quadp (p42, positive)

Percentage Accurate: 53.2% → 85.4%

Time: 10.3s

Alternatives: 5

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: 53.2% 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: 85.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e+153)
   (- (/ b a))
   (if (<= b 4.2e-57)
     (/ (- (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 <= -6e+153) {
		tmp = -(b / a);
	} else if (b <= 4.2e-57) {
		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 <= (-6d+153)) then
        tmp = -(b / a)
    else if (b <= 4.2d-57) 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 <= -6e+153) {
		tmp = -(b / a);
	} else if (b <= 4.2e-57) {
		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 <= -6e+153:
		tmp = -(b / a)
	elif b <= 4.2e-57:
		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 <= -6e+153)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 4.2e-57)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6e+153)
		tmp = -(b / a);
	elseif (b <= 4.2e-57)
		tmp = (sqrt(((b * b) - (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, -6e+153], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 4.2e-57], 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 < -6.00000000000000037e153

   1. Initial program 46.6%

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

      1. *-commutative 46.6%

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

   3. Simplified 46.6%

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

      1. fma-neg 46.6%

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

      2. distribute-lft-neg-in 46.6%

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

      3. metadata-eval 46.6%

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

      4. associate-*r* 46.6%

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

      5. *-commutative 46.6%

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

      6. add-sqr-sqrt 46.6%

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

      7. pow2 46.6%

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

      8. fma-undefine 46.6%

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

      9. add-sqr-sqrt 33.3%

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

      10. hypot-define 56.9%

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

      11. associate-*r* 56.9%

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

      12. *-commutative 56.9%

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

   6. Applied egg-rr 56.9%

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

   7. Taylor expanded in b around -inf 91.6%

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

      1. associate-*r/ 91.6%

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

      2. mul-1-neg 91.6%

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

   9. Simplified 91.6%

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

   if -6.00000000000000037e153 < b < 4.1999999999999999e-57

   1. Initial program 86.3%

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

   if 4.1999999999999999e-57 < b

   1. Initial program 8.6%

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

      1. *-commutative 8.6%

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

   3. Simplified 8.6%

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

      1. add-cube-cbrt 8.6%

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

      2. pow3 8.6%

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

      3. *-commutative 8.6%

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

      4. associate-*l* 8.6%

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

   6. Applied egg-rr 8.6%

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

      1. pow1/3 5.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({\left(a \cdot \left(c \cdot 4\right)\right)}^{0.3333333333333333}\right)}}^{3}}}{a \cdot 2} \]

      2. associate-*r* 5.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\color{blue}{\left(\left(a \cdot c\right) \cdot 4\right)}}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      3. add-cube-cbrt 5.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\left(\color{blue}{\left(\left(\sqrt[3]{a \cdot c} \cdot \sqrt[3]{a \cdot c}\right) \cdot \sqrt[3]{a \cdot c}\right)} \cdot 4\right)}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      4. unpow3 5.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\left(\color{blue}{{\left(\sqrt[3]{a \cdot c}\right)}^{3}} \cdot 4\right)}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      5. *-commutative 5.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\color{blue}{\left(4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}\right)}}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      6. unpow-prod-down 5.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({4}^{0.3333333333333333} \cdot {\left({\left(\sqrt[3]{a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}}}{a \cdot 2} \]

      7. pow1/3 8.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({4}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{{\left(\sqrt[3]{a \cdot c}\right)}^{3}}}\right)}^{3}}}{a \cdot 2} \]

      8. rem-cbrt-cube 8.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({4}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{a \cdot c}}\right)}^{3}}}{a \cdot 2} \]

   8. Applied egg-rr 8.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({4}^{0.3333333333333333} \cdot \sqrt[3]{a \cdot c}\right)}}^{3}}}{a \cdot 2} \]

   9. Taylor expanded in b around inf 28.9%

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

       1. associate-/l* 29.1%

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

       2. rem-cube-cbrt 29.1%

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

       3. associate-/l* 29.1%

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

   11. Simplified 29.1%

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

   12. Taylor expanded in b around 0 93.9%

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

       1. associate-*r/ 93.9%

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

       2. neg-mul-1 93.9%

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

   14. Simplified 93.9%

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

4. Final simplification 89.8%

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

Alternative 2: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if b < -5.8000000000000001e-127

   1. Initial program 80.1%

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

      1. *-commutative 80.1%

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

   3. Simplified 80.1%

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

      1. fma-neg 80.1%

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

      2. distribute-lft-neg-in 80.1%

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

      3. metadata-eval 80.1%

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

      4. associate-*r* 80.1%

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

      5. *-commutative 80.1%

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

      6. add-sqr-sqrt 79.8%

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

      7. pow2 79.8%

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

      8. fma-undefine 79.8%

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

      9. add-sqr-sqrt 61.9%

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

      10. hypot-define 69.8%

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

      11. associate-*r* 69.8%

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

      12. *-commutative 69.8%

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

   6. Applied egg-rr 69.8%

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

   7. Taylor expanded in b around -inf 81.3%

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

      1. associate-*r/ 81.3%

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

      2. mul-1-neg 81.3%

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

   9. Simplified 81.3%

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

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

   1. Initial program 75.6%

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

      1. *-commutative 75.6%

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

   3. Simplified 75.6%

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

      1. add-cube-cbrt 74.8%

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

      2. pow3 74.8%

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

      3. *-commutative 74.8%

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

      4. associate-*l* 74.8%

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

   6. Applied egg-rr 74.8%

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

   7. Taylor expanded in a around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 71.0%

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

      5. distribute-lft-neg-in 71.0%

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

      6. metadata-eval 71.0%

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

      7. rem-cube-cbrt 71.6%

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

   9. Simplified 71.6%

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

   if 5.2e-47 < b

   1. Initial program 8.6%

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

      1. *-commutative 8.6%

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

   3. Simplified 8.6%

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

      1. add-cube-cbrt 8.6%

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

      2. pow3 8.6%

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

      3. *-commutative 8.6%

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

      4. associate-*l* 8.6%

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

   6. Applied egg-rr 8.6%

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

      1. pow1/3 5.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({\left(a \cdot \left(c \cdot 4\right)\right)}^{0.3333333333333333}\right)}}^{3}}}{a \cdot 2} \]

      2. associate-*r* 5.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\color{blue}{\left(\left(a \cdot c\right) \cdot 4\right)}}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      3. add-cube-cbrt 5.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\left(\color{blue}{\left(\left(\sqrt[3]{a \cdot c} \cdot \sqrt[3]{a \cdot c}\right) \cdot \sqrt[3]{a \cdot c}\right)} \cdot 4\right)}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      4. unpow3 5.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\left(\color{blue}{{\left(\sqrt[3]{a \cdot c}\right)}^{3}} \cdot 4\right)}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      5. *-commutative 5.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\color{blue}{\left(4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}\right)}}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      6. unpow-prod-down 5.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({4}^{0.3333333333333333} \cdot {\left({\left(\sqrt[3]{a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}}}{a \cdot 2} \]

      7. pow1/3 8.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({4}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{{\left(\sqrt[3]{a \cdot c}\right)}^{3}}}\right)}^{3}}}{a \cdot 2} \]

      8. rem-cbrt-cube 8.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({4}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{a \cdot c}}\right)}^{3}}}{a \cdot 2} \]

   8. Applied egg-rr 8.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({4}^{0.3333333333333333} \cdot \sqrt[3]{a \cdot c}\right)}}^{3}}}{a \cdot 2} \]

   9. Taylor expanded in b around inf 28.9%

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

       1. associate-/l* 29.1%

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

       2. rem-cube-cbrt 29.1%

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

       3. associate-/l* 29.1%

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

   11. Simplified 29.1%

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

   12. Taylor expanded in b around 0 93.9%

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

       1. associate-*r/ 93.9%

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

       2. neg-mul-1 93.9%

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

   14. Simplified 93.9%

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

4. Final simplification 83.6%

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

Alternative 3: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-186}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-105}:\\ \;\;\;\;-0.5 \cdot \left(-\sqrt{c \cdot \frac{-4}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e-186)
   (- (/ b a))
   (if (<= b 5.2e-105) (* -0.5 (- (sqrt (* c (/ -4.0 a))))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e-186) {
		tmp = -(b / a);
	} else if (b <= 5.2e-105) {
		tmp = -0.5 * -sqrt((c * (-4.0 / a)));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.2d-186)) then
        tmp = -(b / a)
    else if (b <= 5.2d-105) then
        tmp = (-0.5d0) * -sqrt((c * ((-4.0d0) / a)))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e-186) {
		tmp = -(b / a);
	} else if (b <= 5.2e-105) {
		tmp = -0.5 * -Math.sqrt((c * (-4.0 / a)));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.2e-186:
		tmp = -(b / a)
	elif b <= 5.2e-105:
		tmp = -0.5 * -math.sqrt((c * (-4.0 / a)))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e-186)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 5.2e-105)
		tmp = Float64(-0.5 * Float64(-sqrt(Float64(c * Float64(-4.0 / a)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.2e-186)
		tmp = -(b / a);
	elseif (b <= 5.2e-105)
		tmp = -0.5 * -sqrt((c * (-4.0 / a)));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e-186], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 5.2e-105], N[(-0.5 * (-N[Sqrt[N[(c * N[(-4.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.19999999999999986e-186

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

   3. Simplified 79.4%

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

      1. fma-neg 79.4%

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

      2. distribute-lft-neg-in 79.4%

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

      3. metadata-eval 79.4%

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

      4. associate-*r* 79.4%

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

      5. *-commutative 79.4%

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

      6. add-sqr-sqrt 79.0%

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

      7. pow2 79.0%

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

      8. fma-undefine 79.0%

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

      9. add-sqr-sqrt 62.7%

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

      10. hypot-define 71.4%

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

      11. associate-*r* 71.4%

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

      12. *-commutative 71.4%

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

   6. Applied egg-rr 71.4%

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

   7. Taylor expanded in b around -inf 77.5%

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

      1. associate-*r/ 77.5%

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

      2. mul-1-neg 77.5%

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

   9. Simplified 77.5%

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

   if -5.19999999999999986e-186 < b < 5.1999999999999997e-105

   1. Initial program 78.9%

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

      1. *-commutative 78.9%

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

   3. Simplified 78.9%

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

      1. add-cube-cbrt 78.2%

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

      2. pow3 78.1%

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

      3. *-commutative 78.1%

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

      4. associate-*l* 78.1%

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

   6. Applied egg-rr 78.1%

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

      1. pow1/3 0.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({\left(a \cdot \left(c \cdot 4\right)\right)}^{0.3333333333333333}\right)}}^{3}}}{a \cdot 2} \]

      2. associate-*r* 0.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\color{blue}{\left(\left(a \cdot c\right) \cdot 4\right)}}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      3. add-cube-cbrt 0.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\left(\color{blue}{\left(\left(\sqrt[3]{a \cdot c} \cdot \sqrt[3]{a \cdot c}\right) \cdot \sqrt[3]{a \cdot c}\right)} \cdot 4\right)}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      4. unpow3 0.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\left(\color{blue}{{\left(\sqrt[3]{a \cdot c}\right)}^{3}} \cdot 4\right)}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      5. *-commutative 0.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\color{blue}{\left(4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}\right)}}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      6. unpow-prod-down 0.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({4}^{0.3333333333333333} \cdot {\left({\left(\sqrt[3]{a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}}}{a \cdot 2} \]

      7. pow1/3 78.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({4}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{{\left(\sqrt[3]{a \cdot c}\right)}^{3}}}\right)}^{3}}}{a \cdot 2} \]

      8. rem-cbrt-cube 78.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({4}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{a \cdot c}}\right)}^{3}}}{a \cdot 2} \]

   8. Applied egg-rr 78.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({4}^{0.3333333333333333} \cdot \sqrt[3]{a \cdot c}\right)}}^{3}}}{a \cdot 2} \]

   9. Taylor expanded in a around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 30.2%

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

       4. associate-/l* 30.2%

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

       5. rem-cube-cbrt 30.2%

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

       6. rem-cube-cbrt 30.4%

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

       7. metadata-eval 30.4%

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

   11. Simplified 30.4%

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

   if 5.1999999999999997e-105 < b

   1. Initial program 13.3%

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

      1. *-commutative 13.3%

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

   3. Simplified 13.3%

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

      1. add-cube-cbrt 13.2%

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

      2. pow3 13.2%

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

      3. *-commutative 13.2%

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

      4. associate-*l* 13.2%

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

   6. Applied egg-rr 13.2%

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

      1. pow1/3 5.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({\left(a \cdot \left(c \cdot 4\right)\right)}^{0.3333333333333333}\right)}}^{3}}}{a \cdot 2} \]

      2. associate-*r* 5.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\color{blue}{\left(\left(a \cdot c\right) \cdot 4\right)}}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      3. add-cube-cbrt 5.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\left(\color{blue}{\left(\left(\sqrt[3]{a \cdot c} \cdot \sqrt[3]{a \cdot c}\right) \cdot \sqrt[3]{a \cdot c}\right)} \cdot 4\right)}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      4. unpow3 5.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\left(\color{blue}{{\left(\sqrt[3]{a \cdot c}\right)}^{3}} \cdot 4\right)}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      5. *-commutative 5.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\color{blue}{\left(4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}\right)}}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      6. unpow-prod-down 5.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({4}^{0.3333333333333333} \cdot {\left({\left(\sqrt[3]{a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}}}{a \cdot 2} \]

      7. pow1/3 13.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({4}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{{\left(\sqrt[3]{a \cdot c}\right)}^{3}}}\right)}^{3}}}{a \cdot 2} \]

      8. rem-cbrt-cube 13.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({4}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{a \cdot c}}\right)}^{3}}}{a \cdot 2} \]

   8. Applied egg-rr 13.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({4}^{0.3333333333333333} \cdot \sqrt[3]{a \cdot c}\right)}}^{3}}}{a \cdot 2} \]

   9. Taylor expanded in b around inf 27.4%

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

       1. associate-/l* 27.5%

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

       2. rem-cube-cbrt 27.5%

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

       3. associate-/l* 27.5%

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

   11. Simplified 27.5%

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

   12. Taylor expanded in b around 0 89.7%

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

       1. associate-*r/ 89.7%

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

       2. neg-mul-1 89.7%

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

   14. Simplified 89.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-186}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-105}:\\ \;\;\;\;-0.5 \cdot \left(-\sqrt{c \cdot \frac{-4}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.85e-290)
		tmp = Float64(-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 <= 1.85e-290)
		tmp = -(b / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 1.84999999999999989e-290

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

   3. Simplified 79.9%

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

      1. fma-neg 79.9%

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

      2. distribute-lft-neg-in 79.9%

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

      3. metadata-eval 79.9%

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

      4. associate-*r* 79.9%

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

      5. *-commutative 79.9%

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

      6. add-sqr-sqrt 79.5%

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

      7. pow2 79.5%

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

      8. fma-undefine 79.5%

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

      9. add-sqr-sqrt 65.5%

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

      10. hypot-define 73.0%

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

      11. associate-*r* 73.0%

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

      12. *-commutative 73.0%

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

   6. Applied egg-rr 73.0%

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

   7. Taylor expanded in b around -inf 67.3%

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

      1. associate-*r/ 67.3%

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

      2. mul-1-neg 67.3%

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

   9. Simplified 67.3%

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

   if 1.84999999999999989e-290 < b

   1. Initial program 27.0%

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

      1. *-commutative 27.0%

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

   3. Simplified 27.0%

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

      1. add-cube-cbrt 26.7%

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

      2. pow3 26.7%

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

      3. *-commutative 26.7%

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

      4. associate-*l* 26.7%

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

   6. Applied egg-rr 26.7%

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

      1. pow1/3 4.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({\left(a \cdot \left(c \cdot 4\right)\right)}^{0.3333333333333333}\right)}}^{3}}}{a \cdot 2} \]

      2. associate-*r* 4.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\color{blue}{\left(\left(a \cdot c\right) \cdot 4\right)}}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      3. add-cube-cbrt 4.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\left(\color{blue}{\left(\left(\sqrt[3]{a \cdot c} \cdot \sqrt[3]{a \cdot c}\right) \cdot \sqrt[3]{a \cdot c}\right)} \cdot 4\right)}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      4. unpow3 4.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\left(\color{blue}{{\left(\sqrt[3]{a \cdot c}\right)}^{3}} \cdot 4\right)}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      5. *-commutative 4.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\color{blue}{\left(4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}\right)}}^{0.3333333333333333}\right)}^{3}}}{a \cdot 2} \]

      6. unpow-prod-down 4.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({4}^{0.3333333333333333} \cdot {\left({\left(\sqrt[3]{a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}}}{a \cdot 2} \]

      7. pow1/3 26.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({4}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{{\left(\sqrt[3]{a \cdot c}\right)}^{3}}}\right)}^{3}}}{a \cdot 2} \]

      8. rem-cbrt-cube 26.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({4}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{a \cdot c}}\right)}^{3}}}{a \cdot 2} \]

   8. Applied egg-rr 26.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({4}^{0.3333333333333333} \cdot \sqrt[3]{a \cdot c}\right)}}^{3}}}{a \cdot 2} \]

   9. Taylor expanded in b around inf 22.5%

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

       1. associate-/l* 22.6%

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

       2. rem-cube-cbrt 22.6%

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

       3. associate-/l* 22.6%

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

   11. Simplified 22.6%

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

   12. Taylor expanded in b around 0 73.4%

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

       1. associate-*r/ 73.4%

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

       2. neg-mul-1 73.4%

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

   14. Simplified 73.4%

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

4. Final simplification 70.5%

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

Alternative 5: 35.9% accurate, 29.0× 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(-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 52.2%

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

   1. *-commutative 52.2%

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

3. Simplified 52.2%

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

   1. fma-neg 52.2%

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

   2. distribute-lft-neg-in 52.2%

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

   3. metadata-eval 52.2%

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

   4. associate-*r* 52.2%

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

   5. *-commutative 52.2%

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

   6. add-sqr-sqrt 51.0%

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

   7. pow2 51.0%

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

   8. fma-undefine 51.0%

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

   9. add-sqr-sqrt 43.6%

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

   10. hypot-define 49.1%

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

   11. associate-*r* 49.1%

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

   12. *-commutative 49.1%

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

6. Applied egg-rr 49.1%

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

7. Taylor expanded in b around -inf 33.4%

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

   1. associate-*r/ 33.4%

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

   2. mul-1-neg 33.4%

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

9. Simplified 33.4%

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

10. Final simplification 33.4%

    \[\leadsto -\frac{b}{a} \]
11. 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 2024111 
(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)))