quadp (p42, positive)

Percentage Accurate: 53.1% → 85.5%

Time: 13.0s

Alternatives: 10

Speedup: 11.6×

Specification

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 53.1% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+104}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e+104)
   (/ b (- 0.0 a))
   (if (<= b 6.8e-90)
     (/ (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (* a 2.0))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e+104) {
		tmp = b / (0.0 - a);
	} else if (b <= 6.8e-90) {
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.15d+104)) then
        tmp = b / (0.0d0 - a)
    else if (b <= 6.8d-90) then
        tmp = (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b) / (a * 2.0d0)
    else
        tmp = 0.0d0 - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e+104) {
		tmp = b / (0.0 - a);
	} else if (b <= 6.8e-90) {
		tmp = (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e+104:
		tmp = b / (0.0 - a)
	elif b <= 6.8e-90:
		tmp = (math.sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e+104)
		tmp = Float64(b / Float64(0.0 - a));
	elseif (b <= 6.8e-90)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.15e+104)
		tmp = b / (0.0 - a);
	elseif (b <= 6.8e-90)
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e+104], N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e-90], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14999999999999992e104

   1. Initial program 55.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 55.3%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(-1 \cdot a\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

      6. neg-lowering-neg.f64 97.2%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{neg.f64}\left(a\right)\right) \]

   7. Simplified 97.2%

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

   if -1.14999999999999992e104 < b < 6.79999999999999988e-90

   1. Initial program 82.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 82.9%

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

   if 6.79999999999999988e-90 < b

   1. Initial program 12.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 12.6%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 86.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 86.6%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+104}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-90}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.15e+54)
   (/ b (- 0.0 a))
   (if (<= b 3.8e-90)
     (/ 0.5 (/ a (- (sqrt (+ (* b b) (* a (* c -4.0)))) b)))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.15e+54) {
		tmp = b / (0.0 - a);
	} else if (b <= 3.8e-90) {
		tmp = 0.5 / (a / (sqrt(((b * b) + (a * (c * -4.0)))) - b));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.15d+54)) then
        tmp = b / (0.0d0 - a)
    else if (b <= 3.8d-90) then
        tmp = 0.5d0 / (a / (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b))
    else
        tmp = 0.0d0 - (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+54) {
		tmp = b / (0.0 - a);
	} else if (b <= 3.8e-90) {
		tmp = 0.5 / (a / (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.15e+54:
		tmp = b / (0.0 - a)
	elif b <= 3.8e-90:
		tmp = 0.5 / (a / (math.sqrt(((b * b) + (a * (c * -4.0)))) - b))
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.15e+54)
		tmp = Float64(b / Float64(0.0 - a));
	elseif (b <= 3.8e-90)
		tmp = Float64(0.5 / Float64(a / Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b)));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.15e+54)
		tmp = b / (0.0 - a);
	elseif (b <= 3.8e-90)
		tmp = 0.5 / (a / (sqrt(((b * b) + (a * (c * -4.0)))) - b));
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.15e+54], N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e-90], N[(0.5 / N[(a / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.14999999999999988e54

   1. Initial program 59.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 59.7%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(-1 \cdot a\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

      6. neg-lowering-neg.f64 97.5%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{neg.f64}\left(a\right)\right) \]

   7. Simplified 97.5%

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

   if -2.14999999999999988e54 < b < 3.8e-90

   1. Initial program 81.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 81.6%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      13. *-lowering-*.f64 81.5%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right)\right) \]

   6. Applied egg-rr 81.5%

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

   if 3.8e-90 < b

   1. Initial program 12.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 12.6%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 86.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 86.6%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-90}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-90}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.15e+54)
   (/ b (- 0.0 a))
   (if (<= b 6.8e-90)
     (* (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (/ 0.5 a))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.15e+54) {
		tmp = b / (0.0 - a);
	} else if (b <= 6.8e-90) {
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.15d+54)) then
        tmp = b / (0.0d0 - a)
    else if (b <= 6.8d-90) then
        tmp = (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b) * (0.5d0 / a)
    else
        tmp = 0.0d0 - (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+54) {
		tmp = b / (0.0 - a);
	} else if (b <= 6.8e-90) {
		tmp = (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.15e+54:
		tmp = b / (0.0 - a)
	elif b <= 6.8e-90:
		tmp = (math.sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.15e+54)
		tmp = Float64(b / Float64(0.0 - a));
	elseif (b <= 6.8e-90)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.15e+54)
		tmp = b / (0.0 - a);
	elseif (b <= 6.8e-90)
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.15e+54], N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e-90], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.14999999999999988e54

   1. Initial program 59.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 59.7%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(-1 \cdot a\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

      6. neg-lowering-neg.f64 97.5%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{neg.f64}\left(a\right)\right) \]

   7. Simplified 97.5%

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

   if -2.14999999999999988e54 < b < 6.79999999999999988e-90

   1. Initial program 81.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 81.6%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} - b\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}} - b\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} - b\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right) \]

      13. *-lowering-*.f64 81.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right) \]

   6. Applied egg-rr 81.5%

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

   if 6.79999999999999988e-90 < b

   1. Initial program 12.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 12.6%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 86.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 86.6%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-90}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{b \cdot \left(-1 - c \cdot \left(-2 \cdot \frac{a}{b \cdot b}\right)\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.3e-32)
   (/ (- (* b (- -1.0 (* c (* -2.0 (/ a (* b b)))))) b) (* a 2.0))
   (if (<= b 1.8e-90)
     (/ (/ 0.5 a) (/ 1.0 (- (sqrt (* a (* c -4.0))) b)))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-32) {
		tmp = ((b * (-1.0 - (c * (-2.0 * (a / (b * b)))))) - b) / (a * 2.0);
	} else if (b <= 1.8e-90) {
		tmp = (0.5 / a) / (1.0 / (sqrt((a * (c * -4.0))) - b));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.3d-32)) then
        tmp = ((b * ((-1.0d0) - (c * ((-2.0d0) * (a / (b * b)))))) - b) / (a * 2.0d0)
    else if (b <= 1.8d-90) then
        tmp = (0.5d0 / a) / (1.0d0 / (sqrt((a * (c * (-4.0d0)))) - b))
    else
        tmp = 0.0d0 - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-32) {
		tmp = ((b * (-1.0 - (c * (-2.0 * (a / (b * b)))))) - b) / (a * 2.0);
	} else if (b <= 1.8e-90) {
		tmp = (0.5 / a) / (1.0 / (Math.sqrt((a * (c * -4.0))) - b));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.3e-32:
		tmp = ((b * (-1.0 - (c * (-2.0 * (a / (b * b)))))) - b) / (a * 2.0)
	elif b <= 1.8e-90:
		tmp = (0.5 / a) / (1.0 / (math.sqrt((a * (c * -4.0))) - b))
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.3e-32)
		tmp = Float64(Float64(Float64(b * Float64(-1.0 - Float64(c * Float64(-2.0 * Float64(a / Float64(b * b)))))) - b) / Float64(a * 2.0));
	elseif (b <= 1.8e-90)
		tmp = Float64(Float64(0.5 / a) / Float64(1.0 / Float64(sqrt(Float64(a * Float64(c * -4.0))) - b)));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.3e-32)
		tmp = ((b * (-1.0 - (c * (-2.0 * (a / (b * b)))))) - b) / (a * 2.0);
	elseif (b <= 1.8e-90)
		tmp = (0.5 / a) / (1.0 / (sqrt((a * (c * -4.0))) - b));
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.3e-32], N[(N[(N[(b * N[(-1.0 - N[(c * N[(-2.0 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-90], N[(N[(0.5 / a), $MachinePrecision] / N[(1.0 / N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.30000000000000025e-32

   1. Initial program 67.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 67.5%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(b \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}, b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(-1 \cdot b\right) \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(b \cdot -1\right) \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \left(-1 \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{2}}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{-2 \cdot \left(a \cdot c\right)}{b \cdot b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\frac{-2 \cdot \left(a \cdot c\right)}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\frac{\left(-2 \cdot a\right) \cdot c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\left(-2 \cdot a\right) \cdot \frac{c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\left(a \cdot -2\right) \cdot \frac{c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{a \cdot \left(-2 \cdot \frac{c}{b}\right)}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot \left(-2 \cdot \frac{c}{b}\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(-2 \cdot \frac{c}{b}\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{c}{b} \cdot -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{c}{b}\right), -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      18. /-lowering-/.f64 94.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(c, b\right), -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   7. Simplified 94.9%

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

   8. Taylor expanded in c around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{\left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}} + \frac{1}{c}\right)\right)\right)}\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right) + c \cdot \frac{1}{c}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      2. rgt-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right) + 1\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right) + -1 \cdot 1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right) + -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-2, \left(\frac{a}{{b}^{2}}\right)\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(a, \left({b}^{2}\right)\right)\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(a, \left(b \cdot b\right)\right)\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      11. *-lowering-*.f64 94.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   10. Simplified 94.9%

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

   if -3.30000000000000025e-32 < b < 1.7999999999999999e-90

   1. Initial program 77.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 77.3%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right) \]

      13. *-lowering-*.f64 77.2%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right)\right) \]

   6. Applied egg-rr 77.2%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right), b\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right), b\right)\right)\right) \]

      2. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right), b\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right), b\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right), b\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right), b\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right), b\right)\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

      8. *-lowering-*.f64 70.9%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right), b\right)\right)\right) \]

   9. Simplified 70.9%

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

       1. div-inv N/A

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

       2. associate-/r* N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \color{blue}{\left(\frac{1}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\frac{\color{blue}{1}}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\frac{-1 \cdot -1}{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)}} - b}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(\left(-1 \cdot -1\right), \color{blue}{\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)}\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(1, \left(\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)}} - b\right)\right)\right) \]

       8. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\sqrt{a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right)\right) \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right), b\right)\right)\right) \]

       11. *-lowering-*.f64 71.0%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right), b\right)\right)\right) \]

   11. Applied egg-rr 71.0%

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

   if 1.7999999999999999e-90 < b

   1. Initial program 12.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 12.6%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 86.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 86.6%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{b \cdot \left(-1 - c \cdot \left(-2 \cdot \frac{a}{b \cdot b}\right)\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{b \cdot \left(-1 - c \cdot \left(-2 \cdot \frac{a}{b \cdot b}\right)\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-33)
   (/ (- (* b (- -1.0 (* c (* -2.0 (/ a (* b b)))))) b) (* a 2.0))
   (if (<= b 6.5e-91)
     (/ (- (sqrt (* -4.0 (* a c))) b) (* a 2.0))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-33) {
		tmp = ((b * (-1.0 - (c * (-2.0 * (a / (b * b)))))) - b) / (a * 2.0);
	} else if (b <= 6.5e-91) {
		tmp = (sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.5d-33)) then
        tmp = ((b * ((-1.0d0) - (c * ((-2.0d0) * (a / (b * b)))))) - b) / (a * 2.0d0)
    else if (b <= 6.5d-91) then
        tmp = (sqrt(((-4.0d0) * (a * c))) - b) / (a * 2.0d0)
    else
        tmp = 0.0d0 - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-33) {
		tmp = ((b * (-1.0 - (c * (-2.0 * (a / (b * b)))))) - b) / (a * 2.0);
	} else if (b <= 6.5e-91) {
		tmp = (Math.sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e-33:
		tmp = ((b * (-1.0 - (c * (-2.0 * (a / (b * b)))))) - b) / (a * 2.0)
	elif b <= 6.5e-91:
		tmp = (math.sqrt((-4.0 * (a * c))) - b) / (a * 2.0)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-33)
		tmp = Float64(Float64(Float64(b * Float64(-1.0 - Float64(c * Float64(-2.0 * Float64(a / Float64(b * b)))))) - b) / Float64(a * 2.0));
	elseif (b <= 6.5e-91)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e-33)
		tmp = ((b * (-1.0 - (c * (-2.0 * (a / (b * b)))))) - b) / (a * 2.0);
	elseif (b <= 6.5e-91)
		tmp = (sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e-33], N[(N[(N[(b * N[(-1.0 - N[(c * N[(-2.0 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e-91], N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.49999999999999945e-33

   1. Initial program 67.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 67.5%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(b \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}, b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(-1 \cdot b\right) \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(b \cdot -1\right) \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \left(-1 \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{2}}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{-2 \cdot \left(a \cdot c\right)}{b \cdot b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\frac{-2 \cdot \left(a \cdot c\right)}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\frac{\left(-2 \cdot a\right) \cdot c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\left(-2 \cdot a\right) \cdot \frac{c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\left(a \cdot -2\right) \cdot \frac{c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{a \cdot \left(-2 \cdot \frac{c}{b}\right)}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot \left(-2 \cdot \frac{c}{b}\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(-2 \cdot \frac{c}{b}\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{c}{b} \cdot -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{c}{b}\right), -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      18. /-lowering-/.f64 94.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(c, b\right), -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   7. Simplified 94.9%

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

   8. Taylor expanded in c around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{\left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}} + \frac{1}{c}\right)\right)\right)}\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right) + c \cdot \frac{1}{c}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      2. rgt-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right) + 1\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right) + -1 \cdot 1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right) + -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-2, \left(\frac{a}{{b}^{2}}\right)\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(a, \left({b}^{2}\right)\right)\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(a, \left(b \cdot b\right)\right)\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      11. *-lowering-*.f64 94.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   10. Simplified 94.9%

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

   if -8.49999999999999945e-33 < b < 6.5000000000000001e-91

   1. Initial program 77.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 77.3%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      3. *-lowering-*.f64 70.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   7. Simplified 70.9%

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

   if 6.5000000000000001e-91 < b

   1. Initial program 12.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 12.6%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 86.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 86.6%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{b \cdot \left(-1 - c \cdot \left(-2 \cdot \frac{a}{b \cdot b}\right)\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{b \cdot \left(-1 - c \cdot \left(-2 \cdot \frac{a}{b \cdot b}\right)\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-90}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e-33)
   (/ (- (* b (- -1.0 (* c (* -2.0 (/ a (* b b)))))) b) (* a 2.0))
   (if (<= b 5.8e-90)
     (* (/ 0.5 a) (- (sqrt (* c (* a -4.0))) b))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-33) {
		tmp = ((b * (-1.0 - (c * (-2.0 * (a / (b * b)))))) - b) / (a * 2.0);
	} else if (b <= 5.8e-90) {
		tmp = (0.5 / a) * (sqrt((c * (a * -4.0))) - b);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.8d-33)) then
        tmp = ((b * ((-1.0d0) - (c * ((-2.0d0) * (a / (b * b)))))) - b) / (a * 2.0d0)
    else if (b <= 5.8d-90) then
        tmp = (0.5d0 / a) * (sqrt((c * (a * (-4.0d0)))) - b)
    else
        tmp = 0.0d0 - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-33) {
		tmp = ((b * (-1.0 - (c * (-2.0 * (a / (b * b)))))) - b) / (a * 2.0);
	} else if (b <= 5.8e-90) {
		tmp = (0.5 / a) * (Math.sqrt((c * (a * -4.0))) - b);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.8e-33:
		tmp = ((b * (-1.0 - (c * (-2.0 * (a / (b * b)))))) - b) / (a * 2.0)
	elif b <= 5.8e-90:
		tmp = (0.5 / a) * (math.sqrt((c * (a * -4.0))) - b)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e-33)
		tmp = Float64(Float64(Float64(b * Float64(-1.0 - Float64(c * Float64(-2.0 * Float64(a / Float64(b * b)))))) - b) / Float64(a * 2.0));
	elseif (b <= 5.8e-90)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(c * Float64(a * -4.0))) - b));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.8e-33)
		tmp = ((b * (-1.0 - (c * (-2.0 * (a / (b * b)))))) - b) / (a * 2.0);
	elseif (b <= 5.8e-90)
		tmp = (0.5 / a) * (sqrt((c * (a * -4.0))) - b);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e-33], N[(N[(N[(b * N[(-1.0 - N[(c * N[(-2.0 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e-90], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.80000000000000017e-33

   1. Initial program 67.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 67.5%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(b \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}, b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(-1 \cdot b\right) \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(b \cdot -1\right) \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \left(-1 \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{2}}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{-2 \cdot \left(a \cdot c\right)}{b \cdot b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\frac{-2 \cdot \left(a \cdot c\right)}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\frac{\left(-2 \cdot a\right) \cdot c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\left(-2 \cdot a\right) \cdot \frac{c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\left(a \cdot -2\right) \cdot \frac{c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{a \cdot \left(-2 \cdot \frac{c}{b}\right)}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot \left(-2 \cdot \frac{c}{b}\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(-2 \cdot \frac{c}{b}\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{c}{b} \cdot -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{c}{b}\right), -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      18. /-lowering-/.f64 94.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(c, b\right), -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   7. Simplified 94.9%

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

   8. Taylor expanded in c around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \color{blue}{\left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}} + \frac{1}{c}\right)\right)\right)}\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right) + c \cdot \frac{1}{c}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      2. rgt-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right) + 1\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right) + -1 \cdot 1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right) + -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\left(-1 \cdot \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \left(c \cdot \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \left(-2 \cdot \frac{a}{{b}^{2}}\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-2, \left(\frac{a}{{b}^{2}}\right)\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(a, \left({b}^{2}\right)\right)\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(a, \left(b \cdot b\right)\right)\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      11. *-lowering-*.f64 94.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), -1\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   10. Simplified 94.9%

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

   if -1.80000000000000017e-33 < b < 5.79999999999999967e-90

   1. Initial program 77.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 77.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} - b\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}} - b\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} - b\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right) \]

      13. *-lowering-*.f64 77.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right) \]

   6. Applied egg-rr 77.3%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right), b\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right), b\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(c \cdot a\right) \cdot -4\right)\right), b\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(c \cdot \left(a \cdot -4\right)\right)\right), b\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(c \cdot \left(-4 \cdot a\right)\right)\right), b\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-4 \cdot a\right)\right)\right), b\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right), b\right)\right) \]

      7. *-lowering-*.f64 70.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right), b\right)\right) \]

   9. Simplified 70.9%

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

   if 5.79999999999999967e-90 < b

   1. Initial program 12.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 12.6%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 86.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 86.6%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{b \cdot \left(-1 - c \cdot \left(-2 \cdot \frac{a}{b \cdot b}\right)\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-90}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (- (/ c b) (/ b a)) (- 0.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = 0.0d0 - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 71.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 71.8%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(b \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}, b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(-1 \cdot b\right) \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(b \cdot -1\right) \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \left(-1 \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{2}}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{-2 \cdot \left(a \cdot c\right)}{b \cdot b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\frac{-2 \cdot \left(a \cdot c\right)}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\frac{\left(-2 \cdot a\right) \cdot c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\left(-2 \cdot a\right) \cdot \frac{c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\left(a \cdot -2\right) \cdot \frac{c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{a \cdot \left(-2 \cdot \frac{c}{b}\right)}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot \left(-2 \cdot \frac{c}{b}\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(-2 \cdot \frac{c}{b}\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{c}{b} \cdot -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{c}{b}\right), -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      18. /-lowering-/.f64 71.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(c, b\right), -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   7. Simplified 71.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot 2}\right), \color{blue}{\left(b \cdot \left(-1 \cdot \left(1 + \frac{a \cdot \left(\frac{c}{b} \cdot -2\right)}{b}\right)\right) - b\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2 \cdot a}\right), \left(b \cdot \color{blue}{\left(-1 \cdot \left(1 + \frac{a \cdot \left(\frac{c}{b} \cdot -2\right)}{b}\right)\right)} - b\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{b \cdot \left(-1 \cdot \left(1 + \frac{a \cdot \left(\frac{c}{b} \cdot -2\right)}{b}\right)\right)} - b\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \left(\color{blue}{b} \cdot \left(-1 \cdot \left(1 + \frac{a \cdot \left(\frac{c}{b} \cdot -2\right)}{b}\right)\right) - b\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{b \cdot \left(-1 \cdot \left(1 + \frac{a \cdot \left(\frac{c}{b} \cdot -2\right)}{b}\right)\right)} - b\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\left(b \cdot \left(-1 \cdot \left(1 + \frac{a \cdot \left(\frac{c}{b} \cdot -2\right)}{b}\right)\right)\right), \color{blue}{b}\right)\right) \]

   9. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(b \cdot \left(-1 + \frac{\frac{a}{\frac{b}{c}} \cdot 2}{b}\right) - b\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a}\right), \color{blue}{\left(b \cdot \left(-1 + \frac{\frac{a}{\frac{b}{c}} \cdot 2}{b}\right) - b\right)}\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \left(\color{blue}{b \cdot \left(-1 + \frac{\frac{a}{\frac{b}{c}} \cdot 2}{b}\right)} - b\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{\_.f64}\left(\left(b \cdot \left(-1 + \frac{\frac{a}{\frac{b}{c}} \cdot 2}{b}\right)\right), \color{blue}{b}\right)\right) \]

   11. Applied egg-rr 71.3%

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

   12. Taylor expanded in a around inf

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

          \[\leadsto \frac{c}{b} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

       3. unsub-neg N/A

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

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \color{blue}{\left(\frac{b}{a}\right)}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{\color{blue}{b}}{a}\right)\right) \]

       6. /-lowering-/.f64 71.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   14. Simplified 71.5%

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

   if -9.999999999999969e-311 < b

   1. Initial program 25.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 25.7%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 71.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 71.4%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 67.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-232}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.6e-232) (/ b (- 0.0 a)) (- 0.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6e-232) {
		tmp = b / (0.0 - a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.6d-232) then
        tmp = b / (0.0d0 - a)
    else
        tmp = 0.0d0 - (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-232) {
		tmp = b / (0.0 - a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.6e-232:
		tmp = b / (0.0 - a)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.6e-232)
		tmp = Float64(b / Float64(0.0 - a));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.6e-232)
		tmp = b / (0.0 - a);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.6e-232], N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.59999999999999996e-232

   1. Initial program 73.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 73.3%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(-1 \cdot a\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

      6. neg-lowering-neg.f64 67.1%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{neg.f64}\left(a\right)\right) \]

   7. Simplified 67.1%

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

   if 2.59999999999999996e-232 < b

   1. Initial program 20.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 20.6%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 76.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 76.1%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-232}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{-40}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.15e-40) (/ b (- 0.0 a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.15e-40) {
		tmp = b / (0.0 - 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 <= 2.15d-40) then
        tmp = b / (0.0d0 - 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 <= 2.15e-40) {
		tmp = b / (0.0 - a);
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.15e-40:
		tmp = b / (0.0 - a)
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.15e-40)
		tmp = Float64(b / Float64(0.0 - 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-40)
		tmp = b / (0.0 - a);
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.15e-40], N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1500000000000001e-40

   1. Initial program 69.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 69.0%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(-1 \cdot a\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

      6. neg-lowering-neg.f64 54.4%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{neg.f64}\left(a\right)\right) \]

   7. Simplified 54.4%

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

   if 2.1500000000000001e-40 < b

   1. Initial program 7.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   3. Simplified 7.1%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(b \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}, b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(-1 \cdot b\right) \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(b \cdot -1\right) \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \left(-1 \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{2}}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{-2 \cdot \left(a \cdot c\right)}{b \cdot b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\frac{-2 \cdot \left(a \cdot c\right)}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\frac{\left(-2 \cdot a\right) \cdot c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\left(-2 \cdot a\right) \cdot \frac{c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\left(a \cdot -2\right) \cdot \frac{c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{a \cdot \left(-2 \cdot \frac{c}{b}\right)}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot \left(-2 \cdot \frac{c}{b}\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(-2 \cdot \frac{c}{b}\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{c}{b} \cdot -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{c}{b}\right), -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      18. /-lowering-/.f64 2.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(c, b\right), -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   7. Simplified 2.1%

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

   8. Taylor expanded in b around 0

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

      1. /-lowering-/.f64 33.4%

         \[\leadsto \mathsf{/.f64}\left(c, \color{blue}{b}\right) \]

   10. Simplified 33.4%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{-40}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 10.9% 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 49.6%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

3. Simplified 49.6%

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

5. Taylor expanded in b around -inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(b \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}, b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(-1 \cdot b\right) \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(b \cdot -1\right) \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \left(-1 \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(-1 \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{2}}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{-2 \cdot \left(a \cdot c\right)}{b \cdot b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\frac{-2 \cdot \left(a \cdot c\right)}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\frac{\left(-2 \cdot a\right) \cdot c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   11. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\left(-2 \cdot a\right) \cdot \frac{c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{\left(a \cdot -2\right) \cdot \frac{c}{b}}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   13. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{a \cdot \left(-2 \cdot \frac{c}{b}\right)}{b}\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot \left(-2 \cdot \frac{c}{b}\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(-2 \cdot \frac{c}{b}\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{c}{b} \cdot -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{c}{b}\right), -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   18. /-lowering-/.f64 38.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(c, b\right), -2\right)\right), b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

7. Simplified 38.1%

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

8. Taylor expanded in b around 0

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

   1. /-lowering-/.f64 12.7%

      \[\leadsto \mathsf{/.f64}\left(c, \color{blue}{b}\right) \]

10. Simplified 12.7%

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

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024162 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

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