The quadratic formula (r1)

Percentage Accurate: 53.2% → 85.4%

Time: 15.4s

Alternatives: 8

Speedup: 12.9×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e+153)
   (/ b (- a))
   (if (<= b 4.2e-57)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e+153) {
		tmp = b / -a;
	} else if (b <= 4.2e-57) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6d+153)) then
        tmp = b / -a
    else if (b <= 4.2d-57) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e+153) {
		tmp = b / -a;
	} else if (b <= 4.2e-57) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e+153:
		tmp = b / -a
	elif b <= 4.2e-57:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e+153)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 4.2e-57)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6e+153)
		tmp = b / -a;
	elseif (b <= 4.2e-57)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e+153], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 4.2e-57], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.00000000000000037e153

   1. Initial program 46.6%

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

      1. *-commutative 46.6%

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

   3. Simplified 46.6%

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

      1. *-commutative 46.6%

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

      2. add-sqr-sqrt 23.5%

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

      3. sqrt-unprod 46.1%

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

      4. *-commutative 46.1%

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

      5. *-commutative 46.1%

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

      6. swap-sqr 46.1%

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

      7. metadata-eval 46.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 46.1%

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

      9. swap-sqr 46.1%

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

      10. sqrt-unprod 22.8%

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

      11. add-sqr-sqrt 45.9%

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

      12. add-cube-cbrt 45.9%

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

      13. pow3 45.9%

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

   6. Applied egg-rr 46.6%

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

   7. Taylor expanded in b around -inf 91.6%

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

      1. associate-*r/ 91.6%

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

      2. mul-1-neg 91.6%

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

   9. Simplified 91.6%

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

   if -6.00000000000000037e153 < b < 4.1999999999999999e-57

   1. Initial program 86.3%

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

   if 4.1999999999999999e-57 < b

   1. Initial program 8.6%

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

      1. *-commutative 8.6%

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

   3. Simplified 8.6%

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

      1. *-commutative 8.6%

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

      2. add-sqr-sqrt 5.8%

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

      3. sqrt-unprod 6.7%

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

      4. *-commutative 6.7%

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

      5. *-commutative 6.7%

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

      6. swap-sqr 6.7%

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

      7. metadata-eval 6.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 6.7%

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

      9. swap-sqr 6.7%

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

      10. sqrt-unprod 1.2%

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

      11. add-sqr-sqrt 5.9%

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

      12. add-cube-cbrt 5.9%

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

      13. pow3 5.9%

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

   6. Applied egg-rr 8.6%

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

   8. Applied egg-rr 8.6%

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

   9. Taylor expanded in b around inf 93.9%

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

       1. associate-*r/ 93.9%

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

       2. mul-1-neg 93.9%

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

   11. Simplified 93.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-127}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.8e-127)
   (- (/ c b) (/ b a))
   (if (<= b 5.2e-47)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.8e-127) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.2e-47) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.8d-127)) then
        tmp = (c / b) - (b / a)
    else if (b <= 5.2d-47) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.8e-127) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.2e-47) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.8e-127:
		tmp = (c / b) - (b / a)
	elif b <= 5.2e-47:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.8e-127)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 5.2e-47)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.8e-127)
		tmp = (c / b) - (b / a);
	elseif (b <= 5.2e-47)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.8e-127], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-47], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.8000000000000001e-127

   1. Initial program 80.1%

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

      1. *-commutative 80.1%

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

   3. Simplified 80.1%

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

      1. *-commutative 80.1%

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

      2. add-sqr-sqrt 42.2%

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

      3. sqrt-unprod 68.9%

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

      4. *-commutative 68.9%

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

      5. *-commutative 68.9%

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

      6. swap-sqr 68.9%

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

      7. metadata-eval 68.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 68.9%

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

      9. swap-sqr 68.9%

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

      10. sqrt-unprod 33.0%

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

      11. add-sqr-sqrt 64.9%

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

      12. add-cube-cbrt 64.9%

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

      13. pow3 64.9%

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

   6. Applied egg-rr 79.9%

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

   8. Applied egg-rr 79.9%

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

   9. Taylor expanded in b around -inf 81.4%

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

       1. mul-1-neg 81.4%

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

       2. *-commutative 81.4%

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

       3. distribute-rgt-neg-in 81.4%

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

       4. +-commutative 81.4%

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

       5. mul-1-neg 81.4%

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

       6. unsub-neg 81.4%

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

   11. Simplified 81.4%

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

   12. Taylor expanded in a around 0 79.5%

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

       1. +-commutative 79.5%

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

       2. *-commutative 79.5%

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

       3. associate-*r/ 81.6%

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

       4. mul-1-neg 81.6%

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

       5. unsub-neg 81.6%

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

       6. div-sub 81.6%

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

       7. associate-/l* 81.6%

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

       8. associate-/l/ 77.4%

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

       9. associate-/r* 81.7%

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

       10. *-inverses 81.7%

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

       11. associate-*r/ 81.7%

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

       12. *-rgt-identity 81.7%

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

   14. Simplified 81.7%

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

   if -5.8000000000000001e-127 < b < 5.2e-47

   1. Initial program 75.6%

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

      1. *-commutative 75.6%

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

   3. Simplified 75.6%

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

      1. *-commutative 75.6%

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

      2. add-sqr-sqrt 37.4%

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

      3. sqrt-unprod 24.6%

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

      4. *-commutative 24.6%

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

      5. *-commutative 24.6%

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

      6. swap-sqr 24.6%

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

      7. metadata-eval 24.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 24.6%

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

      9. swap-sqr 24.6%

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

      10. sqrt-unprod 0.6%

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

      11. add-sqr-sqrt 2.9%

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

      12. add-cube-cbrt 2.9%

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

      13. pow3 2.9%

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

   6. Applied egg-rr 74.8%

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

   7. Taylor expanded in c around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 71.0%

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

      5. distribute-lft-neg-in 71.0%

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

      6. metadata-eval 71.0%

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

      7. rem-cube-cbrt 71.6%

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

   9. Simplified 71.6%

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

   if 5.2e-47 < b

   1. Initial program 8.6%

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

      1. *-commutative 8.6%

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

   3. Simplified 8.6%

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

      1. *-commutative 8.6%

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

      2. add-sqr-sqrt 5.8%

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

      3. sqrt-unprod 6.7%

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

      4. *-commutative 6.7%

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

      5. *-commutative 6.7%

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

      6. swap-sqr 6.7%

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

      7. metadata-eval 6.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 6.7%

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

      9. swap-sqr 6.7%

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

      10. sqrt-unprod 1.2%

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

      11. add-sqr-sqrt 5.9%

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

      12. add-cube-cbrt 5.9%

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

      13. pow3 5.9%

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

   6. Applied egg-rr 8.6%

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

   8. Applied egg-rr 8.6%

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

   9. Taylor expanded in b around inf 93.9%

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

       1. associate-*r/ 93.9%

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

       2. mul-1-neg 93.9%

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

   11. Simplified 93.9%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-127}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-208}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-167}:\\ \;\;\;\;-0.5 \cdot \sqrt{c \cdot \frac{-4}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.6e-208)
   (- (/ c b) (/ b a))
   (if (<= b 4.5e-167) (* -0.5 (sqrt (* c (/ -4.0 a)))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-208) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.5e-167) {
		tmp = -0.5 * sqrt((c * (-4.0 / a)));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.6d-208)) then
        tmp = (c / b) - (b / a)
    else if (b <= 4.5d-167) then
        tmp = (-0.5d0) * sqrt((c * ((-4.0d0) / a)))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-208) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.5e-167) {
		tmp = -0.5 * Math.sqrt((c * (-4.0 / a)));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.6e-208:
		tmp = (c / b) - (b / a)
	elif b <= 4.5e-167:
		tmp = -0.5 * math.sqrt((c * (-4.0 / a)))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.6e-208)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4.5e-167)
		tmp = Float64(-0.5 * sqrt(Float64(c * Float64(-4.0 / a))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.6e-208)
		tmp = (c / b) - (b / a);
	elseif (b <= 4.5e-167)
		tmp = -0.5 * sqrt((c * (-4.0 / a)));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.6e-208], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.5e-167], N[(-0.5 * N[Sqrt[N[(c * N[(-4.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.60000000000000017e-208

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

   3. Simplified 79.7%

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

      1. *-commutative 79.7%

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

      2. add-sqr-sqrt 43.6%

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

      3. sqrt-unprod 66.1%

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

      4. *-commutative 66.1%

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

      5. *-commutative 66.1%

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

      6. swap-sqr 66.1%

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

      7. metadata-eval 66.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 66.1%

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

      9. swap-sqr 66.1%

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

      10. sqrt-unprod 29.7%

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

      11. add-sqr-sqrt 59.5%

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

      12. add-cube-cbrt 59.5%

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

      13. pow3 59.5%

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

   6. Applied egg-rr 79.4%

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

   8. Applied egg-rr 79.6%

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

   9. Taylor expanded in b around -inf 74.4%

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

       1. mul-1-neg 74.4%

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

       2. *-commutative 74.4%

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

       3. distribute-rgt-neg-in 74.4%

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

       4. +-commutative 74.4%

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

       5. mul-1-neg 74.4%

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

       6. unsub-neg 74.4%

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

   11. Simplified 74.4%

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

   12. Taylor expanded in a around 0 74.5%

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

       1. +-commutative 74.5%

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

       2. *-commutative 74.5%

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

       3. associate-*r/ 76.5%

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

       4. mul-1-neg 76.5%

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

       5. unsub-neg 76.5%

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

       6. div-sub 76.5%

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

       7. associate-/l* 76.5%

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

       8. associate-/l/ 69.9%

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

       9. associate-/r* 76.5%

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

       10. *-inverses 76.5%

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

       11. associate-*r/ 76.5%

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

       12. *-rgt-identity 76.5%

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

   14. Simplified 76.5%

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

   if -2.60000000000000017e-208 < b < 4.5000000000000001e-167

   1. Initial program 74.4%

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

      1. *-commutative 74.4%

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

   3. Simplified 74.4%

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

      1. *-commutative 74.4%

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

      2. add-sqr-sqrt 14.2%

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

      3. sqrt-unprod 14.5%

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

      4. *-commutative 14.5%

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

      5. *-commutative 14.5%

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

      6. swap-sqr 14.5%

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

      7. metadata-eval 14.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 14.5%

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

      9. swap-sqr 14.5%

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

      10. sqrt-unprod 0.3%

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

      11. add-sqr-sqrt 0.7%

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

      12. add-cube-cbrt 0.7%

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

      13. pow3 0.7%

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

   6. Applied egg-rr 73.7%

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

   7. Taylor expanded in c around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      3. rem-square-sqrt 28.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      4. associate-/l* 28.0%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{c \cdot \frac{{\left(\sqrt[3]{-4}\right)}^{3}}{a}}}\right) \]

      5. rem-cube-cbrt 28.2%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{c \cdot \frac{\color{blue}{-4}}{a}}\right) \]

   9. Simplified 28.2%

      \[\leadsto \color{blue}{-0.5 \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)} \]
   10. Step-by-step derivation

       1. pow1 28.2%

          \[\leadsto \color{blue}{{\left(-0.5 \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)\right)}^{1}} \]

       2. *-commutative 28.2%

          \[\leadsto {\color{blue}{\left(\left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right) \cdot -0.5\right)}}^{1} \]

       3. add-sqr-sqrt 0.7%

          \[\leadsto {\left(\color{blue}{\left(\sqrt{-1 \cdot \sqrt{c \cdot \frac{-4}{a}}} \cdot \sqrt{-1 \cdot \sqrt{c \cdot \frac{-4}{a}}}\right)} \cdot -0.5\right)}^{1} \]

       4. sqrt-unprod 53.5%

          \[\leadsto {\left(\color{blue}{\sqrt{\left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right) \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)}} \cdot -0.5\right)}^{1} \]

       5. mul-1-neg 53.5%

          \[\leadsto {\left(\sqrt{\color{blue}{\left(-\sqrt{c \cdot \frac{-4}{a}}\right)} \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)} \cdot -0.5\right)}^{1} \]

       6. mul-1-neg 53.5%

          \[\leadsto {\left(\sqrt{\left(-\sqrt{c \cdot \frac{-4}{a}}\right) \cdot \color{blue}{\left(-\sqrt{c \cdot \frac{-4}{a}}\right)}} \cdot -0.5\right)}^{1} \]

       7. sqr-neg 53.5%

          \[\leadsto {\left(\sqrt{\color{blue}{\sqrt{c \cdot \frac{-4}{a}} \cdot \sqrt{c \cdot \frac{-4}{a}}}} \cdot -0.5\right)}^{1} \]

       8. add-sqr-sqrt 53.5%

          \[\leadsto {\left(\sqrt{\color{blue}{c \cdot \frac{-4}{a}}} \cdot -0.5\right)}^{1} \]

       9. associate-*r/ 53.3%

          \[\leadsto {\left(\sqrt{\color{blue}{\frac{c \cdot -4}{a}}} \cdot -0.5\right)}^{1} \]

   11. Applied egg-rr 53.3%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{c \cdot -4}{a}} \cdot -0.5\right)}^{1}} \]
   12. Step-by-step derivation

       1. unpow1 53.3%

          \[\leadsto \color{blue}{\sqrt{\frac{c \cdot -4}{a}} \cdot -0.5} \]

       2. *-commutative 53.3%

          \[\leadsto \color{blue}{-0.5 \cdot \sqrt{\frac{c \cdot -4}{a}}} \]

       3. associate-/l* 53.5%

          \[\leadsto -0.5 \cdot \sqrt{\color{blue}{c \cdot \frac{-4}{a}}} \]

   13. Simplified 53.5%

       \[\leadsto \color{blue}{-0.5 \cdot \sqrt{c \cdot \frac{-4}{a}}} \]

   if 4.5000000000000001e-167 < b

   1. Initial program 21.8%

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

      1. *-commutative 21.8%

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

   3. Simplified 21.8%

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

      1. *-commutative 21.8%

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

      2. add-sqr-sqrt 16.1%

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

      3. sqrt-unprod 11.2%

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

      4. *-commutative 11.2%

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

      5. *-commutative 11.2%

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

      6. swap-sqr 11.2%

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

      7. metadata-eval 11.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 11.2%

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

      9. swap-sqr 11.2%

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

      10. sqrt-unprod 1.0%

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

      11. add-sqr-sqrt 4.9%

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

      12. add-cube-cbrt 4.9%

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

      13. pow3 4.9%

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

   6. Applied egg-rr 21.6%

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

   8. Applied egg-rr 21.8%

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

   9. Taylor expanded in b around inf 81.6%

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

       1. associate-*r/ 81.6%

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

       2. mul-1-neg 81.6%

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

   11. Simplified 81.6%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-208}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-167}:\\ \;\;\;\;-0.5 \cdot \sqrt{c \cdot \frac{-4}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.42 \cdot 10^{-188}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.42e-188)
   (- (/ c b) (/ b a))
   (if (<= b 4.8e-105) (sqrt (/ (- c) a)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.42e-188) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.8e-105) {
		tmp = sqrt((-c / a));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.42d-188)) then
        tmp = (c / b) - (b / a)
    else if (b <= 4.8d-105) then
        tmp = sqrt((-c / a))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.42e-188) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.8e-105) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.42e-188:
		tmp = (c / b) - (b / a)
	elif b <= 4.8e-105:
		tmp = math.sqrt((-c / a))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.42e-188)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4.8e-105)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.42e-188)
		tmp = (c / b) - (b / a);
	elseif (b <= 4.8e-105)
		tmp = sqrt((-c / a));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.42e-188], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.8e-105], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.4200000000000001e-188

   1. Initial program 79.6%

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

      1. *-commutative 79.6%

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

   3. Simplified 79.6%

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

      1. *-commutative 79.6%

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

      2. add-sqr-sqrt 43.1%

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

      3. sqrt-unprod 65.8%

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

      4. *-commutative 65.8%

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

      5. *-commutative 65.8%

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

      6. swap-sqr 65.8%

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

      7. metadata-eval 65.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 65.8%

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

      9. swap-sqr 65.8%

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

      10. sqrt-unprod 30.0%

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

      11. add-sqr-sqrt 60.1%

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

      12. add-cube-cbrt 60.1%

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

      13. pow3 60.1%

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

   6. Applied egg-rr 79.2%

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

   8. Applied egg-rr 79.4%

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

   9. Taylor expanded in b around -inf 75.0%

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

       1. mul-1-neg 75.0%

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

       2. *-commutative 75.0%

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

       3. distribute-rgt-neg-in 75.0%

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

       4. +-commutative 75.0%

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

       5. mul-1-neg 75.0%

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

       6. unsub-neg 75.0%

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

   11. Simplified 75.0%

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

   12. Taylor expanded in a around 0 75.2%

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

       1. +-commutative 75.2%

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

       2. *-commutative 75.2%

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

       3. associate-*r/ 77.1%

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

       4. mul-1-neg 77.1%

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

       5. unsub-neg 77.1%

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

       6. div-sub 77.1%

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

       7. associate-/l* 77.1%

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

       8. associate-/l/ 70.5%

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

       9. associate-/r* 77.2%

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

       10. *-inverses 77.2%

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

       11. associate-*r/ 77.2%

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

       12. *-rgt-identity 77.2%

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

   14. Simplified 77.2%

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

   if -1.4200000000000001e-188 < b < 4.8000000000000003e-105

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

   3. Simplified 78.4%

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

      1. *-commutative 78.4%

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

      2. add-sqr-sqrt 33.8%

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

      3. sqrt-unprod 21.1%

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

      4. *-commutative 21.1%

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

      5. *-commutative 21.1%

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

      6. swap-sqr 21.1%

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

      7. metadata-eval 21.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 21.1%

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

      9. swap-sqr 21.1%

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

      10. sqrt-unprod 0.2%

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

      11. add-sqr-sqrt 0.7%

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

      12. add-cube-cbrt 0.7%

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

      13. pow3 0.7%

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

   6. Applied egg-rr 77.7%

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

   7. Taylor expanded in c around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      3. rem-square-sqrt 30.8%

         \[\leadsto -0.5 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      4. associate-/l* 30.8%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{c \cdot \frac{{\left(\sqrt[3]{-4}\right)}^{3}}{a}}}\right) \]

      5. rem-cube-cbrt 31.0%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{c \cdot \frac{\color{blue}{-4}}{a}}\right) \]

   9. Simplified 31.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 31.0%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\sqrt{c \cdot \frac{-4}{a}} \cdot \sqrt{c \cdot \frac{-4}{a}}}}\right) \]

       2. sqr-neg 31.0%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\left(-\sqrt{c \cdot \frac{-4}{a}}\right) \cdot \left(-\sqrt{c \cdot \frac{-4}{a}}\right)}}\right) \]

       3. mul-1-neg 31.0%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)} \cdot \left(-\sqrt{c \cdot \frac{-4}{a}}\right)}\right) \]

       4. mul-1-neg 31.0%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right) \cdot \color{blue}{\left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)}}\right) \]

       5. sqrt-unprod 0.8%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\sqrt{-1 \cdot \sqrt{c \cdot \frac{-4}{a}}} \cdot \sqrt{-1 \cdot \sqrt{c \cdot \frac{-4}{a}}}\right)}\right) \]

       6. add-sqr-sqrt 40.8%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \color{blue}{\left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)}\right) \]

       7. mul-1-neg 40.8%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \color{blue}{\left(-\sqrt{c \cdot \frac{-4}{a}}\right)}\right) \]

       8. sqrt-prod 50.9%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \left(-\color{blue}{\sqrt{c} \cdot \sqrt{\frac{-4}{a}}}\right)\right) \]

       9. distribute-rgt-neg-in 50.9%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\sqrt{c} \cdot \left(-\sqrt{\frac{-4}{a}}\right)\right)}\right) \]

   11. Applied egg-rr 50.9%

       \[\leadsto -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\sqrt{c} \cdot \left(-\sqrt{\frac{-4}{a}}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. distribute-rgt-neg-out 50.9%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \color{blue}{\left(-\sqrt{c} \cdot \sqrt{\frac{-4}{a}}\right)}\right) \]

       2. distribute-lft-neg-out 50.9%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\left(-\sqrt{c}\right) \cdot \sqrt{\frac{-4}{a}}\right)}\right) \]

       3. *-commutative 50.9%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\sqrt{\frac{-4}{a}} \cdot \left(-\sqrt{c}\right)\right)}\right) \]

   13. Simplified 50.9%

       \[\leadsto -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\sqrt{\frac{-4}{a}} \cdot \left(-\sqrt{c}\right)\right)}\right) \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \color{blue}{\sqrt{-0.5 \cdot \left(-1 \cdot \left(\sqrt{\frac{-4}{a}} \cdot \left(-\sqrt{c}\right)\right)\right)} \cdot \sqrt{-0.5 \cdot \left(-1 \cdot \left(\sqrt{\frac{-4}{a}} \cdot \left(-\sqrt{c}\right)\right)\right)}} \]

       2. sqrt-unprod 1.2%

          \[\leadsto \color{blue}{\sqrt{\left(-0.5 \cdot \left(-1 \cdot \left(\sqrt{\frac{-4}{a}} \cdot \left(-\sqrt{c}\right)\right)\right)\right) \cdot \left(-0.5 \cdot \left(-1 \cdot \left(\sqrt{\frac{-4}{a}} \cdot \left(-\sqrt{c}\right)\right)\right)\right)}} \]

       3. *-commutative 1.2%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-1 \cdot \left(\sqrt{\frac{-4}{a}} \cdot \left(-\sqrt{c}\right)\right)\right) \cdot -0.5\right)} \cdot \left(-0.5 \cdot \left(-1 \cdot \left(\sqrt{\frac{-4}{a}} \cdot \left(-\sqrt{c}\right)\right)\right)\right)} \]

       4. *-commutative 1.2%

          \[\leadsto \sqrt{\left(\left(-1 \cdot \left(\sqrt{\frac{-4}{a}} \cdot \left(-\sqrt{c}\right)\right)\right) \cdot -0.5\right) \cdot \color{blue}{\left(\left(-1 \cdot \left(\sqrt{\frac{-4}{a}} \cdot \left(-\sqrt{c}\right)\right)\right) \cdot -0.5\right)}} \]

       5. swap-sqr 1.2%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-1 \cdot \left(\sqrt{\frac{-4}{a}} \cdot \left(-\sqrt{c}\right)\right)\right) \cdot \left(-1 \cdot \left(\sqrt{\frac{-4}{a}} \cdot \left(-\sqrt{c}\right)\right)\right)\right) \cdot \left(-0.5 \cdot -0.5\right)}} \]

   15. Applied egg-rr 30.9%

       \[\leadsto \color{blue}{\sqrt{\left(c \cdot \frac{-4}{a}\right) \cdot 0.25}} \]
   16. Step-by-step derivation

       1. associate-*l* 30.9%

          \[\leadsto \sqrt{\color{blue}{c \cdot \left(\frac{-4}{a} \cdot 0.25\right)}} \]

       2. associate-*l/ 30.9%

          \[\leadsto \sqrt{c \cdot \color{blue}{\frac{-4 \cdot 0.25}{a}}} \]

       3. metadata-eval 30.9%

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

       4. metadata-eval 30.9%

          \[\leadsto \sqrt{c \cdot \frac{\color{blue}{\frac{-2}{2}}}{a}} \]

       5. associate-/r* 30.9%

          \[\leadsto \sqrt{c \cdot \color{blue}{\frac{-2}{2 \cdot a}}} \]

       6. *-commutative 30.9%

          \[\leadsto \sqrt{c \cdot \frac{-2}{\color{blue}{a \cdot 2}}} \]

       7. associate-/l* 30.9%

          \[\leadsto \sqrt{\color{blue}{\frac{c \cdot -2}{a \cdot 2}}} \]

       8. times-frac 30.9%

          \[\leadsto \sqrt{\color{blue}{\frac{c}{a} \cdot \frac{-2}{2}}} \]

       9. metadata-eval 30.9%

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

       10. associate-*l/ 30.9%

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

       11. *-commutative 30.9%

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

       12. mul-1-neg 30.9%

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

   17. Simplified 30.9%

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

   if 4.8000000000000003e-105 < b

   1. Initial program 13.3%

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

      1. *-commutative 13.3%

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

   3. Simplified 13.3%

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

      1. *-commutative 13.3%

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

      2. add-sqr-sqrt 8.7%

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

      3. sqrt-unprod 8.7%

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

      4. *-commutative 8.7%

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

      5. *-commutative 8.7%

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

      6. swap-sqr 8.7%

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

      7. metadata-eval 8.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 8.7%

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

      9. swap-sqr 8.7%

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

      10. sqrt-unprod 1.1%

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

      11. add-sqr-sqrt 5.5%

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

      12. add-cube-cbrt 5.5%

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

      13. pow3 5.5%

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

   6. Applied egg-rr 13.2%

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

   8. Applied egg-rr 13.3%

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

   9. Taylor expanded in b around inf 89.7%

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

       1. associate-*r/ 89.7%

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

       2. mul-1-neg 89.7%

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

   11. Simplified 89.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.42 \cdot 10^{-188}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 79.2%

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

      1. *-commutative 79.2%

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

   3. Simplified 79.2%

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

      1. *-commutative 79.2%

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

      2. add-sqr-sqrt 41.3%

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

      3. sqrt-unprod 61.7%

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

      4. *-commutative 61.7%

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

      5. *-commutative 61.7%

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

      6. swap-sqr 61.7%

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

      7. metadata-eval 61.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 61.7%

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

      9. swap-sqr 61.7%

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

      10. sqrt-unprod 27.1%

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

      11. add-sqr-sqrt 54.1%

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

      12. add-cube-cbrt 54.1%

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

      13. pow3 54.1%

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

   6. Applied egg-rr 78.8%

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

   8. Applied egg-rr 79.0%

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

   9. Taylor expanded in b around -inf 67.7%

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

       1. mul-1-neg 67.7%

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

       2. *-commutative 67.7%

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

       3. distribute-rgt-neg-in 67.7%

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

       4. +-commutative 67.7%

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

       5. mul-1-neg 67.7%

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

       6. unsub-neg 67.7%

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

   11. Simplified 67.7%

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

   12. Taylor expanded in a around 0 68.0%

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

       1. +-commutative 68.0%

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

       2. *-commutative 68.0%

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

       3. associate-*r/ 69.7%

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

       4. mul-1-neg 69.7%

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

       5. unsub-neg 69.7%

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

       6. div-sub 69.7%

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

       7. associate-/l* 69.7%

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

       8. associate-/l/ 63.7%

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

       9. associate-/r* 69.8%

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

       10. *-inverses 69.8%

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

       11. associate-*r/ 69.8%

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

       12. *-rgt-identity 69.8%

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

   14. Simplified 69.8%

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

   if -1.999999999999994e-310 < b

   1. Initial program 29.1%

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

      1. *-commutative 29.1%

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

   3. Simplified 29.1%

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

      1. *-commutative 29.1%

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

      2. add-sqr-sqrt 15.5%

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

      3. sqrt-unprod 11.3%

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

      4. *-commutative 11.3%

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

      5. *-commutative 11.3%

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

      6. swap-sqr 11.3%

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

      7. metadata-eval 11.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 11.3%

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

      9. swap-sqr 11.3%

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

      10. sqrt-unprod 0.9%

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

      11. add-sqr-sqrt 4.3%

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

      12. add-cube-cbrt 4.3%

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

      13. pow3 4.3%

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

   6. Applied egg-rr 28.8%

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

   8. Applied egg-rr 29.1%

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

   9. Taylor expanded in b around inf 71.3%

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

       1. associate-*r/ 71.3%

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

       2. mul-1-neg 71.3%

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

   11. Simplified 71.3%

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

4. Final simplification 70.6%

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

Alternative 6: 67.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-290}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.85e-290) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-290) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.85d-290) then
        tmp = b / -a
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-290) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.85e-290:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.85e-290)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.85e-290)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.85e-290], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.84999999999999989e-290

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

   3. Simplified 79.9%

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

      1. *-commutative 79.9%

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

      2. add-sqr-sqrt 40.8%

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

      3. sqrt-unprod 60.5%

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

      4. *-commutative 60.5%

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

      5. *-commutative 60.5%

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

      6. swap-sqr 60.5%

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

      7. metadata-eval 60.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 60.5%

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

      9. swap-sqr 60.5%

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

      10. sqrt-unprod 26.2%

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

      11. add-sqr-sqrt 52.3%

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

      12. add-cube-cbrt 52.3%

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

      13. pow3 52.3%

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

   6. Applied egg-rr 79.5%

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

   7. Taylor expanded in b around -inf 67.3%

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

      1. associate-*r/ 67.3%

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

      2. mul-1-neg 67.3%

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

   9. Simplified 67.3%

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

   if 1.84999999999999989e-290 < b

   1. Initial program 27.0%

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

      1. *-commutative 27.0%

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

   3. Simplified 27.0%

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

      1. *-commutative 27.0%

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

      2. add-sqr-sqrt 15.2%

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

      3. sqrt-unprod 10.9%

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

      4. *-commutative 10.9%

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

      5. *-commutative 10.9%

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

      6. swap-sqr 10.9%

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

      7. metadata-eval 10.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 10.9%

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

      9. swap-sqr 10.9%

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

      10. sqrt-unprod 0.9%

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

      11. add-sqr-sqrt 4.5%

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

      12. add-cube-cbrt 4.5%

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

      13. pow3 4.5%

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

   6. Applied egg-rr 26.7%

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

   8. Applied egg-rr 27.0%

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

   9. Taylor expanded in b around inf 73.4%

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

       1. associate-*r/ 73.4%

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

       2. mul-1-neg 73.4%

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

   11. Simplified 73.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-290}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 2.15e-29) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.15e-29) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.15d-29) then
        tmp = b / -a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.15e-29) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.15e-29:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.15e-29)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.15e-29)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.15e-29], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1499999999999999e-29

   1. Initial program 77.7%

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

      1. *-commutative 77.7%

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

   3. Simplified 77.7%

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

      1. *-commutative 77.7%

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

      2. add-sqr-sqrt 39.9%

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

      3. sqrt-unprod 50.6%

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

      4. *-commutative 50.6%

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

      5. *-commutative 50.6%

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

      6. swap-sqr 50.6%

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

      7. metadata-eval 50.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 50.6%

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

      9. swap-sqr 50.6%

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

      10. sqrt-unprod 19.8%

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

      11. add-sqr-sqrt 39.7%

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

      12. add-cube-cbrt 39.7%

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

      13. pow3 39.7%

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

   6. Applied egg-rr 77.3%

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

   7. Taylor expanded in b around -inf 51.6%

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

      1. associate-*r/ 51.6%

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

      2. mul-1-neg 51.6%

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

   9. Simplified 51.6%

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

   if 2.1499999999999999e-29 < b

   1. Initial program 8.2%

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

      1. *-commutative 8.2%

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

   3. Simplified 8.2%

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

      1. *-commutative 8.2%

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

      2. add-sqr-sqrt 5.8%

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

      3. sqrt-unprod 6.8%

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

      4. *-commutative 6.8%

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

      5. *-commutative 6.8%

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

      6. swap-sqr 6.8%

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

      7. metadata-eval 6.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

      8. metadata-eval 6.8%

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

      9. swap-sqr 6.8%

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

      10. sqrt-unprod 1.2%

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

      11. add-sqr-sqrt 5.9%

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

      12. add-cube-cbrt 5.9%

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

      13. pow3 5.9%

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

   6. Applied egg-rr 8.1%

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

   8. Applied egg-rr 8.2%

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

   9. Taylor expanded in b around -inf 2.2%

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

       1. mul-1-neg 2.2%

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

       2. *-commutative 2.2%

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

       3. distribute-rgt-neg-in 2.2%

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

       4. +-commutative 2.2%

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

       5. mul-1-neg 2.2%

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

       6. unsub-neg 2.2%

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

   11. Simplified 2.2%

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

   12. Taylor expanded in a around inf 27.6%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 10.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

   1. *-commutative 52.2%

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

3. Simplified 52.2%

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

   1. *-commutative 52.2%

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

   2. add-sqr-sqrt 27.4%

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

   3. sqrt-unprod 34.5%

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

   4. *-commutative 34.5%

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

   5. *-commutative 34.5%

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

   6. swap-sqr 34.5%

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

   7. metadata-eval 34.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{16}}}}{a \cdot 2} \]

   8. metadata-eval 34.5%

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

   9. swap-sqr 34.5%

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

   10. sqrt-unprod 12.9%

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

   11. add-sqr-sqrt 27.3%

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

   12. add-cube-cbrt 27.3%

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

   13. pow3 27.3%

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

6. Applied egg-rr 51.9%

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

8. Applied egg-rr 52.1%

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

9. Taylor expanded in b around -inf 32.3%

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

    1. mul-1-neg 32.3%

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

    2. *-commutative 32.3%

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

    3. distribute-rgt-neg-in 32.3%

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

    4. +-commutative 32.3%

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

    5. mul-1-neg 32.3%

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

    6. unsub-neg 32.3%

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

11. Simplified 32.3%

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

12. Taylor expanded in a around inf 12.3%

    \[\leadsto \color{blue}{\frac{c}{b}} \]
13. Add Preprocessing

Developer target: 70.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - t\_0}{2 \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (< b 0.0)
     (/ (+ (- b) t_0) (* 2.0 a))
     (/ c (* a (/ (- (- b) t_0) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b < 0.0d0) then
        tmp = (-b + t_0) / (2.0d0 * a)
    else
        tmp = c / (a * ((-b - t_0) / (2.0d0 * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b < 0.0:
		tmp = (-b + t_0) / (2.0 * a)
	else:
		tmp = c / (a * ((-b - t_0) / (2.0 * a)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c / N[(a * N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024111 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :alt
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))