Cubic critical

Percentage Accurate: 51.7% → 85.7%

Time: 13.2s

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: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{+148}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.26e+148)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 2e-117)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.26e+148) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 2e-117) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.26d+148)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 2d-117) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.26e+148) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 2e-117) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.26e+148:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 2e-117:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.26e+148)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 2e-117)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.26e+148)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 2e-117)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.26e+148], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2e-117], 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[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.25999999999999997e148

   1. Initial program 42.0%

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

      1. sqr-neg 42.0%

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

      2. sqr-neg 42.0%

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

      3. associate-*l* 42.0%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in b around -inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. associate-*l/ 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -1.25999999999999997e148 < b < 2.00000000000000006e-117

   1. Initial program 89.6%

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

   if 2.00000000000000006e-117 < b

   1. Initial program 17.9%

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

      1. sqr-neg 17.9%

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

      2. sqr-neg 17.9%

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

      3. associate-*l* 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in b around inf 69.5%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-/l* 69.5%

         \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{a \cdot c}{b}}{3 \cdot a}} \]

      2. associate-/l* 72.9%

         \[\leadsto -1.5 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{3 \cdot a} \]

      3. *-commutative 72.9%

         \[\leadsto -1.5 \cdot \frac{a \cdot \frac{c}{b}}{\color{blue}{a \cdot 3}} \]

   7. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 46.3%

         \[\leadsto \color{blue}{\sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \cdot \sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}}} \]

      2. sqrt-unprod 40.0%

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

      3. *-commutative 40.0%

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

      4. *-commutative 40.0%

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

      5. swap-sqr 40.0%

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

      6. pow2 40.0%

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

      7. times-frac 40.5%

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

      8. *-inverses 40.5%

         \[\leadsto \sqrt{{\left(\color{blue}{1} \cdot \frac{\frac{c}{b}}{3}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      9. *-un-lft-identity 40.5%

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

      10. div-inv 40.4%

          \[\leadsto \sqrt{{\color{blue}{\left(\frac{c}{b} \cdot \frac{1}{3}\right)}}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      11. metadata-eval 40.4%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot \color{blue}{0.3333333333333333}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      12. metadata-eval 40.4%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot \color{blue}{2.25}} \]

   9. Applied egg-rr 40.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot 2.25}} \]
   10. Step-by-step derivation

       1. *-commutative 40.4%

          \[\leadsto \sqrt{\color{blue}{2.25 \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}}} \]

       2. metadata-eval 40.4%

          \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot -1.5\right)} \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}} \]

       3. unpow2 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \color{blue}{\left(\left(\frac{c}{b} \cdot 0.3333333333333333\right) \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)}} \]

       4. associate-*l/ 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{c \cdot 0.3333333333333333}{b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       5. metadata-eval 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{c \cdot \color{blue}{\frac{1}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       6. div-inv 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{c}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       7. clear-num 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       8. associate-/r* 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{1}{\frac{3}{c} \cdot b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       9. associate-*l/ 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{c \cdot 0.3333333333333333}{b}}\right)} \]

       10. metadata-eval 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{c \cdot \color{blue}{\frac{1}{3}}}{b}\right)} \]

       11. div-inv 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{c}{3}}}{b}\right)} \]

       12. clear-num 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b}\right)} \]

       13. associate-/r* 40.4%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{1}{\frac{3}{c} \cdot b}}\right)} \]

       14. swap-sqr 40.4%

           \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right) \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)}} \]

       15. div-inv 40.5%

           \[\leadsto \sqrt{\color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)} \]

       16. div-inv 40.5%

           \[\leadsto \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       17. sqrt-unprod 51.1%

           \[\leadsto \color{blue}{\sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       18. add-sqr-sqrt 82.1%

           \[\leadsto \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \]

       19. associate-/r* 84.0%

           \[\leadsto \color{blue}{\frac{\frac{-1.5}{\frac{3}{c}}}{b}} \]

   11. Applied egg-rr 84.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{+148}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+148}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e+148)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 2.1e-117)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e+148) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 2.1e-117) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3d+148)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 2.1d-117) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    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 <= -3e+148) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 2.1e-117) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e+148:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 2.1e-117:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e+148)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 2.1e-117)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e+148)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 2.1e-117)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e+148], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.1e-117], 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[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.00000000000000015e148

   1. Initial program 42.0%

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

      1. sqr-neg 42.0%

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

      2. sqr-neg 42.0%

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

      3. associate-*l* 42.0%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in b around -inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. associate-*l/ 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -3.00000000000000015e148 < b < 2.0999999999999999e-117

   1. Initial program 89.6%

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

      1. sqr-neg 89.6%

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

      2. sqr-neg 89.6%

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

      3. associate-*l* 89.6%

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

   3. Simplified 89.6%

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

   if 2.0999999999999999e-117 < b

   1. Initial program 17.9%

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

      1. sqr-neg 17.9%

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

      2. sqr-neg 17.9%

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

      3. associate-*l* 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in b around inf 69.5%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-/l* 69.5%

         \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{a \cdot c}{b}}{3 \cdot a}} \]

      2. associate-/l* 72.9%

         \[\leadsto -1.5 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{3 \cdot a} \]

      3. *-commutative 72.9%

         \[\leadsto -1.5 \cdot \frac{a \cdot \frac{c}{b}}{\color{blue}{a \cdot 3}} \]

   7. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 46.3%

         \[\leadsto \color{blue}{\sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \cdot \sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}}} \]

      2. sqrt-unprod 40.0%

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

      3. *-commutative 40.0%

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

      4. *-commutative 40.0%

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

      5. swap-sqr 40.0%

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

      6. pow2 40.0%

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

      7. times-frac 40.5%

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

      8. *-inverses 40.5%

         \[\leadsto \sqrt{{\left(\color{blue}{1} \cdot \frac{\frac{c}{b}}{3}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      9. *-un-lft-identity 40.5%

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

      10. div-inv 40.4%

          \[\leadsto \sqrt{{\color{blue}{\left(\frac{c}{b} \cdot \frac{1}{3}\right)}}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      11. metadata-eval 40.4%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot \color{blue}{0.3333333333333333}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      12. metadata-eval 40.4%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot \color{blue}{2.25}} \]

   9. Applied egg-rr 40.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot 2.25}} \]
   10. Step-by-step derivation

       1. *-commutative 40.4%

          \[\leadsto \sqrt{\color{blue}{2.25 \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}}} \]

       2. metadata-eval 40.4%

          \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot -1.5\right)} \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}} \]

       3. unpow2 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \color{blue}{\left(\left(\frac{c}{b} \cdot 0.3333333333333333\right) \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)}} \]

       4. associate-*l/ 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{c \cdot 0.3333333333333333}{b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       5. metadata-eval 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{c \cdot \color{blue}{\frac{1}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       6. div-inv 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{c}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       7. clear-num 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       8. associate-/r* 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{1}{\frac{3}{c} \cdot b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       9. associate-*l/ 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{c \cdot 0.3333333333333333}{b}}\right)} \]

       10. metadata-eval 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{c \cdot \color{blue}{\frac{1}{3}}}{b}\right)} \]

       11. div-inv 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{c}{3}}}{b}\right)} \]

       12. clear-num 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b}\right)} \]

       13. associate-/r* 40.4%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{1}{\frac{3}{c} \cdot b}}\right)} \]

       14. swap-sqr 40.4%

           \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right) \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)}} \]

       15. div-inv 40.5%

           \[\leadsto \sqrt{\color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)} \]

       16. div-inv 40.5%

           \[\leadsto \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       17. sqrt-unprod 51.1%

           \[\leadsto \color{blue}{\sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       18. add-sqr-sqrt 82.1%

           \[\leadsto \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \]

       19. associate-/r* 84.0%

           \[\leadsto \color{blue}{\frac{\frac{-1.5}{\frac{3}{c}}}{b}} \]

   11. Applied egg-rr 84.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+148}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-121}:\\ \;\;\;\;\frac{b \cdot \left(\frac{2}{a} - \frac{c}{{b}^{2}} \cdot 1.5\right)}{-3}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e-121)
   (/ (* b (- (/ 2.0 a) (* (/ c (pow b 2.0)) 1.5))) -3.0)
   (if (<= b 2.4e-119)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-121) {
		tmp = (b * ((2.0 / a) - ((c / pow(b, 2.0)) * 1.5))) / -3.0;
	} else if (b <= 2.4e-119) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.8d-121)) then
        tmp = (b * ((2.0d0 / a) - ((c / (b ** 2.0d0)) * 1.5d0))) / (-3.0d0)
    else if (b <= 2.4d-119) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-121) {
		tmp = (b * ((2.0 / a) - ((c / Math.pow(b, 2.0)) * 1.5))) / -3.0;
	} else if (b <= 2.4e-119) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.8e-121:
		tmp = (b * ((2.0 / a) - ((c / math.pow(b, 2.0)) * 1.5))) / -3.0
	elif b <= 2.4e-119:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e-121)
		tmp = Float64(Float64(b * Float64(Float64(2.0 / a) - Float64(Float64(c / (b ^ 2.0)) * 1.5))) / -3.0);
	elseif (b <= 2.4e-119)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.8e-121)
		tmp = (b * ((2.0 / a) - ((c / (b ^ 2.0)) * 1.5))) / -3.0;
	elseif (b <= 2.4e-119)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e-121], N[(N[(b * N[(N[(2.0 / a), $MachinePrecision] - N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 2.4e-119], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.79999999999999992e-121

   1. Initial program 72.5%

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

      1. sqr-neg 72.5%

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

      2. sqr-neg 72.5%

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

      3. associate-*l* 72.5%

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

   3. Simplified 72.5%

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

      1. frac-2neg 72.5%

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

      2. div-inv 72.4%

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

   6. Applied egg-rr 72.4%

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

      1. fma-undefine 72.4%

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

      2. unpow2 72.4%

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

      3. associate-*l* 72.4%

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

      4. *-commutative 72.4%

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

      5. +-commutative 72.4%

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

      6. fma-define 72.4%

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

      7. associate-/r* 72.4%

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

   8. Simplified 72.4%

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

      1. associate-*r/ 72.4%

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

   10. Applied egg-rr 72.4%

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

       1. associate-*r/ 72.4%

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

       2. *-rgt-identity 72.4%

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

       3. *-commutative 72.4%

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

   12. Simplified 72.4%

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

   13. Taylor expanded in b around -inf 81.0%

       \[\leadsto \frac{\color{blue}{-1 \cdot \left(b \cdot \left(1.5 \cdot \frac{c}{{b}^{2}} - 2 \cdot \frac{1}{a}\right)\right)}}{-3} \]
   14. Step-by-step derivation

       1. associate-*r* 81.0%

          \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\right) \cdot \left(1.5 \cdot \frac{c}{{b}^{2}} - 2 \cdot \frac{1}{a}\right)}}{-3} \]

       2. neg-mul-1 81.0%

          \[\leadsto \frac{\color{blue}{\left(-b\right)} \cdot \left(1.5 \cdot \frac{c}{{b}^{2}} - 2 \cdot \frac{1}{a}\right)}{-3} \]

       3. *-commutative 81.0%

          \[\leadsto \frac{\left(-b\right) \cdot \left(\color{blue}{\frac{c}{{b}^{2}} \cdot 1.5} - 2 \cdot \frac{1}{a}\right)}{-3} \]

       4. associate-*r/ 81.0%

          \[\leadsto \frac{\left(-b\right) \cdot \left(\frac{c}{{b}^{2}} \cdot 1.5 - \color{blue}{\frac{2 \cdot 1}{a}}\right)}{-3} \]

       5. metadata-eval 81.0%

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

   15. Simplified 81.0%

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

   if -1.79999999999999992e-121 < b < 2.40000000000000009e-119

   1. Initial program 83.3%

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

      1. sqr-neg 83.3%

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

      2. sqr-neg 83.3%

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

      3. associate-*l* 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in b around 0 80.3%

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

      1. *-commutative 80.3%

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

      2. metadata-eval 80.3%

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

      3. distribute-rgt-neg-in 80.3%

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

      4. *-commutative 80.3%

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

      5. associate-*r* 80.4%

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

      6. neg-sub0 80.4%

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

      7. *-commutative 80.4%

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

   7. Applied egg-rr 80.4%

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

      1. neg-sub0 80.4%

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

      2. *-commutative 80.4%

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

      3. distribute-rgt-neg-in 80.4%

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

      4. distribute-rgt-neg-in 80.4%

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

      5. metadata-eval 80.4%

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

   9. Simplified 80.4%

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

   if 2.40000000000000009e-119 < b

   1. Initial program 17.9%

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

      1. sqr-neg 17.9%

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

      2. sqr-neg 17.9%

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

      3. associate-*l* 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in b around inf 69.5%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-/l* 69.5%

         \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{a \cdot c}{b}}{3 \cdot a}} \]

      2. associate-/l* 72.9%

         \[\leadsto -1.5 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{3 \cdot a} \]

      3. *-commutative 72.9%

         \[\leadsto -1.5 \cdot \frac{a \cdot \frac{c}{b}}{\color{blue}{a \cdot 3}} \]

   7. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 46.3%

         \[\leadsto \color{blue}{\sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \cdot \sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}}} \]

      2. sqrt-unprod 40.0%

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

      3. *-commutative 40.0%

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

      4. *-commutative 40.0%

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

      5. swap-sqr 40.0%

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

      6. pow2 40.0%

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

      7. times-frac 40.5%

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

      8. *-inverses 40.5%

         \[\leadsto \sqrt{{\left(\color{blue}{1} \cdot \frac{\frac{c}{b}}{3}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      9. *-un-lft-identity 40.5%

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

      10. div-inv 40.4%

          \[\leadsto \sqrt{{\color{blue}{\left(\frac{c}{b} \cdot \frac{1}{3}\right)}}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      11. metadata-eval 40.4%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot \color{blue}{0.3333333333333333}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      12. metadata-eval 40.4%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot \color{blue}{2.25}} \]

   9. Applied egg-rr 40.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot 2.25}} \]
   10. Step-by-step derivation

       1. *-commutative 40.4%

          \[\leadsto \sqrt{\color{blue}{2.25 \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}}} \]

       2. metadata-eval 40.4%

          \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot -1.5\right)} \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}} \]

       3. unpow2 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \color{blue}{\left(\left(\frac{c}{b} \cdot 0.3333333333333333\right) \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)}} \]

       4. associate-*l/ 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{c \cdot 0.3333333333333333}{b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       5. metadata-eval 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{c \cdot \color{blue}{\frac{1}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       6. div-inv 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{c}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       7. clear-num 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       8. associate-/r* 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{1}{\frac{3}{c} \cdot b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       9. associate-*l/ 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{c \cdot 0.3333333333333333}{b}}\right)} \]

       10. metadata-eval 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{c \cdot \color{blue}{\frac{1}{3}}}{b}\right)} \]

       11. div-inv 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{c}{3}}}{b}\right)} \]

       12. clear-num 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b}\right)} \]

       13. associate-/r* 40.4%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{1}{\frac{3}{c} \cdot b}}\right)} \]

       14. swap-sqr 40.4%

           \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right) \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)}} \]

       15. div-inv 40.5%

           \[\leadsto \sqrt{\color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)} \]

       16. div-inv 40.5%

           \[\leadsto \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       17. sqrt-unprod 51.1%

           \[\leadsto \color{blue}{\sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       18. add-sqr-sqrt 82.1%

           \[\leadsto \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \]

       19. associate-/r* 84.0%

           \[\leadsto \color{blue}{\frac{\frac{-1.5}{\frac{3}{c}}}{b}} \]

   11. Applied egg-rr 84.1%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-121}:\\ \;\;\;\;\frac{b \cdot \left(\frac{2}{a} - \frac{c}{{b}^{2}} \cdot 1.5\right)}{-3}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e-121)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 2.1e-117)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-121) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 2.1e-117) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.5d-121)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 2.1d-117) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-121) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 2.1e-117) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.5e-121:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 2.1e-117:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e-121)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 2.1e-117)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.5e-121)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 2.1e-117)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e-121], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 2.1e-117], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5e-121

   1. Initial program 72.5%

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

      1. sqr-neg 72.5%

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

      2. sqr-neg 72.5%

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

      3. associate-*l* 72.5%

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

   3. Simplified 72.5%

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

      1. frac-2neg 72.5%

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

      2. div-inv 72.4%

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

   6. Applied egg-rr 72.4%

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

      1. fma-undefine 72.4%

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

      2. unpow2 72.4%

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

      3. associate-*l* 72.4%

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

      4. *-commutative 72.4%

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

      5. +-commutative 72.4%

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

      6. fma-define 72.4%

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

      7. associate-/r* 72.4%

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

   8. Simplified 72.4%

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

      1. associate-*r/ 72.4%

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

   10. Applied egg-rr 72.4%

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

       1. associate-*r/ 72.4%

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

       2. *-rgt-identity 72.4%

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

       3. *-commutative 72.4%

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

   12. Simplified 72.4%

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

   13. Taylor expanded in b around -inf 81.0%

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

       1. *-commutative 81.0%

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

   15. Simplified 81.0%

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

   if -1.5e-121 < b < 2.0999999999999999e-117

   1. Initial program 83.3%

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

      1. sqr-neg 83.3%

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

      2. sqr-neg 83.3%

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

      3. associate-*l* 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in b around 0 80.3%

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

      1. *-commutative 80.3%

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

      2. metadata-eval 80.3%

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

      3. distribute-rgt-neg-in 80.3%

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

      4. *-commutative 80.3%

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

      5. associate-*r* 80.4%

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

      6. neg-sub0 80.4%

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

      7. *-commutative 80.4%

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

   7. Applied egg-rr 80.4%

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

      1. neg-sub0 80.4%

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

      2. *-commutative 80.4%

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

      3. distribute-rgt-neg-in 80.4%

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

      4. distribute-rgt-neg-in 80.4%

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

      5. metadata-eval 80.4%

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

   9. Simplified 80.4%

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

   if 2.0999999999999999e-117 < b

   1. Initial program 17.9%

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

      1. sqr-neg 17.9%

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

      2. sqr-neg 17.9%

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

      3. associate-*l* 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in b around inf 69.5%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-/l* 69.5%

         \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{a \cdot c}{b}}{3 \cdot a}} \]

      2. associate-/l* 72.9%

         \[\leadsto -1.5 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{3 \cdot a} \]

      3. *-commutative 72.9%

         \[\leadsto -1.5 \cdot \frac{a \cdot \frac{c}{b}}{\color{blue}{a \cdot 3}} \]

   7. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 46.3%

         \[\leadsto \color{blue}{\sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \cdot \sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}}} \]

      2. sqrt-unprod 40.0%

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

      3. *-commutative 40.0%

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

      4. *-commutative 40.0%

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

      5. swap-sqr 40.0%

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

      6. pow2 40.0%

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

      7. times-frac 40.5%

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

      8. *-inverses 40.5%

         \[\leadsto \sqrt{{\left(\color{blue}{1} \cdot \frac{\frac{c}{b}}{3}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      9. *-un-lft-identity 40.5%

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

      10. div-inv 40.4%

          \[\leadsto \sqrt{{\color{blue}{\left(\frac{c}{b} \cdot \frac{1}{3}\right)}}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      11. metadata-eval 40.4%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot \color{blue}{0.3333333333333333}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      12. metadata-eval 40.4%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot \color{blue}{2.25}} \]

   9. Applied egg-rr 40.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot 2.25}} \]
   10. Step-by-step derivation

       1. *-commutative 40.4%

          \[\leadsto \sqrt{\color{blue}{2.25 \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}}} \]

       2. metadata-eval 40.4%

          \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot -1.5\right)} \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}} \]

       3. unpow2 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \color{blue}{\left(\left(\frac{c}{b} \cdot 0.3333333333333333\right) \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)}} \]

       4. associate-*l/ 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{c \cdot 0.3333333333333333}{b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       5. metadata-eval 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{c \cdot \color{blue}{\frac{1}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       6. div-inv 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{c}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       7. clear-num 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       8. associate-/r* 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{1}{\frac{3}{c} \cdot b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       9. associate-*l/ 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{c \cdot 0.3333333333333333}{b}}\right)} \]

       10. metadata-eval 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{c \cdot \color{blue}{\frac{1}{3}}}{b}\right)} \]

       11. div-inv 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{c}{3}}}{b}\right)} \]

       12. clear-num 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b}\right)} \]

       13. associate-/r* 40.4%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{1}{\frac{3}{c} \cdot b}}\right)} \]

       14. swap-sqr 40.4%

           \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right) \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)}} \]

       15. div-inv 40.5%

           \[\leadsto \sqrt{\color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)} \]

       16. div-inv 40.5%

           \[\leadsto \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       17. sqrt-unprod 51.1%

           \[\leadsto \color{blue}{\sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       18. add-sqr-sqrt 82.1%

           \[\leadsto \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \]

       19. associate-/r* 84.0%

           \[\leadsto \color{blue}{\frac{\frac{-1.5}{\frac{3}{c}}}{b}} \]

   11. Applied egg-rr 84.1%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-123}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.25e-123)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 1.22e-117)
     (/ (- (sqrt (* (* a c) -3.0)) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e-123) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 1.22e-117) {
		tmp = (sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.25d-123)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 1.22d-117) then
        tmp = (sqrt(((a * c) * (-3.0d0))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e-123) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 1.22e-117) {
		tmp = (Math.sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.25e-123:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 1.22e-117:
		tmp = (math.sqrt(((a * c) * -3.0)) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.25e-123)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 1.22e-117)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) * -3.0)) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.25e-123)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 1.22e-117)
		tmp = (sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.25e-123], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 1.22e-117], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.25000000000000007e-123

   1. Initial program 72.5%

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

      1. sqr-neg 72.5%

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

      2. sqr-neg 72.5%

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

      3. associate-*l* 72.5%

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

   3. Simplified 72.5%

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

      1. frac-2neg 72.5%

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

      2. div-inv 72.4%

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

   6. Applied egg-rr 72.4%

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

      1. fma-undefine 72.4%

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

      2. unpow2 72.4%

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

      3. associate-*l* 72.4%

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

      4. *-commutative 72.4%

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

      5. +-commutative 72.4%

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

      6. fma-define 72.4%

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

      7. associate-/r* 72.4%

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

   8. Simplified 72.4%

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

      1. associate-*r/ 72.4%

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

   10. Applied egg-rr 72.4%

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

       1. associate-*r/ 72.4%

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

       2. *-rgt-identity 72.4%

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

       3. *-commutative 72.4%

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

   12. Simplified 72.4%

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

   13. Taylor expanded in b around -inf 81.0%

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

       1. *-commutative 81.0%

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

   15. Simplified 81.0%

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

   if -1.25000000000000007e-123 < b < 1.21999999999999997e-117

   1. Initial program 83.3%

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

      1. sqr-neg 83.3%

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

      2. sqr-neg 83.3%

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

      3. associate-*l* 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in b around 0 80.3%

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

   if 1.21999999999999997e-117 < b

   1. Initial program 17.9%

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

      1. sqr-neg 17.9%

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

      2. sqr-neg 17.9%

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

      3. associate-*l* 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in b around inf 69.5%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-/l* 69.5%

         \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{a \cdot c}{b}}{3 \cdot a}} \]

      2. associate-/l* 72.9%

         \[\leadsto -1.5 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{3 \cdot a} \]

      3. *-commutative 72.9%

         \[\leadsto -1.5 \cdot \frac{a \cdot \frac{c}{b}}{\color{blue}{a \cdot 3}} \]

   7. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 46.3%

         \[\leadsto \color{blue}{\sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \cdot \sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}}} \]

      2. sqrt-unprod 40.0%

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

      3. *-commutative 40.0%

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

      4. *-commutative 40.0%

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

      5. swap-sqr 40.0%

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

      6. pow2 40.0%

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

      7. times-frac 40.5%

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

      8. *-inverses 40.5%

         \[\leadsto \sqrt{{\left(\color{blue}{1} \cdot \frac{\frac{c}{b}}{3}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      9. *-un-lft-identity 40.5%

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

      10. div-inv 40.4%

          \[\leadsto \sqrt{{\color{blue}{\left(\frac{c}{b} \cdot \frac{1}{3}\right)}}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      11. metadata-eval 40.4%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot \color{blue}{0.3333333333333333}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      12. metadata-eval 40.4%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot \color{blue}{2.25}} \]

   9. Applied egg-rr 40.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot 2.25}} \]
   10. Step-by-step derivation

       1. *-commutative 40.4%

          \[\leadsto \sqrt{\color{blue}{2.25 \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}}} \]

       2. metadata-eval 40.4%

          \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot -1.5\right)} \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}} \]

       3. unpow2 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \color{blue}{\left(\left(\frac{c}{b} \cdot 0.3333333333333333\right) \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)}} \]

       4. associate-*l/ 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{c \cdot 0.3333333333333333}{b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       5. metadata-eval 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{c \cdot \color{blue}{\frac{1}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       6. div-inv 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{c}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       7. clear-num 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       8. associate-/r* 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{1}{\frac{3}{c} \cdot b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       9. associate-*l/ 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{c \cdot 0.3333333333333333}{b}}\right)} \]

       10. metadata-eval 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{c \cdot \color{blue}{\frac{1}{3}}}{b}\right)} \]

       11. div-inv 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{c}{3}}}{b}\right)} \]

       12. clear-num 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b}\right)} \]

       13. associate-/r* 40.4%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{1}{\frac{3}{c} \cdot b}}\right)} \]

       14. swap-sqr 40.4%

           \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right) \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)}} \]

       15. div-inv 40.5%

           \[\leadsto \sqrt{\color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)} \]

       16. div-inv 40.5%

           \[\leadsto \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       17. sqrt-unprod 51.1%

           \[\leadsto \color{blue}{\sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       18. add-sqr-sqrt 82.1%

           \[\leadsto \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \]

       19. associate-/r* 84.0%

           \[\leadsto \color{blue}{\frac{\frac{-1.5}{\frac{3}{c}}}{b}} \]

   11. Applied egg-rr 84.1%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-123}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{b - \sqrt{\left(a \cdot c\right) \cdot -3}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.25e-121)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 2.1e-117)
     (/ (/ (- b (sqrt (* (* a c) -3.0))) a) -3.0)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.25e-121) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 2.1e-117) {
		tmp = ((b - sqrt(((a * c) * -3.0))) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.25d-121)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 2.1d-117) then
        tmp = ((b - sqrt(((a * c) * (-3.0d0)))) / a) / (-3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.25e-121) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 2.1e-117) {
		tmp = ((b - Math.sqrt(((a * c) * -3.0))) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.25e-121:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 2.1e-117:
		tmp = ((b - math.sqrt(((a * c) * -3.0))) / a) / -3.0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.25e-121)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 2.1e-117)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(Float64(a * c) * -3.0))) / a) / -3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.25e-121)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 2.1e-117)
		tmp = ((b - sqrt(((a * c) * -3.0))) / a) / -3.0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.25e-121], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 2.1e-117], N[(N[(N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2500000000000002e-121

   1. Initial program 72.5%

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

      1. sqr-neg 72.5%

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

      2. sqr-neg 72.5%

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

      3. associate-*l* 72.5%

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

   3. Simplified 72.5%

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

      1. frac-2neg 72.5%

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

      2. div-inv 72.4%

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

   6. Applied egg-rr 72.4%

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

      1. fma-undefine 72.4%

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

      2. unpow2 72.4%

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

      3. associate-*l* 72.4%

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

      4. *-commutative 72.4%

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

      5. +-commutative 72.4%

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

      6. fma-define 72.4%

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

      7. associate-/r* 72.4%

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

   8. Simplified 72.4%

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

      1. associate-*r/ 72.4%

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

   10. Applied egg-rr 72.4%

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

       1. associate-*r/ 72.4%

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

       2. *-rgt-identity 72.4%

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

       3. *-commutative 72.4%

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

   12. Simplified 72.4%

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

   13. Taylor expanded in b around -inf 81.0%

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

       1. *-commutative 81.0%

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

   15. Simplified 81.0%

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

   if -2.2500000000000002e-121 < b < 2.0999999999999999e-117

   1. Initial program 83.3%

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

      1. sqr-neg 83.3%

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

      2. sqr-neg 83.3%

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

      3. associate-*l* 83.1%

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

   3. Simplified 83.1%

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

      1. frac-2neg 83.1%

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

      2. div-inv 83.0%

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

   6. Applied egg-rr 82.9%

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

      1. fma-undefine 82.9%

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

      2. unpow2 82.9%

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

      3. associate-*l* 83.0%

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

      4. *-commutative 83.0%

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

      5. +-commutative 83.0%

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

      6. fma-define 83.0%

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

      7. associate-/r* 82.9%

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

   8. Simplified 82.9%

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

      1. associate-*r/ 83.0%

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

   10. Applied egg-rr 83.0%

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

       1. associate-*r/ 83.1%

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

       2. *-rgt-identity 83.1%

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

       3. *-commutative 83.1%

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

   12. Simplified 83.1%

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

   13. Taylor expanded in c around inf 80.2%

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

   if 2.0999999999999999e-117 < b

   1. Initial program 17.9%

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

      1. sqr-neg 17.9%

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

      2. sqr-neg 17.9%

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

      3. associate-*l* 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in b around inf 69.5%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-/l* 69.5%

         \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{a \cdot c}{b}}{3 \cdot a}} \]

      2. associate-/l* 72.9%

         \[\leadsto -1.5 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{3 \cdot a} \]

      3. *-commutative 72.9%

         \[\leadsto -1.5 \cdot \frac{a \cdot \frac{c}{b}}{\color{blue}{a \cdot 3}} \]

   7. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 46.3%

         \[\leadsto \color{blue}{\sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \cdot \sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}}} \]

      2. sqrt-unprod 40.0%

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

      3. *-commutative 40.0%

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

      4. *-commutative 40.0%

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

      5. swap-sqr 40.0%

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

      6. pow2 40.0%

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

      7. times-frac 40.5%

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

      8. *-inverses 40.5%

         \[\leadsto \sqrt{{\left(\color{blue}{1} \cdot \frac{\frac{c}{b}}{3}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      9. *-un-lft-identity 40.5%

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

      10. div-inv 40.4%

          \[\leadsto \sqrt{{\color{blue}{\left(\frac{c}{b} \cdot \frac{1}{3}\right)}}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      11. metadata-eval 40.4%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot \color{blue}{0.3333333333333333}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      12. metadata-eval 40.4%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot \color{blue}{2.25}} \]

   9. Applied egg-rr 40.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot 2.25}} \]
   10. Step-by-step derivation

       1. *-commutative 40.4%

          \[\leadsto \sqrt{\color{blue}{2.25 \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}}} \]

       2. metadata-eval 40.4%

          \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot -1.5\right)} \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}} \]

       3. unpow2 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \color{blue}{\left(\left(\frac{c}{b} \cdot 0.3333333333333333\right) \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)}} \]

       4. associate-*l/ 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{c \cdot 0.3333333333333333}{b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       5. metadata-eval 40.4%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{c \cdot \color{blue}{\frac{1}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       6. div-inv 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{c}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       7. clear-num 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       8. associate-/r* 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{1}{\frac{3}{c} \cdot b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       9. associate-*l/ 40.5%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{c \cdot 0.3333333333333333}{b}}\right)} \]

       10. metadata-eval 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{c \cdot \color{blue}{\frac{1}{3}}}{b}\right)} \]

       11. div-inv 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{c}{3}}}{b}\right)} \]

       12. clear-num 40.5%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b}\right)} \]

       13. associate-/r* 40.4%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{1}{\frac{3}{c} \cdot b}}\right)} \]

       14. swap-sqr 40.4%

           \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right) \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)}} \]

       15. div-inv 40.5%

           \[\leadsto \sqrt{\color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)} \]

       16. div-inv 40.5%

           \[\leadsto \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       17. sqrt-unprod 51.1%

           \[\leadsto \color{blue}{\sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       18. add-sqr-sqrt 82.1%

           \[\leadsto \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \]

       19. associate-/r* 84.0%

           \[\leadsto \color{blue}{\frac{\frac{-1.5}{\frac{3}{c}}}{b}} \]

   11. Applied egg-rr 84.1%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{b - \sqrt{\left(a \cdot c\right) \cdot -3}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (/ (* b 2.0) a) -3.0) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    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-310) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = ((b * 2.0) / a) / -3.0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = ((b * 2.0) / a) / -3.0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 75.8%

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

      1. sqr-neg 75.8%

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

      2. sqr-neg 75.8%

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

      3. associate-*l* 75.8%

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

   3. Simplified 75.8%

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

      1. frac-2neg 75.8%

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

      2. div-inv 75.7%

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

   6. Applied egg-rr 75.7%

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

      1. fma-undefine 75.7%

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

      2. unpow2 75.7%

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

      3. associate-*l* 75.7%

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

      4. *-commutative 75.7%

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

      5. +-commutative 75.7%

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

      6. fma-define 75.8%

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

      7. associate-/r* 75.7%

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

   8. Simplified 75.7%

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

      1. associate-*r/ 75.7%

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

   10. Applied egg-rr 75.7%

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

       1. associate-*r/ 75.7%

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

       2. *-rgt-identity 75.7%

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

       3. *-commutative 75.7%

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

   12. Simplified 75.7%

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

   13. Taylor expanded in b around -inf 69.1%

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

       1. *-commutative 69.1%

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

   15. Simplified 69.1%

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

   if -1.999999999999994e-310 < b

   1. Initial program 27.2%

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

      1. sqr-neg 27.2%

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

      2. sqr-neg 27.2%

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

      3. associate-*l* 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in b around inf 59.6%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-/l* 59.6%

         \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{a \cdot c}{b}}{3 \cdot a}} \]

      2. associate-/l* 63.4%

         \[\leadsto -1.5 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{3 \cdot a} \]

      3. *-commutative 63.4%

         \[\leadsto -1.5 \cdot \frac{a \cdot \frac{c}{b}}{\color{blue}{a \cdot 3}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 40.2%

         \[\leadsto \color{blue}{\sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \cdot \sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}}} \]

      2. sqrt-unprod 34.9%

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

      3. *-commutative 34.9%

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

      4. *-commutative 34.9%

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

      5. swap-sqr 35.0%

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

      6. pow2 35.0%

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

      7. times-frac 35.3%

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

      8. *-inverses 35.3%

         \[\leadsto \sqrt{{\left(\color{blue}{1} \cdot \frac{\frac{c}{b}}{3}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      9. *-un-lft-identity 35.3%

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

      10. div-inv 35.3%

          \[\leadsto \sqrt{{\color{blue}{\left(\frac{c}{b} \cdot \frac{1}{3}\right)}}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      11. metadata-eval 35.3%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot \color{blue}{0.3333333333333333}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      12. metadata-eval 35.3%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot \color{blue}{2.25}} \]

   9. Applied egg-rr 35.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot 2.25}} \]
   10. Step-by-step derivation

       1. *-commutative 35.3%

          \[\leadsto \sqrt{\color{blue}{2.25 \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}}} \]

       2. metadata-eval 35.3%

          \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot -1.5\right)} \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}} \]

       3. unpow2 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \color{blue}{\left(\left(\frac{c}{b} \cdot 0.3333333333333333\right) \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)}} \]

       4. associate-*l/ 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{c \cdot 0.3333333333333333}{b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       5. metadata-eval 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{c \cdot \color{blue}{\frac{1}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       6. div-inv 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{c}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       7. clear-num 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       8. associate-/r* 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{1}{\frac{3}{c} \cdot b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       9. associate-*l/ 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{c \cdot 0.3333333333333333}{b}}\right)} \]

       10. metadata-eval 35.3%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{c \cdot \color{blue}{\frac{1}{3}}}{b}\right)} \]

       11. div-inv 35.3%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{c}{3}}}{b}\right)} \]

       12. clear-num 35.3%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b}\right)} \]

       13. associate-/r* 35.3%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{1}{\frac{3}{c} \cdot b}}\right)} \]

       14. swap-sqr 35.3%

           \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right) \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)}} \]

       15. div-inv 35.3%

           \[\leadsto \sqrt{\color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)} \]

       16. div-inv 35.3%

           \[\leadsto \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       17. sqrt-unprod 44.2%

           \[\leadsto \color{blue}{\sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       18. add-sqr-sqrt 71.1%

           \[\leadsto \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \]

       19. associate-/r* 72.8%

           \[\leadsto \color{blue}{\frac{\frac{-1.5}{\frac{3}{c}}}{b}} \]

   11. Applied egg-rr 72.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (* b -0.6666666666666666) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		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-310)) 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-310) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		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-310)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		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-310], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 75.8%

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

      1. sqr-neg 75.8%

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

      2. sqr-neg 75.8%

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

      3. associate-*l* 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in b around -inf 69.0%

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

      1. *-commutative 69.0%

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

   7. Simplified 69.0%

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

      1. associate-*l/ 69.1%

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

   9. Applied egg-rr 69.1%

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

   if -1.999999999999994e-310 < b

   1. Initial program 27.2%

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

      1. sqr-neg 27.2%

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

      2. sqr-neg 27.2%

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

      3. associate-*l* 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in b around inf 59.6%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-/l* 59.6%

         \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{a \cdot c}{b}}{3 \cdot a}} \]

      2. associate-/l* 63.4%

         \[\leadsto -1.5 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{3 \cdot a} \]

      3. *-commutative 63.4%

         \[\leadsto -1.5 \cdot \frac{a \cdot \frac{c}{b}}{\color{blue}{a \cdot 3}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 40.2%

         \[\leadsto \color{blue}{\sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \cdot \sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}}} \]

      2. sqrt-unprod 34.9%

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

      3. *-commutative 34.9%

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

      4. *-commutative 34.9%

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

      5. swap-sqr 35.0%

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

      6. pow2 35.0%

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

      7. times-frac 35.3%

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

      8. *-inverses 35.3%

         \[\leadsto \sqrt{{\left(\color{blue}{1} \cdot \frac{\frac{c}{b}}{3}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      9. *-un-lft-identity 35.3%

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

      10. div-inv 35.3%

          \[\leadsto \sqrt{{\color{blue}{\left(\frac{c}{b} \cdot \frac{1}{3}\right)}}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      11. metadata-eval 35.3%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot \color{blue}{0.3333333333333333}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      12. metadata-eval 35.3%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot \color{blue}{2.25}} \]

   9. Applied egg-rr 35.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot 2.25}} \]
   10. Step-by-step derivation

       1. *-commutative 35.3%

          \[\leadsto \sqrt{\color{blue}{2.25 \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}}} \]

       2. metadata-eval 35.3%

          \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot -1.5\right)} \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}} \]

       3. unpow2 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \color{blue}{\left(\left(\frac{c}{b} \cdot 0.3333333333333333\right) \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)}} \]

       4. associate-*l/ 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{c \cdot 0.3333333333333333}{b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       5. metadata-eval 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{c \cdot \color{blue}{\frac{1}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       6. div-inv 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{c}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       7. clear-num 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       8. associate-/r* 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{1}{\frac{3}{c} \cdot b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       9. associate-*l/ 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{c \cdot 0.3333333333333333}{b}}\right)} \]

       10. metadata-eval 35.3%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{c \cdot \color{blue}{\frac{1}{3}}}{b}\right)} \]

       11. div-inv 35.3%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{c}{3}}}{b}\right)} \]

       12. clear-num 35.3%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b}\right)} \]

       13. associate-/r* 35.3%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{1}{\frac{3}{c} \cdot b}}\right)} \]

       14. swap-sqr 35.3%

           \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right) \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)}} \]

       15. div-inv 35.3%

           \[\leadsto \sqrt{\color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)} \]

       16. div-inv 35.3%

           \[\leadsto \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       17. sqrt-unprod 44.2%

           \[\leadsto \color{blue}{\sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       18. add-sqr-sqrt 71.1%

           \[\leadsto \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \]

       19. associate-/r* 72.8%

           \[\leadsto \color{blue}{\frac{\frac{-1.5}{\frac{3}{c}}}{b}} \]

   11. Applied egg-rr 72.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = b / (a * -1.5);
	} 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-310)) then
        tmp = b / (a * (-1.5d0))
    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-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = b / (a * -1.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 75.8%

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

      1. sqr-neg 75.8%

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

      2. sqr-neg 75.8%

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

      3. associate-*l* 75.8%

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

   3. Simplified 75.8%

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

      1. frac-2neg 75.8%

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

      2. div-inv 75.7%

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

   6. Applied egg-rr 75.7%

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

      1. fma-undefine 75.7%

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

      2. unpow2 75.7%

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

      3. associate-*l* 75.7%

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

      4. *-commutative 75.7%

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

      5. +-commutative 75.7%

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

      6. fma-define 75.8%

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

      7. associate-/r* 75.7%

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

   8. Simplified 75.7%

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

   9. Taylor expanded in b around -inf 69.0%

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

       1. associate-*r/ 69.1%

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

       2. *-commutative 69.1%

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

       3. associate-/l* 69.0%

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

   11. Simplified 69.0%

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

       1. clear-num 68.9%

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

       2. un-div-inv 69.0%

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

       3. div-inv 69.0%

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

       4. metadata-eval 69.0%

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

   13. Applied egg-rr 69.0%

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

   if -1.999999999999994e-310 < b

   1. Initial program 27.2%

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

      1. sqr-neg 27.2%

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

      2. sqr-neg 27.2%

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

      3. associate-*l* 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in b around inf 59.6%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-/l* 59.6%

         \[\leadsto \color{blue}{-1.5 \cdot \frac{\frac{a \cdot c}{b}}{3 \cdot a}} \]

      2. associate-/l* 63.4%

         \[\leadsto -1.5 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{3 \cdot a} \]

      3. *-commutative 63.4%

         \[\leadsto -1.5 \cdot \frac{a \cdot \frac{c}{b}}{\color{blue}{a \cdot 3}} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 40.2%

         \[\leadsto \color{blue}{\sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}} \cdot \sqrt{-1.5 \cdot \frac{a \cdot \frac{c}{b}}{a \cdot 3}}} \]

      2. sqrt-unprod 34.9%

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

      3. *-commutative 34.9%

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

      4. *-commutative 34.9%

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

      5. swap-sqr 35.0%

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

      6. pow2 35.0%

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

      7. times-frac 35.3%

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

      8. *-inverses 35.3%

         \[\leadsto \sqrt{{\left(\color{blue}{1} \cdot \frac{\frac{c}{b}}{3}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      9. *-un-lft-identity 35.3%

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

      10. div-inv 35.3%

          \[\leadsto \sqrt{{\color{blue}{\left(\frac{c}{b} \cdot \frac{1}{3}\right)}}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      11. metadata-eval 35.3%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot \color{blue}{0.3333333333333333}\right)}^{2} \cdot \left(-1.5 \cdot -1.5\right)} \]

      12. metadata-eval 35.3%

          \[\leadsto \sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot \color{blue}{2.25}} \]

   9. Applied egg-rr 35.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2} \cdot 2.25}} \]
   10. Step-by-step derivation

       1. *-commutative 35.3%

          \[\leadsto \sqrt{\color{blue}{2.25 \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}}} \]

       2. metadata-eval 35.3%

          \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot -1.5\right)} \cdot {\left(\frac{c}{b} \cdot 0.3333333333333333\right)}^{2}} \]

       3. unpow2 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \color{blue}{\left(\left(\frac{c}{b} \cdot 0.3333333333333333\right) \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)}} \]

       4. associate-*l/ 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{c \cdot 0.3333333333333333}{b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       5. metadata-eval 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{c \cdot \color{blue}{\frac{1}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       6. div-inv 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{c}{3}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       7. clear-num 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       8. associate-/r* 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\color{blue}{\frac{1}{\frac{3}{c} \cdot b}} \cdot \left(\frac{c}{b} \cdot 0.3333333333333333\right)\right)} \]

       9. associate-*l/ 35.3%

          \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{c \cdot 0.3333333333333333}{b}}\right)} \]

       10. metadata-eval 35.3%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{c \cdot \color{blue}{\frac{1}{3}}}{b}\right)} \]

       11. div-inv 35.3%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{c}{3}}}{b}\right)} \]

       12. clear-num 35.3%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \frac{\color{blue}{\frac{1}{\frac{3}{c}}}}{b}\right)} \]

       13. associate-/r* 35.3%

           \[\leadsto \sqrt{\left(-1.5 \cdot -1.5\right) \cdot \left(\frac{1}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{1}{\frac{3}{c} \cdot b}}\right)} \]

       14. swap-sqr 35.3%

           \[\leadsto \sqrt{\color{blue}{\left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right) \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)}} \]

       15. div-inv 35.3%

           \[\leadsto \sqrt{\color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \left(-1.5 \cdot \frac{1}{\frac{3}{c} \cdot b}\right)} \]

       16. div-inv 35.3%

           \[\leadsto \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b} \cdot \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       17. sqrt-unprod 44.2%

           \[\leadsto \color{blue}{\sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}} \cdot \sqrt{\frac{-1.5}{\frac{3}{c} \cdot b}}} \]

       18. add-sqr-sqrt 71.1%

           \[\leadsto \color{blue}{\frac{-1.5}{\frac{3}{c} \cdot b}} \]

       19. associate-/r* 72.8%

           \[\leadsto \color{blue}{\frac{\frac{-1.5}{\frac{3}{c}}}{b}} \]

   11. Applied egg-rr 72.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ b (* a -1.5)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-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 <= (-2d-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 <= -2e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-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 <= -2e-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 <= -2e-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, -2e-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 < -1.999999999999994e-310

   1. Initial program 75.8%

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

      1. sqr-neg 75.8%

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

      2. sqr-neg 75.8%

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

      3. associate-*l* 75.8%

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

   3. Simplified 75.8%

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

      1. frac-2neg 75.8%

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

      2. div-inv 75.7%

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

   6. Applied egg-rr 75.7%

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

      1. fma-undefine 75.7%

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

      2. unpow2 75.7%

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

      3. associate-*l* 75.7%

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

      4. *-commutative 75.7%

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

      5. +-commutative 75.7%

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

      6. fma-define 75.8%

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

      7. associate-/r* 75.7%

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

   8. Simplified 75.7%

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

   9. Taylor expanded in b around -inf 69.0%

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

       1. associate-*r/ 69.1%

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

       2. *-commutative 69.1%

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

       3. associate-/l* 69.0%

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

   11. Simplified 69.0%

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

       1. clear-num 68.9%

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

       2. un-div-inv 69.0%

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

       3. div-inv 69.0%

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

       4. metadata-eval 69.0%

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

   13. Applied egg-rr 69.0%

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

   if -1.999999999999994e-310 < b

   1. Initial program 27.2%

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

      1. sqr-neg 27.2%

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

      2. sqr-neg 27.2%

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

      3. associate-*l* 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in b around inf 72.8%

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

      1. *-commutative 72.8%

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

   7. Simplified 72.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (* b (/ -0.6666666666666666 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-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 <= (-2d-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 <= -2e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-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 <= -2e-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 <= -2e-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, -2e-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 < -1.999999999999994e-310

   1. Initial program 75.8%

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

      1. sqr-neg 75.8%

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

      2. sqr-neg 75.8%

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

      3. associate-*l* 75.8%

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

   3. Simplified 75.8%

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

      1. frac-2neg 75.8%

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

      2. div-inv 75.7%

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

   6. Applied egg-rr 75.7%

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

      1. fma-undefine 75.7%

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

      2. unpow2 75.7%

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

      3. associate-*l* 75.7%

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

      4. *-commutative 75.7%

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

      5. +-commutative 75.7%

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

      6. fma-define 75.8%

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

      7. associate-/r* 75.7%

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

   8. Simplified 75.7%

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

   9. Taylor expanded in b around -inf 69.0%

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

       1. associate-*r/ 69.1%

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

       2. *-commutative 69.1%

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

       3. associate-/l* 69.0%

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

   11. Simplified 69.0%

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

   if -1.999999999999994e-310 < b

   1. Initial program 27.2%

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

      1. sqr-neg 27.2%

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

      2. sqr-neg 27.2%

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

      3. associate-*l* 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in b around inf 72.8%

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

      1. *-commutative 72.8%

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

   7. Simplified 72.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		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-310)) 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-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		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-310)
		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-310)
		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-310], 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 < -1.999999999999994e-310

   1. Initial program 75.8%

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

      1. sqr-neg 75.8%

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

      2. sqr-neg 75.8%

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

      3. associate-*l* 75.8%

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

   3. Simplified 75.8%

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

      1. frac-2neg 75.8%

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

      2. div-inv 75.7%

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

   6. Applied egg-rr 75.7%

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

      1. fma-undefine 75.7%

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

      2. unpow2 75.7%

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

      3. associate-*l* 75.7%

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

      4. *-commutative 75.7%

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

      5. +-commutative 75.7%

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

      6. fma-define 75.8%

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

      7. associate-/r* 75.7%

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

   8. Simplified 75.7%

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

   9. Taylor expanded in b around -inf 69.0%

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

       1. associate-*r/ 69.1%

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

       2. *-commutative 69.1%

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

       3. associate-/l* 69.0%

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

   11. Simplified 69.0%

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

   if -1.999999999999994e-310 < b

   1. Initial program 27.2%

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

      1. sqr-neg 27.2%

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

      2. sqr-neg 27.2%

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

      3. associate-*l* 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in c around 0 66.3%

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

   6. Taylor expanded in a around 0 72.6%

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

Alternative 13: 43.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-264}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.4e-264) (* b (/ -0.6666666666666666 a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-264) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.4d-264) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-264) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.4e-264:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.4e-264)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.4e-264)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.4e-264], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < 2.3999999999999999e-264

   1. Initial program 75.9%

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

      1. sqr-neg 75.9%

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

      2. sqr-neg 75.9%

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

      3. associate-*l* 75.9%

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

   3. Simplified 75.9%

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

      1. frac-2neg 75.9%

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

      2. div-inv 75.8%

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

   6. Applied egg-rr 75.7%

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

      1. fma-undefine 75.7%

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

      2. unpow2 75.7%

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

      3. associate-*l* 75.8%

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

      4. *-commutative 75.8%

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

      5. +-commutative 75.8%

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

      6. fma-define 75.8%

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

      7. associate-/r* 75.7%

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

   8. Simplified 75.7%

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

   9. Taylor expanded in b around -inf 66.9%

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

       1. associate-*r/ 67.0%

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

       2. *-commutative 67.0%

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

       3. associate-/l* 67.0%

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

   11. Simplified 67.0%

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

   if 2.3999999999999999e-264 < b

   1. Initial program 25.6%

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

      1. add-cube-cbrt 25.2%

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

      2. pow3 25.3%

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

   4. Applied egg-rr 25.3%

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

      1. rem-cube-cbrt 25.6%

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

      2. clear-num 25.6%

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

      3. inv-pow 25.6%

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

      4. *-commutative 25.6%

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

      5. neg-mul-1 25.6%

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

      6. fma-define 25.6%

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

      7. pow2 25.6%

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

      8. *-commutative 25.6%

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

      9. *-commutative 25.6%

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

   6. Applied egg-rr 25.6%

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

      1. unpow-1 25.6%

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

   8. Simplified 25.6%

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

   9. Taylor expanded in a around 0 23.6%

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

       1. associate-*r/ 23.6%

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

       2. distribute-rgt1-in 23.6%

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

       3. metadata-eval 23.6%

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

       4. mul0-lft 23.6%

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

       5. metadata-eval 23.6%

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

   11. Simplified 23.6%

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

   12. Taylor expanded in a around 0 23.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 11.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 49.8%

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

   1. add-cube-cbrt 49.1%

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

   2. pow3 49.1%

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

4. Applied egg-rr 49.1%

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

   1. rem-cube-cbrt 49.8%

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

   2. clear-num 49.8%

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

   3. inv-pow 49.8%

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

   4. *-commutative 49.8%

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

   5. neg-mul-1 49.8%

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

   6. fma-define 49.8%

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

   7. pow2 49.8%

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

   8. *-commutative 49.8%

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

   9. *-commutative 49.8%

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

6. Applied egg-rr 49.8%

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

   1. unpow-1 49.8%

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

8. Simplified 49.8%

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

9. Taylor expanded in a around 0 13.5%

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

    1. associate-*r/ 13.5%

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

    2. distribute-rgt1-in 13.5%

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

    3. metadata-eval 13.5%

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

    4. mul0-lft 13.5%

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

    5. metadata-eval 13.5%

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

11. Simplified 13.5%

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

12. Taylor expanded in a around 0 13.5%

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

Reproduce

herbie shell --seed 2024165 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))