quadp (p42, positive)

Percentage Accurate: 52.1% → 84.9%

Time: 15.0s

Alternatives: 9

Speedup: 12.9×

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: 52.1% 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: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+147}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+147)
   (/ b (- a))
   (if (<= b 5.6e-34)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ -1.0 (- (/ b c) (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+147) {
		tmp = b / -a;
	} else if (b <= 5.6e-34) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) - (a / 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 <= (-5d+147)) then
        tmp = b / -a
    else if (b <= 5.6d-34) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = (-1.0d0) / ((b / c) - (a / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+147) {
		tmp = b / -a;
	} else if (b <= 5.6e-34) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+147:
		tmp = b / -a
	elif b <= 5.6e-34:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = -1.0 / ((b / c) - (a / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+147)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 5.6e-34)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) - Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+147)
		tmp = b / -a;
	elseif (b <= 5.6e-34)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = -1.0 / ((b / c) - (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+147], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 5.6e-34], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] - N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.0000000000000002e147

   1. Initial program 42.2%

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

      1. *-commutative 42.2%

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

   3. Simplified 42.2%

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

   5. Taylor expanded in b around -inf 94.0%

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

      1. associate-*r/ 94.0%

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

      2. mul-1-neg 94.0%

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

   7. Simplified 94.0%

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

   if -5.0000000000000002e147 < b < 5.59999999999999994e-34

   1. Initial program 81.9%

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

   if 5.59999999999999994e-34 < b

   1. Initial program 7.4%

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

      1. *-commutative 7.4%

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

   3. Simplified 7.4%

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

      1. cancel-sign-sub-inv 7.4%

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

      2. metadata-eval 7.4%

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

      3. associate-*r* 7.4%

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

      4. *-commutative 7.4%

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

      5. fma-undefine 7.4%

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

      6. add-cube-cbrt 5.7%

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

      7. pow3 5.6%

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

      8. associate-*r* 5.6%

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

      9. *-commutative 5.6%

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

   6. Applied egg-rr 5.6%

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

   8. Applied egg-rr 17.1%

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

      1. associate-*r/ 17.1%

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

      2. metadata-eval 17.1%

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

   10. Simplified 17.1%

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

   11. Taylor expanded in a around 0 0.0%

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

       1. *-lft-identity 0.0%

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

       2. mul-1-neg 0.0%

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

       3. unsub-neg 0.0%

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

       4. associate-*r/ 0.0%

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

       5. rem-square-sqrt 0.0%

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

       6. unpow2 0.0%

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

       7. *-commutative 0.0%

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

       8. times-frac 0.0%

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

       9. unpow2 0.0%

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

       10. rem-square-sqrt 0.0%

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

       11. unpow2 0.0%

           \[\leadsto \frac{-1}{\frac{-4}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c} - 1 \cdot \frac{a}{b}} \]

       12. rem-square-sqrt 94.2%

           \[\leadsto \frac{-1}{\frac{-4}{\color{blue}{-4}} \cdot \frac{b}{c} - 1 \cdot \frac{a}{b}} \]

       13. metadata-eval 94.2%

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

       14. *-lft-identity 94.2%

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

   13. Simplified 94.2%

       \[\leadsto \frac{-1}{\color{blue}{1 \cdot \frac{b}{c} - \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+147}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-82}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.46 \cdot 10^{-31}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\left(a \cdot c\right) \cdot -4} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e-82)
   (- (/ c b) (/ b a))
   (if (<= b 1.46e-31)
     (* (/ 0.5 a) (- (sqrt (* (* a c) -4.0)) b))
     (/ -1.0 (- (/ b c) (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e-82) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.46e-31) {
		tmp = (0.5 / a) * (sqrt(((a * c) * -4.0)) - b);
	} else {
		tmp = -1.0 / ((b / c) - (a / 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 <= (-1.1d-82)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.46d-31) then
        tmp = (0.5d0 / a) * (sqrt(((a * c) * (-4.0d0))) - b)
    else
        tmp = (-1.0d0) / ((b / c) - (a / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e-82) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.46e-31) {
		tmp = (0.5 / a) * (Math.sqrt(((a * c) * -4.0)) - b);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.1e-82:
		tmp = (c / b) - (b / a)
	elif b <= 1.46e-31:
		tmp = (0.5 / a) * (math.sqrt(((a * c) * -4.0)) - b)
	else:
		tmp = -1.0 / ((b / c) - (a / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e-82)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.46e-31)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) - Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.1e-82)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.46e-31)
		tmp = (0.5 / a) * (sqrt(((a * c) * -4.0)) - b);
	else
		tmp = -1.0 / ((b / c) - (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e-82], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.46e-31], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] - N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.09999999999999993e-82

   1. Initial program 69.3%

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

      1. *-commutative 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. *-commutative 89.3%

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

      3. distribute-rgt-neg-in 89.3%

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

      4. +-commutative 89.3%

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

      5. mul-1-neg 89.3%

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

      6. unsub-neg 89.3%

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

   7. Simplified 89.3%

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

   8. Taylor expanded in a around inf 89.6%

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

   if -1.09999999999999993e-82 < b < 1.4600000000000001e-31

   1. Initial program 74.0%

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

      1. *-commutative 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in b around 0 66.6%

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

      1. *-commutative 66.6%

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

      2. associate-*r* 66.6%

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

   7. Simplified 66.6%

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

      1. +-commutative 66.6%

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

      2. unsub-neg 66.6%

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

   9. Applied egg-rr 66.6%

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

       1. div-sub 66.6%

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

       2. sub-neg 66.6%

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

       3. div-inv 66.5%

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

       4. *-commutative 66.5%

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

       5. associate-/r* 66.5%

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

       6. metadata-eval 66.5%

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

       7. div-inv 66.5%

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

       8. *-commutative 66.5%

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

       9. associate-/r* 66.5%

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

       10. metadata-eval 66.5%

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

   11. Applied egg-rr 66.5%

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

       1. sub-neg 66.5%

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

       2. distribute-rgt-out-- 66.5%

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

       3. associate-*r* 66.5%

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

       4. *-commutative 66.5%

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

   13. Simplified 66.5%

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

   if 1.4600000000000001e-31 < b

   1. Initial program 7.4%

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

      1. *-commutative 7.4%

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

   3. Simplified 7.4%

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

      1. cancel-sign-sub-inv 7.4%

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

      2. metadata-eval 7.4%

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

      3. associate-*r* 7.4%

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

      4. *-commutative 7.4%

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

      5. fma-undefine 7.4%

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

      6. add-cube-cbrt 5.7%

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

      7. pow3 5.6%

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

      8. associate-*r* 5.6%

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

      9. *-commutative 5.6%

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

   6. Applied egg-rr 5.6%

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

   8. Applied egg-rr 17.1%

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

      1. associate-*r/ 17.1%

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

      2. metadata-eval 17.1%

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

   10. Simplified 17.1%

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

   11. Taylor expanded in a around 0 0.0%

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

       1. *-lft-identity 0.0%

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

       2. mul-1-neg 0.0%

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

       3. unsub-neg 0.0%

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

       4. associate-*r/ 0.0%

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

       5. rem-square-sqrt 0.0%

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

       6. unpow2 0.0%

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

       7. *-commutative 0.0%

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

       8. times-frac 0.0%

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

       9. unpow2 0.0%

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

       10. rem-square-sqrt 0.0%

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

       11. unpow2 0.0%

           \[\leadsto \frac{-1}{\frac{-4}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c} - 1 \cdot \frac{a}{b}} \]

       12. rem-square-sqrt 94.2%

           \[\leadsto \frac{-1}{\frac{-4}{\color{blue}{-4}} \cdot \frac{b}{c} - 1 \cdot \frac{a}{b}} \]

       13. metadata-eval 94.2%

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

       14. *-lft-identity 94.2%

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

   13. Simplified 94.2%

       \[\leadsto \frac{-1}{\color{blue}{1 \cdot \frac{b}{c} - \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-82}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.46 \cdot 10^{-31}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\left(a \cdot c\right) \cdot -4} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e-82)
   (- (/ c b) (/ b a))
   (if (<= b 2.7e-30)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ -1.0 (- (/ b c) (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e-82) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.7e-30) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) - (a / 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 <= (-2.5d-82)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.7d-30) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (-1.0d0) / ((b / c) - (a / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e-82) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.7e-30) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.5e-82:
		tmp = (c / b) - (b / a)
	elif b <= 2.7e-30:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -1.0 / ((b / c) - (a / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e-82)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.7e-30)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) - Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.5e-82)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.7e-30)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -1.0 / ((b / c) - (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e-82], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.7e-30], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] - N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4999999999999999e-82

   1. Initial program 69.3%

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

      1. *-commutative 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. *-commutative 89.3%

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

      3. distribute-rgt-neg-in 89.3%

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

      4. +-commutative 89.3%

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

      5. mul-1-neg 89.3%

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

      6. unsub-neg 89.3%

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

   7. Simplified 89.3%

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

   8. Taylor expanded in a around inf 89.6%

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

   if -2.4999999999999999e-82 < b < 2.69999999999999987e-30

   1. Initial program 74.0%

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

      1. *-commutative 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in b around 0 66.6%

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

      1. *-commutative 66.6%

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

      2. associate-*r* 66.6%

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

   7. Simplified 66.6%

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

      1. +-commutative 66.6%

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

      2. unsub-neg 66.6%

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

   9. Applied egg-rr 66.6%

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

   if 2.69999999999999987e-30 < b

   1. Initial program 7.4%

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

      1. *-commutative 7.4%

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

   3. Simplified 7.4%

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

      1. cancel-sign-sub-inv 7.4%

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

      2. metadata-eval 7.4%

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

      3. associate-*r* 7.4%

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

      4. *-commutative 7.4%

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

      5. fma-undefine 7.4%

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

      6. add-cube-cbrt 5.7%

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

      7. pow3 5.6%

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

      8. associate-*r* 5.6%

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

      9. *-commutative 5.6%

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

   6. Applied egg-rr 5.6%

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

   8. Applied egg-rr 17.1%

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

      1. associate-*r/ 17.1%

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

      2. metadata-eval 17.1%

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

   10. Simplified 17.1%

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

   11. Taylor expanded in a around 0 0.0%

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

       1. *-lft-identity 0.0%

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

       2. mul-1-neg 0.0%

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

       3. unsub-neg 0.0%

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

       4. associate-*r/ 0.0%

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

       5. rem-square-sqrt 0.0%

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

       6. unpow2 0.0%

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

       7. *-commutative 0.0%

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

       8. times-frac 0.0%

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

       9. unpow2 0.0%

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

       10. rem-square-sqrt 0.0%

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

       11. unpow2 0.0%

           \[\leadsto \frac{-1}{\frac{-4}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c} - 1 \cdot \frac{a}{b}} \]

       12. rem-square-sqrt 94.2%

           \[\leadsto \frac{-1}{\frac{-4}{\color{blue}{-4}} \cdot \frac{b}{c} - 1 \cdot \frac{a}{b}} \]

       13. metadata-eval 94.2%

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

       14. *-lft-identity 94.2%

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

   13. Simplified 94.2%

       \[\leadsto \frac{-1}{\color{blue}{1 \cdot \frac{b}{c} - \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-302}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e-302) (- (/ c b) (/ b a)) (/ -1.0 (- (/ b c) (/ a b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-302) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -1.0 / ((b / c) - (a / 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 <= (-3.6d-302)) then
        tmp = (c / b) - (b / a)
    else
        tmp = (-1.0d0) / ((b / c) - (a / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-302) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.6e-302:
		tmp = (c / b) - (b / a)
	else:
		tmp = -1.0 / ((b / c) - (a / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e-302)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) - Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.6e-302)
		tmp = (c / b) - (b / a);
	else
		tmp = -1.0 / ((b / c) - (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e-302], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] - N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.6000000000000001e-302

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in b around -inf 69.1%

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

      1. mul-1-neg 69.1%

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

      2. *-commutative 69.1%

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

      3. distribute-rgt-neg-in 69.1%

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

      4. +-commutative 69.1%

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

      5. mul-1-neg 69.1%

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

      6. unsub-neg 69.1%

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

   7. Simplified 69.1%

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

   8. Taylor expanded in a around inf 70.6%

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

   if -3.6000000000000001e-302 < b

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

   3. Simplified 26.9%

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

      1. cancel-sign-sub-inv 26.9%

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

      2. metadata-eval 26.9%

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

      3. associate-*r* 26.9%

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

      4. *-commutative 26.9%

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

      5. fma-undefine 26.9%

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

      6. add-cube-cbrt 25.6%

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

      7. pow3 25.5%

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

      8. associate-*r* 25.5%

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

      9. *-commutative 25.5%

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

   6. Applied egg-rr 25.5%

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

   8. Applied egg-rr 33.4%

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

      1. associate-*r/ 33.4%

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

      2. metadata-eval 33.4%

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

   10. Simplified 33.4%

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

   11. Taylor expanded in a around 0 0.0%

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

       1. *-lft-identity 0.0%

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

       2. mul-1-neg 0.0%

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

       3. unsub-neg 0.0%

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

       4. associate-*r/ 0.0%

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

       5. rem-square-sqrt 0.0%

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

       6. unpow2 0.0%

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

       7. *-commutative 0.0%

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

       8. times-frac 0.0%

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

       9. unpow2 0.0%

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

       10. rem-square-sqrt 0.0%

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

       11. unpow2 0.0%

           \[\leadsto \frac{-1}{\frac{-4}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c} - 1 \cdot \frac{a}{b}} \]

       12. rem-square-sqrt 73.0%

           \[\leadsto \frac{-1}{\frac{-4}{\color{blue}{-4}} \cdot \frac{b}{c} - 1 \cdot \frac{a}{b}} \]

       13. metadata-eval 73.0%

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

       14. *-lft-identity 73.0%

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

   13. Simplified 73.0%

       \[\leadsto \frac{-1}{\color{blue}{1 \cdot \frac{b}{c} - \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-302}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.5% accurate, 9.7× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (- (/ c b) (/ b a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 71.6%

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

      1. *-commutative 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in b around -inf 68.1%

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

      1. mul-1-neg 68.1%

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

      2. *-commutative 68.1%

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

      3. distribute-rgt-neg-in 68.1%

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

      4. +-commutative 68.1%

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

      5. mul-1-neg 68.1%

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

      6. unsub-neg 68.1%

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

   7. Simplified 68.1%

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

   8. Taylor expanded in a around inf 69.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 26.5%

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

      1. *-commutative 26.5%

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

   3. Simplified 26.5%

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

   5. Taylor expanded in b around inf 73.8%

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

      1. associate-*r/ 73.8%

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

      2. neg-mul-1 73.8%

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

   7. Simplified 73.8%

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

4. Final simplification 71.6%

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

Alternative 6: 42.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 5.8e-16) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.8e-16) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.8d-16) then
        tmp = b / -a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.8e-16) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.8e-16:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.8e-16)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.8e-16)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.8e-16], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.7999999999999996e-16

   1. Initial program 70.6%

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

      1. *-commutative 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in b around -inf 54.7%

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

      1. associate-*r/ 54.7%

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

      2. mul-1-neg 54.7%

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

   7. Simplified 54.7%

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

   if 5.7999999999999996e-16 < b

   1. Initial program 7.5%

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

      1. *-commutative 7.5%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in b around -inf 2.6%

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

      1. mul-1-neg 2.6%

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

      2. *-commutative 2.6%

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

      3. distribute-rgt-neg-in 2.6%

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

      4. +-commutative 2.6%

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

      5. mul-1-neg 2.6%

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

      6. unsub-neg 2.6%

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

   7. Simplified 2.6%

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

   8. Taylor expanded in a around inf 21.2%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 12.9× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b / -a
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 71.6%

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

      1. *-commutative 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in b around -inf 69.0%

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

      1. associate-*r/ 69.0%

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

      2. mul-1-neg 69.0%

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

   7. Simplified 69.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 26.5%

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

      1. *-commutative 26.5%

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

   3. Simplified 26.5%

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

   5. Taylor expanded in b around inf 73.8%

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

      1. associate-*r/ 73.8%

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

      2. neg-mul-1 73.8%

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

   7. Simplified 73.8%

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

4. Final simplification 71.2%

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

Alternative 8: 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 50.6%

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

   1. *-commutative 50.6%

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

3. Simplified 50.6%

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

6. Applied egg-rr 22.2%

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

   1. unpow-1 22.2%

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

8. Simplified 22.2%

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

9. Taylor expanded in a around 0 2.3%

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

10. Final simplification 2.3%

    \[\leadsto \frac{b}{a} \]
11. Add Preprocessing

Alternative 9: 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 50.6%

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

   1. *-commutative 50.6%

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

3. Simplified 50.6%

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

5. Taylor expanded in b around -inf 37.5%

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

   1. mul-1-neg 37.5%

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

   2. *-commutative 37.5%

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

   3. distribute-rgt-neg-in 37.5%

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

   4. +-commutative 37.5%

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

   5. mul-1-neg 37.5%

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

   6. unsub-neg 37.5%

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

7. Simplified 37.5%

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

8. Taylor expanded in a around inf 9.0%

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

9. Final simplification 9.0%

   \[\leadsto \frac{c}{b} \]
10. Add Preprocessing

Developer target: 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 2024080 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))))))

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