Cubic critical

Percentage Accurate: 52.6% → 86.1%

Time: 18.1s

Alternatives: 11

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: 52.6% 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: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+134}:\\ \;\;\;\;\frac{-1}{\left(-\frac{a}{b}\right) \cdot -1.5}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.02e+134)
   (/ -1.0 (* (- (/ a b)) -1.5))
   (if (<= b 8e-70)
     (/ (- (sqrt (fma b b (* 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 <= -1.02e+134) {
		tmp = -1.0 / (-(a / b) * -1.5);
	} else if (b <= 8e-70) {
		tmp = (sqrt(fma(b, b, (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 <= -1.02e+134)
		tmp = Float64(-1.0 / Float64(Float64(-Float64(a / b)) * -1.5));
	elseif (b <= 8e-70)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.02e+134], N[(-1.0 / N[((-N[(a / b), $MachinePrecision]) * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-70], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $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 < -1.02e134

   1. Initial program 43.8%

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

      1. neg-sub0 43.8%

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

      2. associate-+l- 43.8%

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

      3. sub0-neg 43.8%

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

      4. sub0-neg 43.8%

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

      5. associate-+l- 43.8%

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

      6. neg-sub0 43.8%

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

      7. associate-*l* 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in b around -inf 99.5%

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

      1. *-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. add-sqr-sqrt 48.3%

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

      2. sqrt-unprod 36.9%

         \[\leadsto \color{blue}{\sqrt{\left(\frac{b}{a} \cdot -0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)}} \]

      3. swap-sqr 36.9%

         \[\leadsto \sqrt{\color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(-0.6666666666666666 \cdot -0.6666666666666666\right)}} \]

      4. pow1 36.9%

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

      5. metadata-eval 36.9%

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

      6. pow1 36.9%

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

      7. metadata-eval 36.9%

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

      8. pow-sqr 36.9%

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

      9. metadata-eval 36.9%

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

      10. metadata-eval 36.9%

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

      11. metadata-eval 36.9%

          \[\leadsto \sqrt{{\left(\frac{b}{a}\right)}^{2} \cdot \color{blue}{0.4444444444444444}} \]

   8. Applied egg-rr 36.9%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{b}{a}\right)}^{2} \cdot 0.4444444444444444}} \]
   9. Step-by-step derivation

      1. *-commutative 36.9%

         \[\leadsto \sqrt{\color{blue}{0.4444444444444444 \cdot {\left(\frac{b}{a}\right)}^{2}}} \]

      2. metadata-eval 36.9%

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

      3. unpow2 36.9%

         \[\leadsto \sqrt{\left(-0.6666666666666666 \cdot -0.6666666666666666\right) \cdot \color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right)}} \]

      4. swap-sqr 36.9%

         \[\leadsto \sqrt{\color{blue}{\left(-0.6666666666666666 \cdot \frac{b}{a}\right) \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)}} \]

      5. clear-num 36.9%

         \[\leadsto \sqrt{\left(-0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}}\right) \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)} \]

      6. div-inv 36.9%

         \[\leadsto \sqrt{\color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)} \]

      7. clear-num 36.9%

         \[\leadsto \sqrt{\frac{-0.6666666666666666}{\frac{a}{b}} \cdot \left(-0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}}\right)} \]

      8. div-inv 36.9%

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

      9. sqrt-unprod 48.3%

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

      10. add-sqr-sqrt 99.5%

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

      11. clear-num 99.4%

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

      12. metadata-eval 99.4%

          \[\leadsto \frac{\color{blue}{\frac{2}{2}}}{\frac{\frac{a}{b}}{-0.6666666666666666}} \]

      13. frac-2neg 99.4%

          \[\leadsto \color{blue}{\frac{-\frac{2}{2}}{-\frac{\frac{a}{b}}{-0.6666666666666666}}} \]

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

      16. div-inv 99.7%

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

      17. metadata-eval 99.7%

          \[\leadsto \frac{-1}{-\frac{a}{b} \cdot \color{blue}{-1.5}} \]

      18. distribute-lft-neg-in 99.7%

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

      19. distribute-neg-frac 99.7%

          \[\leadsto \frac{-1}{\color{blue}{\frac{-a}{b}} \cdot -1.5} \]

   10. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\frac{-1}{\frac{-a}{b} \cdot -1.5}} \]

   if -1.02e134 < b < 7.99999999999999997e-70

   1. Initial program 86.1%

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

   3. Simplified 86.1%

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

   if 7.99999999999999997e-70 < b

   1. Initial program 16.3%

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

      1. neg-sub0 16.3%

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

      2. associate-+l- 16.3%

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

      3. sub0-neg 16.3%

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

      4. sub0-neg 16.3%

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

      5. associate-+l- 16.3%

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

      6. neg-sub0 16.3%

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

      7. associate-*l* 16.3%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in b around inf 88.0%

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

      1. associate-*r/ 88.0%

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

   6. Simplified 88.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+134}:\\ \;\;\;\;\frac{-1}{\left(-\frac{a}{b}\right) \cdot -1.5}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{+133}:\\ \;\;\;\;\frac{-1}{\left(-\frac{a}{b}\right) \cdot -1.5}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.9e+133)
   (/ -1.0 (* (- (/ a b)) -1.5))
   (if (<= b 4.5e-68)
     (/ (- (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 <= -3.9e+133) {
		tmp = -1.0 / (-(a / b) * -1.5);
	} else if (b <= 4.5e-68) {
		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 <= (-3.9d+133)) then
        tmp = (-1.0d0) / (-(a / b) * (-1.5d0))
    else if (b <= 4.5d-68) 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 <= -3.9e+133) {
		tmp = -1.0 / (-(a / b) * -1.5);
	} else if (b <= 4.5e-68) {
		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 <= -3.9e+133:
		tmp = -1.0 / (-(a / b) * -1.5)
	elif b <= 4.5e-68:
		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 <= -3.9e+133)
		tmp = Float64(-1.0 / Float64(Float64(-Float64(a / b)) * -1.5));
	elseif (b <= 4.5e-68)
		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 <= -3.9e+133)
		tmp = -1.0 / (-(a / b) * -1.5);
	elseif (b <= 4.5e-68)
		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, -3.9e+133], N[(-1.0 / N[((-N[(a / b), $MachinePrecision]) * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.5e-68], 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 < -3.90000000000000014e133

   1. Initial program 43.8%

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

      1. neg-sub0 43.8%

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

      2. associate-+l- 43.8%

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

      3. sub0-neg 43.8%

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

      4. sub0-neg 43.8%

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

      5. associate-+l- 43.8%

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

      6. neg-sub0 43.8%

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

      7. associate-*l* 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in b around -inf 99.5%

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

      1. *-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. add-sqr-sqrt 48.3%

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

      2. sqrt-unprod 36.9%

         \[\leadsto \color{blue}{\sqrt{\left(\frac{b}{a} \cdot -0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)}} \]

      3. swap-sqr 36.9%

         \[\leadsto \sqrt{\color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(-0.6666666666666666 \cdot -0.6666666666666666\right)}} \]

      4. pow1 36.9%

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

      5. metadata-eval 36.9%

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

      6. pow1 36.9%

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

      7. metadata-eval 36.9%

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

      8. pow-sqr 36.9%

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

      9. metadata-eval 36.9%

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

      10. metadata-eval 36.9%

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

      11. metadata-eval 36.9%

          \[\leadsto \sqrt{{\left(\frac{b}{a}\right)}^{2} \cdot \color{blue}{0.4444444444444444}} \]

   8. Applied egg-rr 36.9%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{b}{a}\right)}^{2} \cdot 0.4444444444444444}} \]
   9. Step-by-step derivation

      1. *-commutative 36.9%

         \[\leadsto \sqrt{\color{blue}{0.4444444444444444 \cdot {\left(\frac{b}{a}\right)}^{2}}} \]

      2. metadata-eval 36.9%

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

      3. unpow2 36.9%

         \[\leadsto \sqrt{\left(-0.6666666666666666 \cdot -0.6666666666666666\right) \cdot \color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right)}} \]

      4. swap-sqr 36.9%

         \[\leadsto \sqrt{\color{blue}{\left(-0.6666666666666666 \cdot \frac{b}{a}\right) \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)}} \]

      5. clear-num 36.9%

         \[\leadsto \sqrt{\left(-0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}}\right) \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)} \]

      6. div-inv 36.9%

         \[\leadsto \sqrt{\color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)} \]

      7. clear-num 36.9%

         \[\leadsto \sqrt{\frac{-0.6666666666666666}{\frac{a}{b}} \cdot \left(-0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}}\right)} \]

      8. div-inv 36.9%

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

      9. sqrt-unprod 48.3%

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

      10. add-sqr-sqrt 99.5%

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

      11. clear-num 99.4%

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

      12. metadata-eval 99.4%

          \[\leadsto \frac{\color{blue}{\frac{2}{2}}}{\frac{\frac{a}{b}}{-0.6666666666666666}} \]

      13. frac-2neg 99.4%

          \[\leadsto \color{blue}{\frac{-\frac{2}{2}}{-\frac{\frac{a}{b}}{-0.6666666666666666}}} \]

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

      16. div-inv 99.7%

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

      17. metadata-eval 99.7%

          \[\leadsto \frac{-1}{-\frac{a}{b} \cdot \color{blue}{-1.5}} \]

      18. distribute-lft-neg-in 99.7%

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

      19. distribute-neg-frac 99.7%

          \[\leadsto \frac{-1}{\color{blue}{\frac{-a}{b}} \cdot -1.5} \]

   10. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\frac{-1}{\frac{-a}{b} \cdot -1.5}} \]

   if -3.90000000000000014e133 < b < 4.49999999999999999e-68

   1. Initial program 86.1%

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

      1. neg-sub0 86.1%

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

      2. associate-+l- 86.1%

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

      3. sub0-neg 86.1%

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

      4. sub0-neg 86.1%

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

      5. associate-+l- 86.1%

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

      6. neg-sub0 86.1%

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

      7. associate-*l* 86.0%

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

   3. Simplified 86.0%

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

   if 4.49999999999999999e-68 < b

   1. Initial program 16.3%

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

      1. neg-sub0 16.3%

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

      2. associate-+l- 16.3%

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

      3. sub0-neg 16.3%

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

      4. sub0-neg 16.3%

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

      5. associate-+l- 16.3%

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

      6. neg-sub0 16.3%

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

      7. associate-*l* 16.3%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in b around inf 88.0%

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

      1. associate-*r/ 88.0%

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

   6. Simplified 88.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{+133}:\\ \;\;\;\;\frac{-1}{\left(-\frac{a}{b}\right) \cdot -1.5}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+134}:\\ \;\;\;\;\frac{-1}{\left(-\frac{a}{b}\right) \cdot -1.5}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+134)
   (/ -1.0 (* (- (/ a b)) -1.5))
   (if (<= b 4.6e-66)
     (/ (- (sqrt (- (* b b) (* 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+134) {
		tmp = -1.0 / (-(a / b) * -1.5);
	} else if (b <= 4.6e-66) {
		tmp = (sqrt(((b * b) - (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+134)) then
        tmp = (-1.0d0) / (-(a / b) * (-1.5d0))
    else if (b <= 4.6d-66) then
        tmp = (sqrt(((b * b) - (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+134) {
		tmp = -1.0 / (-(a / b) * -1.5);
	} else if (b <= 4.6e-66) {
		tmp = (Math.sqrt(((b * b) - (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+134:
		tmp = -1.0 / (-(a / b) * -1.5)
	elif b <= 4.6e-66:
		tmp = (math.sqrt(((b * b) - (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+134)
		tmp = Float64(-1.0 / Float64(Float64(-Float64(a / b)) * -1.5));
	elseif (b <= 4.6e-66)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - 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+134)
		tmp = -1.0 / (-(a / b) * -1.5);
	elseif (b <= 4.6e-66)
		tmp = (sqrt(((b * b) - (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+134], N[(-1.0 / N[((-N[(a / b), $MachinePrecision]) * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.6e-66], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.49999999999999998e134

   1. Initial program 43.8%

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

      1. neg-sub0 43.8%

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

      2. associate-+l- 43.8%

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

      3. sub0-neg 43.8%

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

      4. sub0-neg 43.8%

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

      5. associate-+l- 43.8%

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

      6. neg-sub0 43.8%

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

      7. associate-*l* 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in b around -inf 99.5%

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

      1. *-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. add-sqr-sqrt 48.3%

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

      2. sqrt-unprod 36.9%

         \[\leadsto \color{blue}{\sqrt{\left(\frac{b}{a} \cdot -0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)}} \]

      3. swap-sqr 36.9%

         \[\leadsto \sqrt{\color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(-0.6666666666666666 \cdot -0.6666666666666666\right)}} \]

      4. pow1 36.9%

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

      5. metadata-eval 36.9%

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

      6. pow1 36.9%

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

      7. metadata-eval 36.9%

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

      8. pow-sqr 36.9%

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

      9. metadata-eval 36.9%

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

      10. metadata-eval 36.9%

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

      11. metadata-eval 36.9%

          \[\leadsto \sqrt{{\left(\frac{b}{a}\right)}^{2} \cdot \color{blue}{0.4444444444444444}} \]

   8. Applied egg-rr 36.9%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{b}{a}\right)}^{2} \cdot 0.4444444444444444}} \]
   9. Step-by-step derivation

      1. *-commutative 36.9%

         \[\leadsto \sqrt{\color{blue}{0.4444444444444444 \cdot {\left(\frac{b}{a}\right)}^{2}}} \]

      2. metadata-eval 36.9%

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

      3. unpow2 36.9%

         \[\leadsto \sqrt{\left(-0.6666666666666666 \cdot -0.6666666666666666\right) \cdot \color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right)}} \]

      4. swap-sqr 36.9%

         \[\leadsto \sqrt{\color{blue}{\left(-0.6666666666666666 \cdot \frac{b}{a}\right) \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)}} \]

      5. clear-num 36.9%

         \[\leadsto \sqrt{\left(-0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}}\right) \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)} \]

      6. div-inv 36.9%

         \[\leadsto \sqrt{\color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)} \]

      7. clear-num 36.9%

         \[\leadsto \sqrt{\frac{-0.6666666666666666}{\frac{a}{b}} \cdot \left(-0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}}\right)} \]

      8. div-inv 36.9%

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

      9. sqrt-unprod 48.3%

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

      10. add-sqr-sqrt 99.5%

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

      11. clear-num 99.4%

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

      12. metadata-eval 99.4%

          \[\leadsto \frac{\color{blue}{\frac{2}{2}}}{\frac{\frac{a}{b}}{-0.6666666666666666}} \]

      13. frac-2neg 99.4%

          \[\leadsto \color{blue}{\frac{-\frac{2}{2}}{-\frac{\frac{a}{b}}{-0.6666666666666666}}} \]

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

      16. div-inv 99.7%

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

      17. metadata-eval 99.7%

          \[\leadsto \frac{-1}{-\frac{a}{b} \cdot \color{blue}{-1.5}} \]

      18. distribute-lft-neg-in 99.7%

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

      19. distribute-neg-frac 99.7%

          \[\leadsto \frac{-1}{\color{blue}{\frac{-a}{b}} \cdot -1.5} \]

   10. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\frac{-1}{\frac{-a}{b} \cdot -1.5}} \]

   if -1.49999999999999998e134 < b < 4.59999999999999984e-66

   1. Initial program 86.1%

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

   if 4.59999999999999984e-66 < b

   1. Initial program 16.3%

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

      1. neg-sub0 16.3%

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

      2. associate-+l- 16.3%

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

      3. sub0-neg 16.3%

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

      4. sub0-neg 16.3%

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

      5. associate-+l- 16.3%

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

      6. neg-sub0 16.3%

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

      7. associate-*l* 16.3%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in b around inf 88.0%

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

      1. associate-*r/ 88.0%

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

   6. Simplified 88.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+134}:\\ \;\;\;\;\frac{-1}{\left(-\frac{a}{b}\right) \cdot -1.5}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-69}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.05e-66)
   (/ (- (- (/ (* a 1.5) (/ b c)) b) b) (* a 3.0))
   (if (<= b 1.35e-69)
     (* (/ 0.3333333333333333 a) (+ b (sqrt (* c (* a -3.0)))))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.05e-66) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 1.35e-69) {
		tmp = (0.3333333333333333 / a) * (b + sqrt((c * (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 <= (-1.05d-66)) then
        tmp = ((((a * 1.5d0) / (b / c)) - b) - b) / (a * 3.0d0)
    else if (b <= 1.35d-69) then
        tmp = (0.3333333333333333d0 / a) * (b + sqrt((c * (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 <= -1.05e-66) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 1.35e-69) {
		tmp = (0.3333333333333333 / a) * (b + Math.sqrt((c * (a * -3.0))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.05e-66:
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0)
	elif b <= 1.35e-69:
		tmp = (0.3333333333333333 / a) * (b + math.sqrt((c * (a * -3.0))))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.05e-66)
		tmp = Float64(Float64(Float64(Float64(Float64(a * 1.5) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	elseif (b <= 1.35e-69)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(b + sqrt(Float64(c * 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 <= -1.05e-66)
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	elseif (b <= 1.35e-69)
		tmp = (0.3333333333333333 / a) * (b + sqrt((c * (a * -3.0))));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.05e-66], N[(N[(N[(N[(N[(a * 1.5), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-69], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05e-66

   1. Initial program 67.5%

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

      1. neg-sub0 67.5%

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

      2. associate-+l- 67.5%

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

      3. sub0-neg 67.5%

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

      4. sub0-neg 67.5%

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

      5. associate-+l- 67.5%

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

      6. neg-sub0 67.5%

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

      7. associate-*l* 67.5%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in b around -inf 83.1%

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

      1. +-commutative 83.1%

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

      2. mul-1-neg 83.1%

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

      3. unsub-neg 83.1%

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

      4. associate-/l* 88.0%

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

      5. associate-*r/ 88.0%

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

   6. Simplified 88.0%

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

   if -1.05e-66 < b < 1.3499999999999999e-69

   1. Initial program 83.8%

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

      1. neg-sub0 83.8%

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

      2. associate-+l- 83.8%

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

      3. sub0-neg 83.8%

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

      4. sub0-neg 83.8%

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

      5. associate-+l- 83.8%

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

      6. neg-sub0 83.8%

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

      7. associate-*l* 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in b around 0 81.0%

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

      1. associate-*r* 81.0%

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

      2. *-commutative 81.0%

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

      3. *-commutative 81.0%

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

   6. Simplified 81.0%

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

      1. div-inv 81.1%

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

      2. *-commutative 81.1%

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

      3. metadata-eval 81.1%

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

      4. associate-/r* 80.8%

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

      5. metadata-eval 80.8%

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

      6. metadata-eval 80.8%

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

      7. add-sqr-sqrt 41.4%

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

      8. sqrt-unprod 79.3%

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

      9. sqr-neg 79.3%

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

      10. unpow2 79.3%

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

      11. unpow2 79.3%

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

      12. sqrt-prod 38.5%

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

      13. add-sqr-sqrt 79.1%

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

   8. Applied egg-rr 79.1%

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

   if 1.3499999999999999e-69 < b

   1. Initial program 16.3%

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

      1. neg-sub0 16.3%

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

      2. associate-+l- 16.3%

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

      3. sub0-neg 16.3%

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

      4. sub0-neg 16.3%

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

      5. associate-+l- 16.3%

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

      6. neg-sub0 16.3%

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

      7. associate-*l* 16.3%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in b around inf 88.0%

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

      1. associate-*r/ 88.0%

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

   6. Simplified 88.0%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-69}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 10^{-67}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.05e-23)
   (/ (- (- (/ (* a 1.5) (/ b c)) b) b) (* a 3.0))
   (if (<= b 1e-67)
     (/ (- (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 <= -1.05e-23) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 1e-67) {
		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 <= (-1.05d-23)) then
        tmp = ((((a * 1.5d0) / (b / c)) - b) - b) / (a * 3.0d0)
    else if (b <= 1d-67) 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 <= -1.05e-23) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 1e-67) {
		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 <= -1.05e-23:
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0)
	elif b <= 1e-67:
		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 <= -1.05e-23)
		tmp = Float64(Float64(Float64(Float64(Float64(a * 1.5) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	elseif (b <= 1e-67)
		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 <= -1.05e-23)
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	elseif (b <= 1e-67)
		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, -1.05e-23], N[(N[(N[(N[(N[(a * 1.5), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-67], 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 < -1.05e-23

   1. Initial program 66.7%

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

      1. neg-sub0 66.7%

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

      2. associate-+l- 66.7%

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

      3. sub0-neg 66.7%

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

      4. sub0-neg 66.7%

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

      5. associate-+l- 66.7%

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

      6. neg-sub0 66.7%

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

      7. associate-*l* 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in b around -inf 85.7%

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

      1. +-commutative 85.7%

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

      2. mul-1-neg 85.7%

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

      3. unsub-neg 85.7%

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

      4. associate-/l* 90.9%

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

      5. associate-*r/ 90.9%

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

   6. Simplified 90.9%

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

   if -1.05e-23 < b < 9.99999999999999943e-68

   1. Initial program 83.6%

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

      1. neg-sub0 83.6%

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

      2. associate-+l- 83.6%

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

      3. sub0-neg 83.6%

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

      4. sub0-neg 83.6%

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

      5. associate-+l- 83.6%

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

      6. neg-sub0 83.6%

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

      7. associate-*l* 83.5%

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

   3. Simplified 83.5%

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

   5. Applied egg-rr 82.6%

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

      1. +-commutative 82.6%

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

      2. fma-udef 82.6%

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

      3. associate-+l+ 82.6%

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

      4. *-commutative 82.6%

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

      5. associate-*l* 82.6%

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

      6. *-commutative 82.6%

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

      7. *-commutative 82.6%

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

      8. associate-*l* 82.5%

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

      9. *-commutative 82.5%

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

      10. +-commutative 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in b around 0 78.5%

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

      1. distribute-rgt-out 78.8%

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

      2. metadata-eval 78.8%

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

      3. *-commutative 78.8%

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

      4. associate-*r* 78.8%

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

      5. mul-1-neg 78.8%

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

      6. sub-neg 78.8%

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

   10. Simplified 78.8%

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

   if 9.99999999999999943e-68 < b

   1. Initial program 16.3%

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

      1. neg-sub0 16.3%

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

      2. associate-+l- 16.3%

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

      3. sub0-neg 16.3%

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

      4. sub0-neg 16.3%

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

      5. associate-+l- 16.3%

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

      6. neg-sub0 16.3%

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

      7. associate-*l* 16.3%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in b around inf 88.0%

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

      1. associate-*r/ 88.0%

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

   6. Simplified 88.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 10^{-67}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6: 67.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} 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 <= (-4d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    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 <= -4e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	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 <= -4e-310)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 72.0%

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

      1. neg-sub0 72.0%

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

      2. associate-+l- 72.0%

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

      3. sub0-neg 72.0%

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

      4. sub0-neg 72.0%

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

      5. associate-+l- 72.0%

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

      6. neg-sub0 72.0%

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

      7. associate-*l* 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in b around -inf 63.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -3.999999999999988e-310 < b

   1. Initial program 33.4%

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

      1. neg-sub0 33.4%

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

      2. associate-+l- 33.4%

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

      3. sub0-neg 33.4%

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

      4. sub0-neg 33.4%

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

      5. associate-+l- 33.4%

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

      6. neg-sub0 33.4%

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

      7. associate-*l* 33.4%

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

   3. Simplified 33.4%

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

   4. Taylor expanded in b around inf 70.0%

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

      1. associate-*r/ 70.0%

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

   6. Simplified 70.0%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7: 66.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (* b (/ -0.6666666666666666 a)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 72.0%

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

      1. neg-sub0 72.0%

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

      2. associate-+l- 72.0%

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

      3. sub0-neg 72.0%

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

      4. sub0-neg 72.0%

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

      5. associate-+l- 72.0%

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

      6. neg-sub0 72.0%

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

      7. associate-*l* 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in b around -inf 63.2%

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

      1. *-commutative 63.2%

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

   6. Simplified 63.2%

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

      1. *-commutative 63.2%

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

      2. clear-num 63.1%

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

      3. un-div-inv 63.2%

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

   8. Applied egg-rr 63.2%

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

      1. associate-/r/ 63.2%

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

   10. Applied egg-rr 63.2%

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

   if -3.999999999999988e-310 < b

   1. Initial program 33.4%

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

      1. neg-sub0 33.4%

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

      2. associate-+l- 33.4%

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

      3. sub0-neg 33.4%

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

      4. sub0-neg 33.4%

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

      5. associate-+l- 33.4%

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

      6. neg-sub0 33.4%

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

      7. associate-*l* 33.4%

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

   3. Simplified 33.4%

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

   4. Taylor expanded in b around inf 70.0%

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

      1. associate-*r/ 70.0%

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

      2. associate-/l* 68.8%

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

   6. Simplified 68.8%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]

Alternative 8: 66.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ b (* a -1.5)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = b / (a * -1.5)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = b / (a * -1.5);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 72.0%

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

      1. neg-sub0 72.0%

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

      2. associate-+l- 72.0%

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

      3. sub0-neg 72.0%

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

      4. sub0-neg 72.0%

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

      5. associate-+l- 72.0%

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

      6. neg-sub0 72.0%

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

      7. associate-*l* 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in b around -inf 63.2%

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

      1. *-commutative 63.2%

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

   6. Simplified 63.2%

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

      1. *-commutative 63.2%

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

      2. clear-num 63.1%

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

      3. un-div-inv 63.2%

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

   8. Applied egg-rr 63.2%

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

      1. associate-/r/ 63.2%

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

   10. Applied egg-rr 63.2%

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

       1. clear-num 63.1%

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

       2. associate-*l/ 63.2%

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

       3. *-un-lft-identity 63.2%

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

       4. div-inv 63.2%

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

       5. metadata-eval 63.2%

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

   12. Applied egg-rr 63.2%

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

   if -3.999999999999988e-310 < b

   1. Initial program 33.4%

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

      1. neg-sub0 33.4%

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

      2. associate-+l- 33.4%

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

      3. sub0-neg 33.4%

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

      4. sub0-neg 33.4%

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

      5. associate-+l- 33.4%

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

      6. neg-sub0 33.4%

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

      7. associate-*l* 33.4%

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

   3. Simplified 33.4%

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

   4. Taylor expanded in b around inf 70.0%

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

      1. associate-*r/ 70.0%

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

      2. associate-/l* 68.8%

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

   6. Simplified 68.8%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]

Alternative 9: 67.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-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 <= (-4d-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 <= -4e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-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 <= -4e-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 <= -4e-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, -4e-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 < -3.999999999999988e-310

   1. Initial program 72.0%

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

      1. neg-sub0 72.0%

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

      2. associate-+l- 72.0%

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

      3. sub0-neg 72.0%

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

      4. sub0-neg 72.0%

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

      5. associate-+l- 72.0%

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

      6. neg-sub0 72.0%

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

      7. associate-*l* 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in b around -inf 63.2%

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

      1. *-commutative 63.2%

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

   6. Simplified 63.2%

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

      1. *-commutative 63.2%

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

      2. clear-num 63.1%

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

      3. un-div-inv 63.2%

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

   8. Applied egg-rr 63.2%

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

      1. associate-/r/ 63.2%

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

   10. Applied egg-rr 63.2%

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

       1. clear-num 63.1%

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

       2. associate-*l/ 63.2%

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

       3. *-un-lft-identity 63.2%

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

       4. div-inv 63.2%

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

       5. metadata-eval 63.2%

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

   12. Applied egg-rr 63.2%

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

   if -3.999999999999988e-310 < b

   1. Initial program 33.4%

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

      1. neg-sub0 33.4%

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

      2. associate-+l- 33.4%

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

      3. sub0-neg 33.4%

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

      4. sub0-neg 33.4%

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

      5. associate-+l- 33.4%

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

      6. neg-sub0 33.4%

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

      7. associate-*l* 33.4%

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

   3. Simplified 33.4%

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

   4. Taylor expanded in b around inf 70.0%

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

      1. associate-*r/ 70.0%

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

   6. Simplified 70.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 10: 67.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ (* b -0.6666666666666666) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-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 <= (-4d-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 <= -4e-310) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-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 <= -4e-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 <= -4e-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, -4e-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 < -3.999999999999988e-310

   1. Initial program 72.0%

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

      1. neg-sub0 72.0%

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

      2. associate-+l- 72.0%

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

      3. sub0-neg 72.0%

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

      4. sub0-neg 72.0%

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

      5. associate-+l- 72.0%

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

      6. neg-sub0 72.0%

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

      7. associate-*l* 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in b around -inf 63.2%

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

      1. associate-*r/ 63.3%

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

      2. *-commutative 63.3%

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

   6. Simplified 63.3%

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

   if -3.999999999999988e-310 < b

   1. Initial program 33.4%

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

      1. neg-sub0 33.4%

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

      2. associate-+l- 33.4%

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

      3. sub0-neg 33.4%

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

      4. sub0-neg 33.4%

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

      5. associate-+l- 33.4%

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

      6. neg-sub0 33.4%

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

      7. associate-*l* 33.4%

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

   3. Simplified 33.4%

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

   4. Taylor expanded in b around inf 70.0%

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

      1. associate-*r/ 70.0%

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

   6. Simplified 70.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 11: 34.6% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ b \cdot \frac{-0.6666666666666666}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* b (/ -0.6666666666666666 a)))

C

double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / 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 * ((-0.6666666666666666d0) / a)
end function

Java

public static double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / a);
}

Python

def code(a, b, c):
	return b * (-0.6666666666666666 / a)

Julia

function code(a, b, c)
	return Float64(b * Float64(-0.6666666666666666 / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = b * (-0.6666666666666666 / a);
end

Wolfram

code[a_, b_, c_] := N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

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

   1. neg-sub0 51.0%

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

   2. associate-+l- 51.0%

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

   3. sub0-neg 51.0%

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

   4. sub0-neg 51.0%

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

   5. associate-+l- 51.0%

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

   6. neg-sub0 51.0%

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

   7. associate-*l* 51.0%

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

3. Simplified 51.0%

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

4. Taylor expanded in b around -inf 30.4%

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

   1. *-commutative 30.4%

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

6. Simplified 30.4%

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

   1. *-commutative 30.4%

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

   2. clear-num 30.4%

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

   3. un-div-inv 30.4%

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

8. Applied egg-rr 30.4%

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

   1. associate-/r/ 30.4%

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

10. Applied egg-rr 30.4%

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

11. Final simplification 30.4%

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

Reproduce

herbie shell --seed 2023336 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))