   1. associate-*r/ 61.9%

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

   2. mul-1-neg 61.9%

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

7. Simplified 61.9%

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

8. Final simplification 61.9%

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

Alternative 11: 1.6% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return b / 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 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 58.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 58.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 57.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. pow1/3 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 55.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 55.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 57.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

8. Simplified 57.8%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
9. Step-by-step derivation

   1. clear-num 57.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}} \]

   2. inv-pow 57.8%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1}} \]

   3. neg-mul-1 57.8%

      \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right)}^{-1} \]

   4. fma-define 57.8%

      \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}\right)}}\right)}^{-1} \]

   5. pow1/3 55.3%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

   6. pow-pow 58.1%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

   7. metadata-eval 58.1%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

   8. *-commutative 58.1%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}\right)}^{-1} \]

   9. *-commutative 58.1%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}\right)}^{-1} \]

10. Applied egg-rr 58.1%

    \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}\right)}^{-1}} \]
11. Step-by-step derivation

    1. unpow-1 58.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

    2. associate-/l* 58.1%

       \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}\right)}}} \]

    3. *-commutative 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(a \cdot 4\right) \cdot c}}\right)}} \]

    4. *-commutative 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(4 \cdot a\right)} \cdot c}\right)}} \]

    5. associate-*r* 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}} \]

    6. cancel-sign-sub-inv 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)}}\right)}} \]

    7. metadata-eval 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} + \color{blue}{-4} \cdot \left(a \cdot c\right)}\right)}} \]

    8. +-commutative 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}\right)}} \]

    9. fma-define 58.1%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}\right)}} \]

    10. *-commutative 58.1%

        \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, \color{blue}{c \cdot a}, {b}^{2}\right)}\right)}} \]

12. Simplified 58.1%

    \[\leadsto \color{blue}{\frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}\right)}}} \]

13. Taylor expanded in b around inf 79.1%

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

    1. *-commutative 79.1%

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

15. Simplified 79.1%

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

16. Taylor expanded in a around inf 1.6%

    \[\leadsto \color{blue}{\frac{b}{a}} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))