Cubic critical

Percentage Accurate: 51.7% → 84.2%

Time: 14.3s

Alternatives: 14

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{+45}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(a \cdot \frac{1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.25e+45)
   (/ b (* a -1.5))
   (if (<= b 8e-107)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (/ 1.0 (* b (- (* a (/ 1.5 (pow b 2.0))) (/ 2.0 c)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.25e+45) {
		tmp = b / (a * -1.5);
	} else if (b <= 8e-107) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * ((a * (1.5 / pow(b, 2.0))) - (2.0 / 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 <= (-2.25d+45)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 8d-107) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (b * ((a * (1.5d0 / (b ** 2.0d0))) - (2.0d0 / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.25e+45) {
		tmp = b / (a * -1.5);
	} else if (b <= 8e-107) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * ((a * (1.5 / Math.pow(b, 2.0))) - (2.0 / c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.25e+45:
		tmp = b / (a * -1.5)
	elif b <= 8e-107:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = 1.0 / (b * ((a * (1.5 / math.pow(b, 2.0))) - (2.0 / c)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.25e+45)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 8e-107)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(a * Float64(1.5 / (b ^ 2.0))) - Float64(2.0 / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.25e+45)
		tmp = b / (a * -1.5);
	elseif (b <= 8e-107)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = 1.0 / (b * ((a * (1.5 / (b ^ 2.0))) - (2.0 / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.25e+45], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-107], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(a * N[(1.5 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2499999999999999e45

   1. Initial program 56.6%

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

      1. sqr-neg 56.6%

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

      2. sqr-neg 56.6%

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

      3. associate-*l* 56.6%

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

   3. Simplified 56.6%

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

      1. frac-2neg 56.6%

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

      2. div-inv 56.6%

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

   6. Applied egg-rr 56.6%

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

   7. Taylor expanded in b around -inf 92.9%

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

      1. *-commutative 92.9%

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

      2. associate-*l/ 93.0%

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

      3. associate-/l* 93.0%

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

   9. Simplified 93.0%

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

       1. clear-num 93.0%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 93.1%

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

       3. div-inv 93.1%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 93.1%

          \[\leadsto \frac{b}{a \cdot \color{blue}{-1.5}} \]

   11. Applied egg-rr 93.1%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if -2.2499999999999999e45 < b < 8e-107

   1. Initial program 82.0%

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

   if 8e-107 < b

   1. Initial program 12.2%

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

      1. sqr-neg 12.2%

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

      2. sqr-neg 12.2%

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

      3. associate-*l* 12.2%

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

   3. Simplified 12.2%

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

      1. prod-diff 12.0%

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

      2. *-commutative 12.0%

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

      3. associate-*r* 12.0%

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

      4. *-commutative 12.0%

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

      5. distribute-rgt-neg-in 12.0%

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

      6. *-commutative 12.0%

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

      7. distribute-rgt-neg-in 12.0%

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

      8. metadata-eval 12.0%

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

      9. *-commutative 12.0%

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

      10. fma-undefine 12.0%

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

      11. distribute-lft-neg-in 12.0%

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

      12. *-commutative 12.0%

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

      13. associate-*r* 12.0%

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

      14. *-commutative 12.0%

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

      15. distribute-rgt-neg-in 12.0%

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

      16. *-commutative 12.0%

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

      17. distribute-rgt-neg-in 12.0%

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

      18. metadata-eval 12.0%

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

      19. associate-*r* 12.0%

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

      20. *-commutative 12.0%

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

      21. associate-*l* 12.0%

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

   6. Applied egg-rr 12.0%

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

      1. clear-num 12.0%

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

      2. inv-pow 12.0%

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

   8. Applied egg-rr 12.6%

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

   10. Simplified 12.2%

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

   11. Taylor expanded in b around inf 91.7%

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

       1. associate-*r/ 91.7%

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

       2. *-commutative 91.7%

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

       3. *-lft-identity 91.7%

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

       4. times-frac 91.7%

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

       5. /-rgt-identity 91.7%

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

       6. associate-*r/ 91.7%

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

       7. metadata-eval 91.7%

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

   13. Simplified 91.7%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{+45}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(a \cdot \frac{1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{+45}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(a \cdot \frac{1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.25e+45)
   (/ b (* a -1.5))
   (if (<= b 8.4e-107)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (/ 1.0 (* b (- (* a (/ 1.5 (pow b 2.0))) (/ 2.0 c)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.25e+45) {
		tmp = b / (a * -1.5);
	} else if (b <= 8.4e-107) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * ((a * (1.5 / pow(b, 2.0))) - (2.0 / 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 <= (-2.25d+45)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 8.4d-107) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (b * ((a * (1.5d0 / (b ** 2.0d0))) - (2.0d0 / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.25e+45) {
		tmp = b / (a * -1.5);
	} else if (b <= 8.4e-107) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * ((a * (1.5 / Math.pow(b, 2.0))) - (2.0 / c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.25e+45:
		tmp = b / (a * -1.5)
	elif b <= 8.4e-107:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = 1.0 / (b * ((a * (1.5 / math.pow(b, 2.0))) - (2.0 / c)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.25e+45)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 8.4e-107)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(a * Float64(1.5 / (b ^ 2.0))) - Float64(2.0 / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.25e+45)
		tmp = b / (a * -1.5);
	elseif (b <= 8.4e-107)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = 1.0 / (b * ((a * (1.5 / (b ^ 2.0))) - (2.0 / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.25e+45], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.4e-107], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(a * N[(1.5 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2499999999999999e45

   1. Initial program 56.6%

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

      1. sqr-neg 56.6%

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

      2. sqr-neg 56.6%

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

      3. associate-*l* 56.6%

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

   3. Simplified 56.6%

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

      1. frac-2neg 56.6%

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

      2. div-inv 56.6%

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

   6. Applied egg-rr 56.6%

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

   7. Taylor expanded in b around -inf 92.9%

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

      1. *-commutative 92.9%

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

      2. associate-*l/ 93.0%

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

      3. associate-/l* 93.0%

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

   9. Simplified 93.0%

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

       1. clear-num 93.0%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 93.1%

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

       3. div-inv 93.1%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 93.1%

          \[\leadsto \frac{b}{a \cdot \color{blue}{-1.5}} \]

   11. Applied egg-rr 93.1%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if -2.2499999999999999e45 < b < 8.3999999999999997e-107

   1. Initial program 82.0%

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

      1. sqr-neg 82.0%

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

      2. sqr-neg 82.0%

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

      3. associate-*l* 81.9%

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

   3. Simplified 81.9%

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

   if 8.3999999999999997e-107 < b

   1. Initial program 12.2%

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

      1. sqr-neg 12.2%

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

      2. sqr-neg 12.2%

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

      3. associate-*l* 12.2%

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

   3. Simplified 12.2%

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

      1. prod-diff 12.0%

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

      2. *-commutative 12.0%

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

      3. associate-*r* 12.0%

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

      4. *-commutative 12.0%

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

      5. distribute-rgt-neg-in 12.0%

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

      6. *-commutative 12.0%

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

      7. distribute-rgt-neg-in 12.0%

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

      8. metadata-eval 12.0%

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

      9. *-commutative 12.0%

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

      10. fma-undefine 12.0%

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

      11. distribute-lft-neg-in 12.0%

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

      12. *-commutative 12.0%

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

      13. associate-*r* 12.0%

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

      14. *-commutative 12.0%

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

      15. distribute-rgt-neg-in 12.0%

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

      16. *-commutative 12.0%

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

      17. distribute-rgt-neg-in 12.0%

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

      18. metadata-eval 12.0%

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

      19. associate-*r* 12.0%

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

      20. *-commutative 12.0%

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

      21. associate-*l* 12.0%

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

   6. Applied egg-rr 12.0%

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

      1. clear-num 12.0%

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

      2. inv-pow 12.0%

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

   8. Applied egg-rr 12.6%

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

   10. Simplified 12.2%

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

   11. Taylor expanded in b around inf 91.7%

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

       1. associate-*r/ 91.7%

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

       2. *-commutative 91.7%

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

       3. *-lft-identity 91.7%

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

       4. times-frac 91.7%

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

       5. /-rgt-identity 91.7%

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

       6. associate-*r/ 91.7%

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

       7. metadata-eval 91.7%

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

   13. Simplified 91.7%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{+45}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(a \cdot \frac{1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-119}:\\ \;\;\;\;b \cdot \left(0.6666666666666666 \cdot \frac{-1}{a} - -0.5 \cdot \frac{c}{{b}^{2}}\right)\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(a \cdot \frac{1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-119)
   (* b (- (* 0.6666666666666666 (/ -1.0 a)) (* -0.5 (/ c (pow b 2.0)))))
   (if (<= b 6.2e-107)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ 1.0 (* b (- (* a (/ 1.5 (pow b 2.0))) (/ 2.0 c)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-119) {
		tmp = b * ((0.6666666666666666 * (-1.0 / a)) - (-0.5 * (c / pow(b, 2.0))));
	} else if (b <= 6.2e-107) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * ((a * (1.5 / pow(b, 2.0))) - (2.0 / 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 <= (-2.4d-119)) then
        tmp = b * ((0.6666666666666666d0 * ((-1.0d0) / a)) - ((-0.5d0) * (c / (b ** 2.0d0))))
    else if (b <= 6.2d-107) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (b * ((a * (1.5d0 / (b ** 2.0d0))) - (2.0d0 / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-119) {
		tmp = b * ((0.6666666666666666 * (-1.0 / a)) - (-0.5 * (c / Math.pow(b, 2.0))));
	} else if (b <= 6.2e-107) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * ((a * (1.5 / Math.pow(b, 2.0))) - (2.0 / c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e-119:
		tmp = b * ((0.6666666666666666 * (-1.0 / a)) - (-0.5 * (c / math.pow(b, 2.0))))
	elif b <= 6.2e-107:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = 1.0 / (b * ((a * (1.5 / math.pow(b, 2.0))) - (2.0 / c)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e-119)
		tmp = Float64(b * Float64(Float64(0.6666666666666666 * Float64(-1.0 / a)) - Float64(-0.5 * Float64(c / (b ^ 2.0)))));
	elseif (b <= 6.2e-107)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(a * Float64(1.5 / (b ^ 2.0))) - Float64(2.0 / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.4e-119)
		tmp = b * ((0.6666666666666666 * (-1.0 / a)) - (-0.5 * (c / (b ^ 2.0))));
	elseif (b <= 6.2e-107)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = 1.0 / (b * ((a * (1.5 / (b ^ 2.0))) - (2.0 / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e-119], N[(b * N[(N[(0.6666666666666666 * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.2e-107], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(a * N[(1.5 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.40000000000000009e-119

   1. Initial program 67.0%

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

      1. sqr-neg 67.0%

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

      2. sqr-neg 67.0%

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

      3. associate-*l* 66.9%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in b around -inf 82.4%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   if -2.40000000000000009e-119 < b < 6.20000000000000043e-107

   1. Initial program 75.5%

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

      1. sqr-neg 75.5%

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

      2. sqr-neg 75.5%

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

      3. associate-*l* 75.5%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in b around 0 75.3%

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

      1. *-commutative 75.3%

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

      2. *-commutative 75.3%

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

      3. associate-*r* 75.4%

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

   7. Simplified 75.4%

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

   if 6.20000000000000043e-107 < b

   1. Initial program 12.2%

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

      1. sqr-neg 12.2%

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

      2. sqr-neg 12.2%

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

      3. associate-*l* 12.2%

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

   3. Simplified 12.2%

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

      1. prod-diff 12.0%

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

      2. *-commutative 12.0%

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

      3. associate-*r* 12.0%

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

      4. *-commutative 12.0%

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

      5. distribute-rgt-neg-in 12.0%

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

      6. *-commutative 12.0%

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

      7. distribute-rgt-neg-in 12.0%

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

      8. metadata-eval 12.0%

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

      9. *-commutative 12.0%

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

      10. fma-undefine 12.0%

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

      11. distribute-lft-neg-in 12.0%

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

      12. *-commutative 12.0%

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

      13. associate-*r* 12.0%

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

      14. *-commutative 12.0%

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

      15. distribute-rgt-neg-in 12.0%

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

      16. *-commutative 12.0%

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

      17. distribute-rgt-neg-in 12.0%

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

      18. metadata-eval 12.0%

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

      19. associate-*r* 12.0%

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

      20. *-commutative 12.0%

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

      21. associate-*l* 12.0%

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

   6. Applied egg-rr 12.0%

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

      1. clear-num 12.0%

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

      2. inv-pow 12.0%

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

   8. Applied egg-rr 12.6%

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

   10. Simplified 12.2%

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

   11. Taylor expanded in b around inf 91.7%

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

       1. associate-*r/ 91.7%

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

       2. *-commutative 91.7%

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

       3. *-lft-identity 91.7%

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

       4. times-frac 91.7%

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

       5. /-rgt-identity 91.7%

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

       6. associate-*r/ 91.7%

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

       7. metadata-eval 91.7%

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

   13. Simplified 91.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-119}:\\ \;\;\;\;b \cdot \left(0.6666666666666666 \cdot \frac{-1}{a} - -0.5 \cdot \frac{c}{{b}^{2}}\right)\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(a \cdot \frac{1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(a \cdot \frac{1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.05e-119)
   (/ b (* a -1.5))
   (if (<= b 8.4e-107)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ 1.0 (* b (- (* a (/ 1.5 (pow b 2.0))) (/ 2.0 c)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-119) {
		tmp = b / (a * -1.5);
	} else if (b <= 8.4e-107) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * ((a * (1.5 / pow(b, 2.0))) - (2.0 / 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 <= (-2.05d-119)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 8.4d-107) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (b * ((a * (1.5d0 / (b ** 2.0d0))) - (2.0d0 / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-119) {
		tmp = b / (a * -1.5);
	} else if (b <= 8.4e-107) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * ((a * (1.5 / Math.pow(b, 2.0))) - (2.0 / c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.05e-119:
		tmp = b / (a * -1.5)
	elif b <= 8.4e-107:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = 1.0 / (b * ((a * (1.5 / math.pow(b, 2.0))) - (2.0 / c)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.05e-119)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 8.4e-107)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(a * Float64(1.5 / (b ^ 2.0))) - Float64(2.0 / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.05e-119)
		tmp = b / (a * -1.5);
	elseif (b <= 8.4e-107)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = 1.0 / (b * ((a * (1.5 / (b ^ 2.0))) - (2.0 / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.05e-119], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.4e-107], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(a * N[(1.5 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0500000000000001e-119

   1. Initial program 67.0%

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

      1. sqr-neg 67.0%

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

      2. sqr-neg 67.0%

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

      3. associate-*l* 66.9%

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

   3. Simplified 66.9%

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

      1. frac-2neg 66.9%

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

      2. div-inv 66.9%

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

   6. Applied egg-rr 66.9%

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

   7. Taylor expanded in b around -inf 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-*l/ 82.1%

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

      3. associate-/l* 82.1%

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

   9. Simplified 82.1%

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

       1. clear-num 82.1%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 82.1%

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

       3. div-inv 82.1%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 82.1%

          \[\leadsto \frac{b}{a \cdot \color{blue}{-1.5}} \]

   11. Applied egg-rr 82.1%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if -2.0500000000000001e-119 < b < 8.3999999999999997e-107

   1. Initial program 75.5%

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

      1. sqr-neg 75.5%

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

      2. sqr-neg 75.5%

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

      3. associate-*l* 75.5%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in b around 0 75.3%

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

      1. *-commutative 75.3%

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

      2. *-commutative 75.3%

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

      3. associate-*r* 75.4%

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

   7. Simplified 75.4%

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

   if 8.3999999999999997e-107 < b

   1. Initial program 12.2%

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

      1. sqr-neg 12.2%

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

      2. sqr-neg 12.2%

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

      3. associate-*l* 12.2%

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

   3. Simplified 12.2%

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

      1. prod-diff 12.0%

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

      2. *-commutative 12.0%

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

      3. associate-*r* 12.0%

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

      4. *-commutative 12.0%

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

      5. distribute-rgt-neg-in 12.0%

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

      6. *-commutative 12.0%

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

      7. distribute-rgt-neg-in 12.0%

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

      8. metadata-eval 12.0%

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

      9. *-commutative 12.0%

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

      10. fma-undefine 12.0%

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

      11. distribute-lft-neg-in 12.0%

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

      12. *-commutative 12.0%

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

      13. associate-*r* 12.0%

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

      14. *-commutative 12.0%

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

      15. distribute-rgt-neg-in 12.0%

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

      16. *-commutative 12.0%

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

      17. distribute-rgt-neg-in 12.0%

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

      18. metadata-eval 12.0%

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

      19. associate-*r* 12.0%

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

      20. *-commutative 12.0%

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

      21. associate-*l* 12.0%

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

   6. Applied egg-rr 12.0%

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

      1. clear-num 12.0%

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

      2. inv-pow 12.0%

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

   8. Applied egg-rr 12.6%

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

   10. Simplified 12.2%

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

   11. Taylor expanded in b around inf 91.7%

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

       1. associate-*r/ 91.7%

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

       2. *-commutative 91.7%

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

       3. *-lft-identity 91.7%

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

       4. times-frac 91.7%

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

       5. /-rgt-identity 91.7%

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

       6. associate-*r/ 91.7%

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

       7. metadata-eval 91.7%

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

   13. Simplified 91.7%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(a \cdot \frac{1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-119)
   (/ b (* a -1.5))
   (if (<= b 6.2e-107)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-119) {
		tmp = b / (a * -1.5);
	} else if (b <= 6.2e-107) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.4d-119)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 6.2d-107) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-119) {
		tmp = b / (a * -1.5);
	} else if (b <= 6.2e-107) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e-119:
		tmp = b / (a * -1.5)
	elif b <= 6.2e-107:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e-119)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 6.2e-107)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.4e-119)
		tmp = b / (a * -1.5);
	elseif (b <= 6.2e-107)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e-119], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.2e-107], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.40000000000000009e-119

   1. Initial program 67.0%

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

      1. sqr-neg 67.0%

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

      2. sqr-neg 67.0%

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

      3. associate-*l* 66.9%

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

   3. Simplified 66.9%

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

      1. frac-2neg 66.9%

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

      2. div-inv 66.9%

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

   6. Applied egg-rr 66.9%

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

   7. Taylor expanded in b around -inf 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-*l/ 82.1%

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

      3. associate-/l* 82.1%

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

   9. Simplified 82.1%

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

       1. clear-num 82.1%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 82.1%

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

       3. div-inv 82.1%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 82.1%

          \[\leadsto \frac{b}{a \cdot \color{blue}{-1.5}} \]

   11. Applied egg-rr 82.1%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if -2.40000000000000009e-119 < b < 6.20000000000000043e-107

   1. Initial program 75.5%

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

      1. sqr-neg 75.5%

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

      2. sqr-neg 75.5%

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

      3. associate-*l* 75.5%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in b around 0 75.3%

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

      1. *-commutative 75.3%

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

      2. *-commutative 75.3%

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

      3. associate-*r* 75.4%

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

   7. Simplified 75.4%

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

   if 6.20000000000000043e-107 < b

   1. Initial program 12.2%

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

      1. sqr-neg 12.2%

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

      2. sqr-neg 12.2%

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

      3. associate-*l* 12.2%

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

   3. Simplified 12.2%

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

   5. Taylor expanded in b around inf 91.4%

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

      1. *-commutative 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-119)
   (/ b (* a -1.5))
   (if (<= b 7.2e-107)
     (/ (- (sqrt (* (* a c) -3.0)) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-119) {
		tmp = b / (a * -1.5);
	} else if (b <= 7.2e-107) {
		tmp = (sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.4d-119)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 7.2d-107) then
        tmp = (sqrt(((a * c) * (-3.0d0))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-119) {
		tmp = b / (a * -1.5);
	} else if (b <= 7.2e-107) {
		tmp = (Math.sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e-119:
		tmp = b / (a * -1.5)
	elif b <= 7.2e-107:
		tmp = (math.sqrt(((a * c) * -3.0)) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e-119)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 7.2e-107)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) * -3.0)) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.4e-119)
		tmp = b / (a * -1.5);
	elseif (b <= 7.2e-107)
		tmp = (sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e-119], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e-107], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.40000000000000009e-119

   1. Initial program 67.0%

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

      1. sqr-neg 67.0%

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

      2. sqr-neg 67.0%

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

      3. associate-*l* 66.9%

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

   3. Simplified 66.9%

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

      1. frac-2neg 66.9%

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

      2. div-inv 66.9%

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

   6. Applied egg-rr 66.9%

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

   7. Taylor expanded in b around -inf 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-*l/ 82.1%

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

      3. associate-/l* 82.1%

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

   9. Simplified 82.1%

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

       1. clear-num 82.1%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 82.1%

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

       3. div-inv 82.1%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 82.1%

          \[\leadsto \frac{b}{a \cdot \color{blue}{-1.5}} \]

   11. Applied egg-rr 82.1%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if -2.40000000000000009e-119 < b < 7.19999999999999953e-107

   1. Initial program 75.5%

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

      1. sqr-neg 75.5%

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

      2. sqr-neg 75.5%

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

      3. associate-*l* 75.5%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in b around 0 75.3%

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

   if 7.19999999999999953e-107 < b

   1. Initial program 12.2%

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

      1. sqr-neg 12.2%

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

      2. sqr-neg 12.2%

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

      3. associate-*l* 12.2%

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

   3. Simplified 12.2%

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

   5. Taylor expanded in b around inf 91.4%

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

      1. *-commutative 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-121}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-107}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \sqrt{a \cdot \left(c \cdot -3\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e-121)
   (/ b (* a -1.5))
   (if (<= b 5.1e-107)
     (* (/ 0.3333333333333333 a) (sqrt (* a (* c -3.0))))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e-121) {
		tmp = b / (a * -1.5);
	} else if (b <= 5.1e-107) {
		tmp = (0.3333333333333333 / a) * sqrt((a * (c * -3.0)));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.4d-121)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 5.1d-107) then
        tmp = (0.3333333333333333d0 / a) * sqrt((a * (c * (-3.0d0))))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e-121) {
		tmp = b / (a * -1.5);
	} else if (b <= 5.1e-107) {
		tmp = (0.3333333333333333 / a) * Math.sqrt((a * (c * -3.0)));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.4e-121:
		tmp = b / (a * -1.5)
	elif b <= 5.1e-107:
		tmp = (0.3333333333333333 / a) * math.sqrt((a * (c * -3.0)))
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.4e-121)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 5.1e-107)
		tmp = Float64(Float64(0.3333333333333333 / a) * sqrt(Float64(a * Float64(c * -3.0))));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.4e-121)
		tmp = b / (a * -1.5);
	elseif (b <= 5.1e-107)
		tmp = (0.3333333333333333 / a) * sqrt((a * (c * -3.0)));
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.4e-121], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.1e-107], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.40000000000000042e-121

   1. Initial program 67.0%

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

      1. sqr-neg 67.0%

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

      2. sqr-neg 67.0%

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

      3. associate-*l* 66.9%

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

   3. Simplified 66.9%

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

      1. frac-2neg 66.9%

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

      2. div-inv 66.9%

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

   6. Applied egg-rr 66.9%

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

   7. Taylor expanded in b around -inf 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-*l/ 82.1%

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

      3. associate-/l* 82.1%

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

   9. Simplified 82.1%

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

       1. clear-num 82.1%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 82.1%

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

       3. div-inv 82.1%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 82.1%

          \[\leadsto \frac{b}{a \cdot \color{blue}{-1.5}} \]

   11. Applied egg-rr 82.1%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if -4.40000000000000042e-121 < b < 5.1000000000000002e-107

   1. Initial program 75.5%

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

      1. sqr-neg 75.5%

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

      2. sqr-neg 75.5%

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

      3. associate-*l* 75.5%

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

   3. Simplified 75.5%

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

      1. prod-diff 75.1%

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

      2. *-commutative 75.1%

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

      3. associate-*r* 75.1%

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

      4. *-commutative 75.1%

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

      5. distribute-rgt-neg-in 75.1%

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

      6. *-commutative 75.1%

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

      7. distribute-rgt-neg-in 75.1%

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

      8. metadata-eval 75.1%

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

      9. *-commutative 75.1%

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

      10. fma-undefine 75.1%

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

      11. distribute-lft-neg-in 75.1%

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

      12. *-commutative 75.1%

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

      13. associate-*r* 75.0%

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

      14. *-commutative 75.0%

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

      15. distribute-rgt-neg-in 75.0%

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

      16. *-commutative 75.0%

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

      17. distribute-rgt-neg-in 75.0%

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

      18. metadata-eval 75.0%

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

      19. associate-*r* 75.1%

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

      20. *-commutative 75.1%

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

      21. associate-*l* 75.0%

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

   6. Applied egg-rr 75.0%

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

   7. Taylor expanded in b around 0 74.8%

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

      1. associate-*r* 74.9%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}} \]

      2. associate-*r/ 74.8%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot 1}{a}} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)} \]

      3. metadata-eval 74.8%

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

      4. distribute-rgt-out 75.2%

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

      5. metadata-eval 75.2%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. associate-*r* 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 75.2%

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

   9. Simplified 75.2%

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

   if 5.1000000000000002e-107 < b

   1. Initial program 12.2%

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

      1. sqr-neg 12.2%

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

      2. sqr-neg 12.2%

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

      3. associate-*l* 12.2%

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

   3. Simplified 12.2%

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

   5. Taylor expanded in b around inf 91.4%

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

      1. *-commutative 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-121}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-107}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \sqrt{a \cdot \left(c \cdot -3\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-135}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-197}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{-3 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.05e-135)
   (/ b (* a -1.5))
   (if (<= b 2.7e-197)
     (* 0.3333333333333333 (sqrt (* -3.0 (/ c a))))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-135) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.7e-197) {
		tmp = 0.3333333333333333 * sqrt((-3.0 * (c / a)));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.05d-135)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 2.7d-197) then
        tmp = 0.3333333333333333d0 * sqrt(((-3.0d0) * (c / a)))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-135) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.7e-197) {
		tmp = 0.3333333333333333 * Math.sqrt((-3.0 * (c / a)));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.05e-135:
		tmp = b / (a * -1.5)
	elif b <= 2.7e-197:
		tmp = 0.3333333333333333 * math.sqrt((-3.0 * (c / a)))
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.05e-135)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 2.7e-197)
		tmp = Float64(0.3333333333333333 * sqrt(Float64(-3.0 * Float64(c / a))));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.05e-135)
		tmp = b / (a * -1.5);
	elseif (b <= 2.7e-197)
		tmp = 0.3333333333333333 * sqrt((-3.0 * (c / a)));
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.05e-135], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.7e-197], N[(0.3333333333333333 * N[Sqrt[N[(-3.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.05000000000000005e-135

   1. Initial program 67.0%

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

      1. sqr-neg 67.0%

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

      2. sqr-neg 67.0%

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

      3. associate-*l* 66.9%

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

   3. Simplified 66.9%

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

      1. frac-2neg 66.9%

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

      2. div-inv 66.9%

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

   6. Applied egg-rr 66.9%

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

   7. Taylor expanded in b around -inf 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-*l/ 82.1%

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

      3. associate-/l* 82.1%

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

   9. Simplified 82.1%

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

       1. clear-num 82.1%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 82.1%

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

       3. div-inv 82.1%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 82.1%

          \[\leadsto \frac{b}{a \cdot \color{blue}{-1.5}} \]

   11. Applied egg-rr 82.1%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if -2.05000000000000005e-135 < b < 2.70000000000000017e-197

   1. Initial program 81.3%

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

      1. sqr-neg 81.3%

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

      2. sqr-neg 81.3%

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

      3. associate-*l* 81.2%

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

   3. Simplified 81.2%

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

      1. prod-diff 80.9%

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

      2. *-commutative 80.9%

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

      3. associate-*r* 81.0%

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

      4. *-commutative 81.0%

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

      5. distribute-rgt-neg-in 81.0%

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

      6. *-commutative 81.0%

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

      7. distribute-rgt-neg-in 81.0%

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

      8. metadata-eval 81.0%

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

      9. *-commutative 81.0%

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

      10. fma-undefine 81.0%

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

      11. distribute-lft-neg-in 81.0%

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

      12. *-commutative 81.0%

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

      13. associate-*r* 80.9%

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

      14. *-commutative 80.9%

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

      15. distribute-rgt-neg-in 80.9%

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

      16. *-commutative 80.9%

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

      17. distribute-rgt-neg-in 80.9%

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

      18. metadata-eval 80.9%

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

      19. associate-*r* 81.0%

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

      20. *-commutative 81.0%

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

      21. associate-*l* 80.9%

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

   6. Applied egg-rr 80.9%

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

   7. Taylor expanded in a around inf 36.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{-6 \cdot c + 3 \cdot c}{a}}} \]
   8. Step-by-step derivation

      1. *-commutative 36.0%

         \[\leadsto \color{blue}{\sqrt{\frac{-6 \cdot c + 3 \cdot c}{a}} \cdot 0.3333333333333333} \]

      2. distribute-rgt-out 36.0%

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

      3. metadata-eval 36.0%

         \[\leadsto \sqrt{\frac{c \cdot \color{blue}{-3}}{a}} \cdot 0.3333333333333333 \]

   9. Simplified 36.0%

      \[\leadsto \color{blue}{\sqrt{\frac{c \cdot -3}{a}} \cdot 0.3333333333333333} \]

   10. Taylor expanded in c around 0 36.1%

       \[\leadsto \sqrt{\color{blue}{-3 \cdot \frac{c}{a}}} \cdot 0.3333333333333333 \]

   if 2.70000000000000017e-197 < b

   1. Initial program 22.4%

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

      1. sqr-neg 22.4%

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

      2. sqr-neg 22.4%

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

      3. associate-*l* 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in b around inf 78.5%

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

      1. *-commutative 78.5%

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

   7. Simplified 78.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-135}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-197}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{-3 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-149}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{\left(c \cdot \frac{-3}{a}\right) \cdot 0.1111111111111111}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-149)
   (/ b (* a -1.5))
   (if (<= b 3.4e-200)
     (sqrt (* (* c (/ -3.0 a)) 0.1111111111111111))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-149) {
		tmp = b / (a * -1.5);
	} else if (b <= 3.4e-200) {
		tmp = sqrt(((c * (-3.0 / a)) * 0.1111111111111111));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3d-149)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 3.4d-200) then
        tmp = sqrt(((c * ((-3.0d0) / a)) * 0.1111111111111111d0))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-149) {
		tmp = b / (a * -1.5);
	} else if (b <= 3.4e-200) {
		tmp = Math.sqrt(((c * (-3.0 / a)) * 0.1111111111111111));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e-149:
		tmp = b / (a * -1.5)
	elif b <= 3.4e-200:
		tmp = math.sqrt(((c * (-3.0 / a)) * 0.1111111111111111))
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-149)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 3.4e-200)
		tmp = sqrt(Float64(Float64(c * Float64(-3.0 / a)) * 0.1111111111111111));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e-149)
		tmp = b / (a * -1.5);
	elseif (b <= 3.4e-200)
		tmp = sqrt(((c * (-3.0 / a)) * 0.1111111111111111));
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-149], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.4e-200], N[Sqrt[N[(N[(c * N[(-3.0 / a), $MachinePrecision]), $MachinePrecision] * 0.1111111111111111), $MachinePrecision]], $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.0000000000000002e-149

   1. Initial program 67.2%

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

      1. sqr-neg 67.2%

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

      2. sqr-neg 67.2%

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

      3. associate-*l* 67.2%

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

   3. Simplified 67.2%

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

      1. frac-2neg 67.2%

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

      2. div-inv 67.2%

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

   6. Applied egg-rr 67.2%

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

   7. Taylor expanded in b around -inf 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-*l/ 81.5%

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

      3. associate-/l* 81.5%

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

   9. Simplified 81.5%

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

       1. clear-num 81.4%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 81.4%

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

       3. div-inv 81.5%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 81.5%

          \[\leadsto \frac{b}{a \cdot \color{blue}{-1.5}} \]

   11. Applied egg-rr 81.5%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if -3.0000000000000002e-149 < b < 3.4000000000000003e-200

   1. Initial program 80.7%

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

      1. sqr-neg 80.7%

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

      2. sqr-neg 80.7%

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

      3. associate-*l* 80.6%

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

   3. Simplified 80.6%

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

      1. prod-diff 80.3%

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

      2. *-commutative 80.3%

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

      3. associate-*r* 80.3%

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

      4. *-commutative 80.3%

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

      5. distribute-rgt-neg-in 80.3%

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

      6. *-commutative 80.3%

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

      7. distribute-rgt-neg-in 80.3%

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

      8. metadata-eval 80.3%

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

      9. *-commutative 80.3%

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

      10. fma-undefine 80.4%

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

      11. distribute-lft-neg-in 80.4%

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

      12. *-commutative 80.4%

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

      13. associate-*r* 80.2%

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

      14. *-commutative 80.2%

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

      15. distribute-rgt-neg-in 80.2%

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

      16. *-commutative 80.2%

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

      17. distribute-rgt-neg-in 80.2%

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

      18. metadata-eval 80.2%

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

      19. associate-*r* 80.4%

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

      20. *-commutative 80.4%

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

      21. associate-*l* 80.2%

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

   6. Applied egg-rr 80.2%

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

   7. Taylor expanded in a around inf 36.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{-6 \cdot c + 3 \cdot c}{a}}} \]
   8. Step-by-step derivation

      1. *-commutative 36.9%

         \[\leadsto \color{blue}{\sqrt{\frac{-6 \cdot c + 3 \cdot c}{a}} \cdot 0.3333333333333333} \]

      2. distribute-rgt-out 36.9%

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

      3. metadata-eval 36.9%

         \[\leadsto \sqrt{\frac{c \cdot \color{blue}{-3}}{a}} \cdot 0.3333333333333333 \]

   9. Simplified 36.9%

      \[\leadsto \color{blue}{\sqrt{\frac{c \cdot -3}{a}} \cdot 0.3333333333333333} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 36.8%

          \[\leadsto \color{blue}{\sqrt{\sqrt{\frac{c \cdot -3}{a}} \cdot 0.3333333333333333} \cdot \sqrt{\sqrt{\frac{c \cdot -3}{a}} \cdot 0.3333333333333333}} \]

       2. sqrt-unprod 36.9%

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

       3. swap-sqr 36.9%

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

       4. add-sqr-sqrt 37.0%

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

       5. associate-/l* 37.0%

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

       6. metadata-eval 37.0%

          \[\leadsto \sqrt{\left(c \cdot \frac{-3}{a}\right) \cdot \color{blue}{0.1111111111111111}} \]

   11. Applied egg-rr 37.0%

       \[\leadsto \color{blue}{\sqrt{\left(c \cdot \frac{-3}{a}\right) \cdot 0.1111111111111111}} \]

   if 3.4000000000000003e-200 < b

   1. Initial program 22.4%

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

      1. sqr-neg 22.4%

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

      2. sqr-neg 22.4%

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

      3. associate-*l* 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in b around inf 78.5%

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

      1. *-commutative 78.5%

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

   7. Simplified 78.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-149}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{\left(c \cdot \frac{-3}{a}\right) \cdot 0.1111111111111111}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ b (* a -1.5)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b / (a * -1.5)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = b / (a * -1.5);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 69.9%

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

      1. sqr-neg 69.9%

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

      2. sqr-neg 69.9%

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

      3. associate-*l* 69.8%

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

   3. Simplified 69.8%

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

      1. frac-2neg 69.8%

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

      2. div-inv 69.7%

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

   6. Applied egg-rr 69.8%

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

   7. Taylor expanded in b around -inf 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-*l/ 71.9%

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

      3. associate-/l* 71.9%

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

   9. Simplified 71.9%

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

       1. clear-num 71.8%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 71.8%

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

       3. div-inv 71.9%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 71.9%

          \[\leadsto \frac{b}{a \cdot \color{blue}{-1.5}} \]

   11. Applied egg-rr 71.9%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 28.1%

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

      1. sqr-neg 28.1%

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

      2. sqr-neg 28.1%

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

      3. associate-*l* 28.1%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in b around inf 69.9%

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

      1. *-commutative 69.9%

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

   7. Simplified 69.9%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (* b (/ -0.6666666666666666 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 69.9%

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

      1. sqr-neg 69.9%

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

      2. sqr-neg 69.9%

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

      3. associate-*l* 69.8%

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

   3. Simplified 69.8%

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

      1. frac-2neg 69.8%

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

      2. div-inv 69.7%

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

   6. Applied egg-rr 69.8%

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

   7. Taylor expanded in b around -inf 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-*l/ 71.9%

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

      3. associate-/l* 71.9%

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

   9. Simplified 71.9%

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

   if -4.999999999999985e-310 < b

   1. Initial program 28.1%

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

      1. sqr-neg 28.1%

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

      2. sqr-neg 28.1%

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

      3. associate-*l* 28.1%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in b around inf 69.9%

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

      1. *-commutative 69.9%

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

   7. Simplified 69.9%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-309}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2e-309) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e-309) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2d-309) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e-309) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2e-309:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2e-309)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2e-309)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2e-309], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9999999999999988e-309

   1. Initial program 69.9%

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

      1. sqr-neg 69.9%

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

      2. sqr-neg 69.9%

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

      3. associate-*l* 69.8%

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

   3. Simplified 69.8%

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

      1. frac-2neg 69.8%

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

      2. div-inv 69.7%

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

   6. Applied egg-rr 69.8%

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

   7. Taylor expanded in b around -inf 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-*l/ 71.9%

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

      3. associate-/l* 71.9%

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

   9. Simplified 71.9%

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

   if 1.9999999999999988e-309 < b

   1. Initial program 28.1%

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

      1. sqr-neg 28.1%

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

      2. sqr-neg 28.1%

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

      3. associate-*l* 28.1%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in c around 0 63.1%

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

   6. Taylor expanded in a around 0 69.6%

      \[\leadsto c \cdot \color{blue}{\frac{-0.5}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 41.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+96}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e+96) (* b (/ -0.6666666666666666 a)) (* (/ c b) 0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e+96) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * 0.5;
	}
	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 <= 3d+96) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (c / b) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e+96) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * 0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e+96:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * 0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e+96)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(c / b) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3e+96)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e+96], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3e96

   1. Initial program 60.9%

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

      1. sqr-neg 60.9%

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

      2. sqr-neg 60.9%

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

      3. associate-*l* 60.8%

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

   3. Simplified 60.8%

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

      1. frac-2neg 60.8%

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

      2. div-inv 60.8%

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

   6. Applied egg-rr 60.8%

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

   7. Taylor expanded in b around -inf 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-*l/ 49.1%

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

      3. associate-/l* 49.1%

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

   9. Simplified 49.1%

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

   if 3e96 < b

   1. Initial program 9.9%

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

      1. sqr-neg 9.9%

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

      2. sqr-neg 9.9%

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

      3. associate-*l* 9.9%

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

   3. Simplified 9.9%

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

      1. clear-num 9.9%

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

      2. inv-pow 9.9%

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

   6. Applied egg-rr 1.5%

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

   7. Taylor expanded in b around -inf 40.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+96}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 10.9% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

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) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

   1. sqr-neg 50.3%

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

   2. sqr-neg 50.3%

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

   3. associate-*l* 50.3%

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

3. Simplified 50.3%

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

   1. clear-num 50.2%

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

   2. inv-pow 50.2%

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

6. Applied egg-rr 30.4%

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

7. Taylor expanded in b around -inf 10.6%

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

8. Final simplification 10.6%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))