Quadratic roots, full range

Percentage Accurate: 51.4% → 85.3%

Time: 12.9s

Alternatives: 9

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+118}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e+118)
   (- (/ c b) (/ b a))
   (if (<= b 2.6e-73)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e+118) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.6e-73) {
		tmp = (sqrt(((b * b) - (c * (a * 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 <= (-6d+118)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.6d-73) then
        tmp = (sqrt(((b * b) - (c * (a * 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 <= -6e+118) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.6e-73) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e+118:
		tmp = (c / b) - (b / a)
	elif b <= 2.6e-73:
		tmp = (math.sqrt(((b * b) - (c * (a * 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 <= -6e+118)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.6e-73)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 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 <= -6e+118)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.6e-73)
		tmp = (sqrt(((b * b) - (c * (a * 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, -6e+118], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.6e-73], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 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 < -6e118

   1. Initial program 29.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\left(\frac{a \cdot c}{{b}^{2}}\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(b \cdot b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot {c}^{2}\right), \left({b}^{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   5. Simplified 63.6%

      \[\leadsto \frac{\color{blue}{0 - b \cdot \left(2 + -2 \cdot \left(\frac{a \cdot c}{b \cdot b} + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right)}}{2 \cdot a} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   7. 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 95.9%

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

   8. Simplified 95.9%

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

   if -6e118 < b < 2.6000000000000001e-73

   1. Initial program 91.6%

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

   if 2.6000000000000001e-73 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 90.4%

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

   5. Simplified 90.4%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

      3. /-lowering-/.f64 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+118}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+99}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.6e+99)
   (- (/ c b) (/ b a))
   (if (<= b 8.6e-76)
     (* (/ -0.5 a) (- b (sqrt (+ (* b b) (* c (* a -4.0))))))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e+99) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8.6e-76) {
		tmp = (-0.5 / a) * (b - sqrt(((b * b) + (c * (a * -4.0)))));
	} 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 <= (-2.6d+99)) then
        tmp = (c / b) - (b / a)
    else if (b <= 8.6d-76) then
        tmp = ((-0.5d0) / a) * (b - sqrt(((b * b) + (c * (a * (-4.0d0))))))
    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 <= -2.6e+99) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8.6e-76) {
		tmp = (-0.5 / a) * (b - Math.sqrt(((b * b) + (c * (a * -4.0)))));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.6e+99:
		tmp = (c / b) - (b / a)
	elif b <= 8.6e-76:
		tmp = (-0.5 / a) * (b - math.sqrt(((b * b) + (c * (a * -4.0)))))
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.6e+99)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 8.6e-76)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))));
	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 <= -2.6e+99)
		tmp = (c / b) - (b / a);
	elseif (b <= 8.6e-76)
		tmp = (-0.5 / a) * (b - sqrt(((b * b) + (c * (a * -4.0)))));
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.6e+99], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-76], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.6e99

   1. Initial program 38.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\left(\frac{a \cdot c}{{b}^{2}}\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(b \cdot b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot {c}^{2}\right), \left({b}^{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   5. Simplified 62.7%

      \[\leadsto \frac{\color{blue}{0 - b \cdot \left(2 + -2 \cdot \left(\frac{a \cdot c}{b \cdot b} + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right)}}{2 \cdot a} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   7. 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 96.4%

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

   8. Simplified 96.4%

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

   if -2.6e99 < b < 8.5999999999999998e-76

   1. Initial program 91.0%

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

   4. Applied egg-rr 90.9%

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

   if 8.5999999999999998e-76 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 90.4%

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

   5. Simplified 90.4%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

      3. /-lowering-/.f64 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+99}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-43}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-43)
   (- (/ c b) (/ b a))
   (if (<= b 2.75e-83)
     (/ (- (sqrt (* 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 <= -2e-43) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.75e-83) {
		tmp = (sqrt((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 <= (-2d-43)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.75d-83) then
        tmp = (sqrt((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 <= -2e-43) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.75e-83) {
		tmp = (Math.sqrt((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 <= -2e-43:
		tmp = (c / b) - (b / a)
	elif b <= 2.75e-83:
		tmp = (math.sqrt((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 <= -2e-43)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.75e-83)
		tmp = Float64(Float64(sqrt(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 <= -2e-43)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.75e-83)
		tmp = (sqrt((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, -2e-43], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.75e-83], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $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 < -2.00000000000000015e-43

   1. Initial program 60.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\left(\frac{a \cdot c}{{b}^{2}}\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(b \cdot b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot {c}^{2}\right), \left({b}^{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   5. Simplified 67.2%

      \[\leadsto \frac{\color{blue}{0 - b \cdot \left(2 + -2 \cdot \left(\frac{a \cdot c}{b \cdot b} + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right)}}{2 \cdot a} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   7. 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 90.7%

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

   8. Simplified 90.7%

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

   if -2.00000000000000015e-43 < b < 2.74999999999999982e-83

   1. Initial program 87.8%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 84.6%

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

   5. Simplified 84.6%

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

   if 2.74999999999999982e-83 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 90.4%

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

   5. Simplified 90.4%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

      3. /-lowering-/.f64 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-43}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.4e-44)
   (- (/ c b) (/ b a))
   (if (<= b 2.75e-78)
     (/ (/ (- (sqrt (* -4.0 (* c a))) b) a) 2.0)
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.4e-44) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.75e-78) {
		tmp = ((sqrt((-4.0 * (c * a))) - 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 <= (-5.4d-44)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.75d-78) then
        tmp = ((sqrt(((-4.0d0) * (c * a))) - 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 <= -5.4e-44) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.75e-78) {
		tmp = ((Math.sqrt((-4.0 * (c * a))) - b) / a) / 2.0;
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.4e-44:
		tmp = (c / b) - (b / a)
	elif b <= 2.75e-78:
		tmp = ((math.sqrt((-4.0 * (c * a))) - b) / a) / 2.0
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.4e-44)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.75e-78)
		tmp = Float64(Float64(Float64(sqrt(Float64(-4.0 * Float64(c * a))) - b) / 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 <= -5.4e-44)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.75e-78)
		tmp = ((sqrt((-4.0 * (c * a))) - b) / a) / 2.0;
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.4e-44], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.75e-78], N[(N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.3999999999999998e-44

   1. Initial program 60.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\left(\frac{a \cdot c}{{b}^{2}}\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(b \cdot b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot {c}^{2}\right), \left({b}^{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   5. Simplified 67.2%

      \[\leadsto \frac{\color{blue}{0 - b \cdot \left(2 + -2 \cdot \left(\frac{a \cdot c}{b \cdot b} + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right)}}{2 \cdot a} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   7. 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 90.7%

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

   8. Simplified 90.7%

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

   if -5.3999999999999998e-44 < b < 2.75000000000000009e-78

   1. Initial program 87.8%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 84.6%

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

   5. Simplified 84.6%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

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

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

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

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

      9. associate-*r* N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 84.5%

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

   7. Applied egg-rr 84.5%

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

   if 2.75000000000000009e-78 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 90.4%

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

   5. Simplified 90.4%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

      3. /-lowering-/.f64 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-44}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{-79}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-44)
   (- (/ c b) (/ b a))
   (if (<= b 1.32e-79)
     (* (/ -0.5 a) (- b (sqrt (* c (* a -4.0)))))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-44) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.32e-79) {
		tmp = (-0.5 / a) * (b - sqrt((c * (a * -4.0))));
	} 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 <= (-9d-44)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.32d-79) then
        tmp = ((-0.5d0) / a) * (b - sqrt((c * (a * (-4.0d0)))))
    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 <= -9e-44) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.32e-79) {
		tmp = (-0.5 / a) * (b - Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9e-44:
		tmp = (c / b) - (b / a)
	elif b <= 1.32e-79:
		tmp = (-0.5 / a) * (b - math.sqrt((c * (a * -4.0))))
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9e-44)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.32e-79)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(c * Float64(a * -4.0)))));
	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 <= -9e-44)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.32e-79)
		tmp = (-0.5 / a) * (b - sqrt((c * (a * -4.0))));
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9e-44], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.32e-79], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.9999999999999997e-44

   1. Initial program 60.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\left(\frac{a \cdot c}{{b}^{2}}\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(b \cdot b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot {c}^{2}\right), \left({b}^{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   5. Simplified 67.2%

      \[\leadsto \frac{\color{blue}{0 - b \cdot \left(2 + -2 \cdot \left(\frac{a \cdot c}{b \cdot b} + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right)}}{2 \cdot a} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   7. 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 90.7%

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

   8. Simplified 90.7%

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

   if -8.9999999999999997e-44 < b < 1.32e-79

   1. Initial program 87.8%

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

   4. Applied egg-rr 87.7%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 84.5%

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

   7. Simplified 84.5%

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

   if 1.32e-79 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 90.4%

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

   5. Simplified 90.4%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

      3. /-lowering-/.f64 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-44}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{-79}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-299}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.4e-299) (- 0.0 (/ b a)) (- 0.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.4e-299) {
		tmp = 0.0 - (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 <= 1.4d-299) then
        tmp = 0.0d0 - (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 <= 1.4e-299) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.4e-299:
		tmp = 0.0 - (b / a)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.4e-299)
		tmp = Float64(0.0 - 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 <= 1.4e-299)
		tmp = 0.0 - (b / a);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.4e-299], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4000000000000001e-299

   1. Initial program 71.7%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   4. 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-sub0 N/A

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

      7. --lowering--.f64 63.3%

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

   5. Simplified 63.3%

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

   if 1.4000000000000001e-299 < b

   1. Initial program 33.1%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 70.5%

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

   5. Simplified 70.5%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

      3. /-lowering-/.f64 70.5%

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

   7. Applied egg-rr 70.5%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-299}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.3% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (- 0.0 (/ c b)))

C

double code(double a, double b, double c) {
	return 0.0 - (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 = 0.0d0 - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 - (c / b);
}

Python

def code(a, b, c):
	return 0.0 - (c / b)

Julia

function code(a, b, c)
	return Float64(0.0 - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
4. 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 38.3%

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

5. Simplified 38.3%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

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

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

   3. /-lowering-/.f64 38.3%

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

7. Applied egg-rr 38.3%

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

8. Final simplification 38.3%

   \[\leadsto 0 - \frac{c}{b} \]
9. Add Preprocessing

Alternative 8: 10.7% 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 51.4%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(0 - b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(b \cdot \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \left(2 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   6. distribute-lft-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \left(-2 \cdot \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\left(\frac{a \cdot c}{{b}^{2}}\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left({b}^{2}\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(b \cdot b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\left({a}^{2} \cdot {c}^{2}\right), \left({b}^{4}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

5. Simplified 21.7%

   \[\leadsto \frac{\color{blue}{0 - b \cdot \left(2 + -2 \cdot \left(\frac{a \cdot c}{b \cdot b} + \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right)}}{2 \cdot a} \]

6. Taylor expanded in b around inf

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

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

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

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

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 29.4%

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

8. Simplified 29.4%

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

9. Taylor expanded in b around 0

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

    1. /-lowering-/.f64 13.4%

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

11. Simplified 13.4%

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

Alternative 9: 2.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return b / 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 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

4. Applied egg-rr 35.0%

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

5. Taylor expanded in a around 0

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

   1. /-lowering-/.f64 2.7%

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

7. Simplified 2.7%

   \[\leadsto \color{blue}{\frac{b}{a}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))