quadp (p42, positive)

Percentage Accurate: 51.7% → 85.6%

Time: 15.8s

Alternatives: 9

Speedup: 19.1×

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: 51.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+141}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+141)
   (/ (- b) a)
   (if (<= b 3e-79)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+141) {
		tmp = -b / a;
	} else if (b <= 3e-79) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d+141)) then
        tmp = -b / a
    else if (b <= 3d-79) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+141) {
		tmp = -b / a;
	} else if (b <= 3e-79) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+141:
		tmp = -b / a
	elif b <= 3e-79:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+141)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 3e-79)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e+141)
		tmp = -b / a;
	elseif (b <= 3e-79)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+141], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 3e-79], 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[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000002e141

   1. Initial program 41.4%

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

      1. *-commutative 41.4%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in b around -inf 91.1%

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

      1. associate-*r/ 91.1%

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

      2. mul-1-neg 91.1%

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

   7. Simplified 91.1%

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

   if -1.00000000000000002e141 < b < 3e-79

   1. Initial program 80.6%

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

   if 3e-79 < b

   1. Initial program 12.9%

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

      1. *-commutative 12.9%

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

   3. Simplified 12.9%

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

   5. Taylor expanded in b around inf 87.9%

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

      1. associate-*r/ 87.9%

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

      2. neg-mul-1 87.9%

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

   7. Simplified 87.9%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+141}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{if}\;b \leq -4 \cdot 10^{-59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -5.6 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ (+ b (sqrt (* c (* a -4.0)))) a))))
   (if (<= b -4e-59)
     (- (/ c b) (/ b a))
     (if (<= b -5.6e-76)
       t_0
       (if (<= b -6.5e-97)
         (/ (- b) a)
         (if (<= b 1.28e-76) t_0 (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = 0.5 * ((b + sqrt((c * (a * -4.0)))) / a);
	double tmp;
	if (b <= -4e-59) {
		tmp = (c / b) - (b / a);
	} else if (b <= -5.6e-76) {
		tmp = t_0;
	} else if (b <= -6.5e-97) {
		tmp = -b / a;
	} else if (b <= 1.28e-76) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((b + sqrt((c * (a * (-4.0d0))))) / a)
    if (b <= (-4d-59)) then
        tmp = (c / b) - (b / a)
    else if (b <= (-5.6d-76)) then
        tmp = t_0
    else if (b <= (-6.5d-97)) then
        tmp = -b / a
    else if (b <= 1.28d-76) then
        tmp = t_0
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = 0.5 * ((b + Math.sqrt((c * (a * -4.0)))) / a);
	double tmp;
	if (b <= -4e-59) {
		tmp = (c / b) - (b / a);
	} else if (b <= -5.6e-76) {
		tmp = t_0;
	} else if (b <= -6.5e-97) {
		tmp = -b / a;
	} else if (b <= 1.28e-76) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = 0.5 * ((b + math.sqrt((c * (a * -4.0)))) / a)
	tmp = 0
	if b <= -4e-59:
		tmp = (c / b) - (b / a)
	elif b <= -5.6e-76:
		tmp = t_0
	elif b <= -6.5e-97:
		tmp = -b / a
	elif b <= 1.28e-76:
		tmp = t_0
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(0.5 * Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / a))
	tmp = 0.0
	if (b <= -4e-59)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= -5.6e-76)
		tmp = t_0;
	elseif (b <= -6.5e-97)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.28e-76)
		tmp = t_0;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = 0.5 * ((b + sqrt((c * (a * -4.0)))) / a);
	tmp = 0.0;
	if (b <= -4e-59)
		tmp = (c / b) - (b / a);
	elseif (b <= -5.6e-76)
		tmp = t_0;
	elseif (b <= -6.5e-97)
		tmp = -b / a;
	elseif (b <= 1.28e-76)
		tmp = t_0;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(0.5 * N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4e-59], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -5.6e-76], t$95$0, If[LessEqual[b, -6.5e-97], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.28e-76], t$95$0, N[((-c) / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.0000000000000001e-59

   1. Initial program 71.1%

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

      1. *-commutative 71.1%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in b around -inf 86.0%

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

      1. +-commutative 86.0%

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

      2. mul-1-neg 86.0%

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

      3. unsub-neg 86.0%

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

   7. Simplified 86.0%

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

   if -4.0000000000000001e-59 < b < -5.6000000000000002e-76 or -6.5000000000000004e-97 < b < 1.28e-76

   1. Initial program 72.1%

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

      1. *-commutative 72.1%

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

   3. Simplified 72.1%

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

      1. clear-num 72.0%

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

      2. inv-pow 72.0%

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

      3. add-sqr-sqrt 37.3%

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

      4. sqrt-unprod 71.9%

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

      5. sqr-neg 71.9%

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

      6. sqrt-prod 34.9%

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

      7. add-sqr-sqrt 69.7%

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

      8. fma-neg 69.7%

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

      9. distribute-lft-neg-in 69.7%

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

      10. *-commutative 69.7%

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

      11. associate-*r* 69.7%

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

      12. metadata-eval 69.7%

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

   6. Applied egg-rr 69.7%

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

   7. Taylor expanded in b around 0 69.7%

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

      1. associate-*r* 69.7%

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

   9. Simplified 69.7%

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

       1. expm1-log1p-u 45.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{a \cdot 2}{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}\right)}^{-1}\right)\right)} \]

       2. expm1-udef 26.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{a \cdot 2}{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}\right)}^{-1}\right)} - 1} \]

       3. unpow-1 26.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{a \cdot 2}{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}}}\right)} - 1 \]

       4. clear-num 26.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}{a \cdot 2}}\right)} - 1 \]

       5. *-un-lft-identity 26.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot \left(b + \sqrt{\left(-4 \cdot a\right) \cdot c}\right)}}{a \cdot 2}\right)} - 1 \]

       6. *-commutative 26.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1 \cdot \left(b + \sqrt{\left(-4 \cdot a\right) \cdot c}\right)}{\color{blue}{2 \cdot a}}\right)} - 1 \]

       7. times-frac 26.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{2} \cdot \frac{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}{a}}\right)} - 1 \]

       8. metadata-eval 26.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.5} \cdot \frac{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}{a}\right)} - 1 \]

       9. associate-*l* 26.6%

          \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{b + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{a}\right)} - 1 \]

   11. Applied egg-rr 26.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 45.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\right)\right)} \]

       2. expm1-log1p 69.7%

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

       3. *-commutative 69.7%

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

       4. *-commutative 69.7%

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

       5. associate-*l* 69.7%

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

   13. Simplified 69.7%

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

   if -5.6000000000000002e-76 < b < -6.5000000000000004e-97

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if 1.28e-76 < b

   1. Initial program 12.9%

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

      1. *-commutative 12.9%

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

   3. Simplified 12.9%

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

   5. Taylor expanded in b around inf 87.9%

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

      1. associate-*r/ 87.9%

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

      2. neg-mul-1 87.9%

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

   7. Simplified 87.9%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -5.6 \cdot 10^{-76}:\\ \;\;\;\;0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-76}:\\ \;\;\;\;0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b + \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -2 \cdot 10^{-59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{0.5}{\frac{a}{t_0}}\\ \mathbf{elif}\;b \leq -4.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ b (sqrt (* c (* a -4.0))))))
   (if (<= b -2e-59)
     (- (/ c b) (/ b a))
     (if (<= b -9.5e-76)
       (/ 0.5 (/ a t_0))
       (if (<= b -4.6e-97)
         (/ (- b) a)
         (if (<= b 3.2e-80) (* 0.5 (/ t_0 a)) (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = b + sqrt((c * (a * -4.0)));
	double tmp;
	if (b <= -2e-59) {
		tmp = (c / b) - (b / a);
	} else if (b <= -9.5e-76) {
		tmp = 0.5 / (a / t_0);
	} else if (b <= -4.6e-97) {
		tmp = -b / a;
	} else if (b <= 3.2e-80) {
		tmp = 0.5 * (t_0 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b + sqrt((c * (a * (-4.0d0))))
    if (b <= (-2d-59)) then
        tmp = (c / b) - (b / a)
    else if (b <= (-9.5d-76)) then
        tmp = 0.5d0 / (a / t_0)
    else if (b <= (-4.6d-97)) then
        tmp = -b / a
    else if (b <= 3.2d-80) then
        tmp = 0.5d0 * (t_0 / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b + Math.sqrt((c * (a * -4.0)));
	double tmp;
	if (b <= -2e-59) {
		tmp = (c / b) - (b / a);
	} else if (b <= -9.5e-76) {
		tmp = 0.5 / (a / t_0);
	} else if (b <= -4.6e-97) {
		tmp = -b / a;
	} else if (b <= 3.2e-80) {
		tmp = 0.5 * (t_0 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b + math.sqrt((c * (a * -4.0)))
	tmp = 0
	if b <= -2e-59:
		tmp = (c / b) - (b / a)
	elif b <= -9.5e-76:
		tmp = 0.5 / (a / t_0)
	elif b <= -4.6e-97:
		tmp = -b / a
	elif b <= 3.2e-80:
		tmp = 0.5 * (t_0 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b + sqrt(Float64(c * Float64(a * -4.0))))
	tmp = 0.0
	if (b <= -2e-59)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= -9.5e-76)
		tmp = Float64(0.5 / Float64(a / t_0));
	elseif (b <= -4.6e-97)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 3.2e-80)
		tmp = Float64(0.5 * Float64(t_0 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b + sqrt((c * (a * -4.0)));
	tmp = 0.0;
	if (b <= -2e-59)
		tmp = (c / b) - (b / a);
	elseif (b <= -9.5e-76)
		tmp = 0.5 / (a / t_0);
	elseif (b <= -4.6e-97)
		tmp = -b / a;
	elseif (b <= 3.2e-80)
		tmp = 0.5 * (t_0 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e-59], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -9.5e-76], N[(0.5 / N[(a / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -4.6e-97], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 3.2e-80], N[(0.5 * N[(t$95$0 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -2.0000000000000001e-59

   1. Initial program 71.1%

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

      1. *-commutative 71.1%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in b around -inf 86.0%

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

      1. +-commutative 86.0%

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

      2. mul-1-neg 86.0%

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

      3. unsub-neg 86.0%

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

   7. Simplified 86.0%

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

   if -2.0000000000000001e-59 < b < -9.49999999999999984e-76

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

   3. Simplified 99.2%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

      3. add-sqr-sqrt 99.2%

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

      4. sqrt-unprod 99.2%

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

      5. sqr-neg 99.2%

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

      6. sqrt-prod 0.0%

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

      7. add-sqr-sqrt 88.3%

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

      8. fma-neg 88.3%

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

      9. distribute-lft-neg-in 88.3%

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

      10. *-commutative 88.3%

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

      11. associate-*r* 88.3%

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

      12. metadata-eval 88.3%

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

   6. Applied egg-rr 88.3%

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

   7. Taylor expanded in b around 0 88.3%

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

      1. associate-*r* 88.3%

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

   9. Simplified 88.3%

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

       1. unpow-1 88.3%

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

       2. *-commutative 88.3%

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

       3. *-un-lft-identity 88.3%

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

       4. times-frac 88.3%

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

       5. metadata-eval 88.3%

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

       6. associate-*l* 88.3%

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

   11. Applied egg-rr 88.3%

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

       1. associate-/r* 88.3%

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

       2. metadata-eval 88.3%

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

       3. *-commutative 88.3%

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

       4. *-commutative 88.3%

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

       5. associate-*l* 88.3%

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

   13. Simplified 88.3%

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

   if -9.49999999999999984e-76 < b < -4.59999999999999988e-97

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -4.59999999999999988e-97 < b < 3.1999999999999999e-80

   1. Initial program 69.7%

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

      1. *-commutative 69.7%

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

   3. Simplified 69.7%

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

      1. clear-num 69.6%

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

      2. inv-pow 69.6%

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

      3. add-sqr-sqrt 31.7%

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

      4. sqrt-unprod 69.4%

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

      5. sqr-neg 69.4%

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

      6. sqrt-prod 38.0%

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

      7. add-sqr-sqrt 68.0%

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

      8. fma-neg 68.0%

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

      9. distribute-lft-neg-in 68.0%

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

      10. *-commutative 68.0%

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

      11. associate-*r* 68.0%

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

      12. metadata-eval 68.0%

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

   6. Applied egg-rr 68.0%

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

   7. Taylor expanded in b around 0 68.0%

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

      1. associate-*r* 68.0%

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

   9. Simplified 68.0%

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

       1. expm1-log1p-u 44.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{a \cdot 2}{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}\right)}^{-1}\right)\right)} \]

       2. expm1-udef 27.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{a \cdot 2}{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}\right)}^{-1}\right)} - 1} \]

       3. unpow-1 27.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{a \cdot 2}{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}}}\right)} - 1 \]

       4. clear-num 27.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}{a \cdot 2}}\right)} - 1 \]

       5. *-un-lft-identity 27.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot \left(b + \sqrt{\left(-4 \cdot a\right) \cdot c}\right)}}{a \cdot 2}\right)} - 1 \]

       6. *-commutative 27.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1 \cdot \left(b + \sqrt{\left(-4 \cdot a\right) \cdot c}\right)}{\color{blue}{2 \cdot a}}\right)} - 1 \]

       7. times-frac 27.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{2} \cdot \frac{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}{a}}\right)} - 1 \]

       8. metadata-eval 27.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.5} \cdot \frac{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}{a}\right)} - 1 \]

       9. associate-*l* 27.5%

          \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{b + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{a}\right)} - 1 \]

   11. Applied egg-rr 27.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 44.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\right)\right)} \]

       2. expm1-log1p 68.1%

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

       3. *-commutative 68.1%

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

       4. *-commutative 68.1%

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

       5. associate-*l* 68.1%

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

   13. Simplified 68.1%

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

   if 3.1999999999999999e-80 < b

   1. Initial program 12.9%

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

      1. *-commutative 12.9%

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

   3. Simplified 12.9%

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

   5. Taylor expanded in b around inf 87.9%

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

      1. associate-*r/ 87.9%

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

      2. neg-mul-1 87.9%

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

   7. Simplified 87.9%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{0.5}{\frac{a}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}}\\ \mathbf{elif}\;b \leq -4.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.52 \cdot 10^{-59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))))
   (if (<= b -1.52e-59)
     (- (/ c b) (/ b a))
     (if (<= b -3.8e-76)
       t_0
       (if (<= b -6.5e-97) (/ (- b) a) (if (<= b 3.6e-75) t_0 (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -1.52e-59) {
		tmp = (c / b) - (b / a);
	} else if (b <= -3.8e-76) {
		tmp = t_0;
	} else if (b <= -6.5e-97) {
		tmp = -b / a;
	} else if (b <= 3.6e-75) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    if (b <= (-1.52d-59)) then
        tmp = (c / b) - (b / a)
    else if (b <= (-3.8d-76)) then
        tmp = t_0
    else if (b <= (-6.5d-97)) then
        tmp = -b / a
    else if (b <= 3.6d-75) then
        tmp = t_0
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -1.52e-59) {
		tmp = (c / b) - (b / a);
	} else if (b <= -3.8e-76) {
		tmp = t_0;
	} else if (b <= -6.5e-97) {
		tmp = -b / a;
	} else if (b <= 3.6e-75) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	tmp = 0
	if b <= -1.52e-59:
		tmp = (c / b) - (b / a)
	elif b <= -3.8e-76:
		tmp = t_0
	elif b <= -6.5e-97:
		tmp = -b / a
	elif b <= 3.6e-75:
		tmp = t_0
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (b <= -1.52e-59)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= -3.8e-76)
		tmp = t_0;
	elseif (b <= -6.5e-97)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 3.6e-75)
		tmp = t_0;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	tmp = 0.0;
	if (b <= -1.52e-59)
		tmp = (c / b) - (b / a);
	elseif (b <= -3.8e-76)
		tmp = t_0;
	elseif (b <= -6.5e-97)
		tmp = -b / a;
	elseif (b <= 3.6e-75)
		tmp = t_0;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.52e-59], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -3.8e-76], t$95$0, If[LessEqual[b, -6.5e-97], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 3.6e-75], t$95$0, N[((-c) / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.51999999999999998e-59

   1. Initial program 71.1%

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

      1. *-commutative 71.1%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in b around -inf 86.0%

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

      1. +-commutative 86.0%

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

      2. mul-1-neg 86.0%

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

      3. unsub-neg 86.0%

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

   7. Simplified 86.0%

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

   if -1.51999999999999998e-59 < b < -3.8000000000000002e-76 or -6.5000000000000004e-97 < b < 3.6e-75

   1. Initial program 72.1%

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

      1. *-commutative 72.1%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in b around 0 70.4%

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

      1. associate-*r* 70.4%

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

   7. Simplified 70.4%

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

   if -3.8000000000000002e-76 < b < -6.5000000000000004e-97

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if 3.6e-75 < b

   1. Initial program 12.9%

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

      1. *-commutative 12.9%

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

   3. Simplified 12.9%

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

   5. Taylor expanded in b around inf 87.9%

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

      1. associate-*r/ 87.9%

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

      2. neg-mul-1 87.9%

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

   7. Simplified 87.9%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.52 \cdot 10^{-59}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.9% accurate, 12.8× 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(Float64(-c) / 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 75.8%

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

      1. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in b around -inf 64.3%

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

      1. +-commutative 64.3%

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

      2. mul-1-neg 64.3%

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

      3. unsub-neg 64.3%

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

   7. Simplified 64.3%

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

   if -4.999999999999985e-310 < 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. Taylor expanded in b around inf 68.9%

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

      1. associate-*r/ 68.9%

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

      2. neg-mul-1 68.9%

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

   7. Simplified 68.9%

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

4. Final simplification 66.9%

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

Alternative 6: 42.9% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 7.2e+38) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.2e+38) {
		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 <= 7.2d+38) 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 <= 7.2e+38) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.2e+38:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.2e+38)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7.2e+38)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.2e+38], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.19999999999999938e38

   1. Initial program 66.2%

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

      1. *-commutative 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in b around -inf 41.2%

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

      1. associate-*r/ 41.2%

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

      2. mul-1-neg 41.2%

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

   7. Simplified 41.2%

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

   if 7.19999999999999938e38 < b

   1. Initial program 7.6%

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

      1. *-commutative 7.6%

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

   3. Simplified 7.6%

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

   5. Taylor expanded in b around -inf 2.2%

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

      1. +-commutative 2.2%

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

      2. mul-1-neg 2.2%

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

      3. unsub-neg 2.2%

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

      4. associate-/l* 2.3%

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

      5. associate-*r/ 2.3%

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

      6. *-commutative 2.3%

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

   7. Simplified 2.3%

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

   8. Taylor expanded in b around 0 31.2%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 19.1× 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(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / 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 75.8%

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

      1. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in b around -inf 64.2%

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

      1. associate-*r/ 64.2%

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

      2. mul-1-neg 64.2%

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

   7. Simplified 64.2%

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

   if -4.999999999999985e-310 < 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. Taylor expanded in b around inf 68.9%

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

      1. associate-*r/ 68.9%

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

      2. neg-mul-1 68.9%

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

   7. Simplified 68.9%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
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 48.1%

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

   1. *-commutative 48.1%

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

3. Simplified 48.1%

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

   1. add-sqr-sqrt 47.1%

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

   2. pow2 47.1%

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

   3. pow1/2 47.1%

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

   4. sqrt-pow1 47.1%

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

   5. fma-neg 47.1%

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

   6. distribute-lft-neg-in 47.1%

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

   7. *-commutative 47.1%

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

   8. associate-*r* 47.1%

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

   9. metadata-eval 47.1%

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

   10. metadata-eval 47.1%

       \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a \cdot 2} \]

6. Applied egg-rr 47.1%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)\right)}^{0.25}\right)}^{2}}}{a \cdot 2} \]

7. Taylor expanded in b around -inf 29.2%

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

   1. associate-*r* 29.2%

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

   2. metadata-eval 29.2%

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

   3. mul-1-neg 29.2%

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

   4. add-sqr-sqrt 14.2%

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

   5. sqrt-unprod 14.8%

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

   6. sqr-neg 14.8%

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

   7. sqrt-unprod 1.3%

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

   8. add-sqr-sqrt 2.8%

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

   9. distribute-frac-neg 2.8%

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

   10. expm1-log1p-u 2.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-b}{-a}\right)\right)} \]

   11. frac-2neg 2.1%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{b}{a}}\right)\right) \]

   12. expm1-udef 2.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b}{a}\right)} - 1} \]

9. Applied egg-rr 2.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b}{a}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 2.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{a}\right)\right)} \]

    2. expm1-log1p 2.8%

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

11. Simplified 2.8%

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

12. Final simplification 2.8%

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

Alternative 9: 11.0% 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 48.1%

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

   1. *-commutative 48.1%

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

3. Simplified 48.1%

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

5. Taylor expanded in b around -inf 27.3%

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

   1. +-commutative 27.3%

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

   2. mul-1-neg 27.3%

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

   3. unsub-neg 27.3%

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

   4. associate-/l* 29.0%

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

   5. associate-*r/ 29.0%

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

   6. *-commutative 29.0%

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

7. Simplified 29.0%

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

8. Taylor expanded in b around 0 11.7%

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

9. Final simplification 11.7%

   \[\leadsto \frac{c}{b} \]
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 2024013 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (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)))