quadp (p42, positive)

Percentage Accurate: 52.1% → 84.8%

Time: 12.7s

Alternatives: 9

Speedup: 12.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e+47)
   (/ b (- a))
   (if (<= b 3.5e-98)
     (/ (- (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 <= -2.2e+47) {
		tmp = b / -a;
	} else if (b <= 3.5e-98) {
		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 <= (-2.2d+47)) then
        tmp = b / -a
    else if (b <= 3.5d-98) 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 <= -2.2e+47) {
		tmp = b / -a;
	} else if (b <= 3.5e-98) {
		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 <= -2.2e+47:
		tmp = b / -a
	elif b <= 3.5e-98:
		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 <= -2.2e+47)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 3.5e-98)
		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 <= -2.2e+47)
		tmp = b / -a;
	elseif (b <= 3.5e-98)
		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, -2.2e+47], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 3.5e-98], 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 < -2.1999999999999999e47

   1. Initial program 56.8%

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

      1. *-commutative 56.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in b around -inf 94.8%

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

      1. associate-*r/ 94.8%

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

      2. mul-1-neg 94.8%

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

   7. Simplified 94.8%

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

   if -2.1999999999999999e47 < b < 3.5000000000000002e-98

   1. Initial program 81.4%

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

   if 3.5000000000000002e-98 < b

   1. Initial program 19.3%

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

      1. *-commutative 19.3%

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

   3. Simplified 19.3%

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

   5. Taylor expanded in b around inf 84.5%

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

      1. associate-*r/ 84.5%

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

      2. neg-mul-1 84.5%

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

   7. Simplified 84.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.35 \cdot 10^{-79}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.35e-79)
   (/ b (- a))
   (if (<= b 3.7e-97)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.35e-79) {
		tmp = b / -a;
	} else if (b <= 3.7e-97) {
		tmp = (sqrt((c * (a * -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 <= (-2.35d-79)) then
        tmp = b / -a
    else if (b <= 3.7d-97) then
        tmp = (sqrt((c * (a * (-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 <= -2.35e-79) {
		tmp = b / -a;
	} else if (b <= 3.7e-97) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.35e-79:
		tmp = b / -a
	elif b <= 3.7e-97:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.35e-79)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 3.7e-97)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -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 <= -2.35e-79)
		tmp = b / -a;
	elseif (b <= 3.7e-97)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.35e-79], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 3.7e-97], N[(N[(N[Sqrt[N[(c * N[(a * -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 < -2.3500000000000001e-79

   1. Initial program 62.8%

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

      1. *-commutative 62.8%

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

   3. Simplified 62.8%

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

   5. Taylor expanded in b around -inf 90.6%

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

      1. associate-*r/ 90.6%

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

      2. mul-1-neg 90.6%

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

   7. Simplified 90.6%

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

   if -2.3500000000000001e-79 < b < 3.69999999999999976e-97

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

   3. Simplified 79.7%

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

      1. add-exp-log 74.5%

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

      2. sub-neg 74.5%

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

      3. +-commutative 74.5%

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

      4. *-commutative 74.5%

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

      5. distribute-rgt-neg-in 74.5%

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

      6. fma-define 74.5%

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

      7. metadata-eval 74.5%

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

      8. pow2 74.5%

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

   6. Applied egg-rr 74.5%

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

   7. Taylor expanded in b around 0 72.4%

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

      1. *-commutative 72.4%

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

   9. Simplified 72.4%

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

       1. +-commutative 72.4%

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

       2. unsub-neg 72.4%

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

       3. rem-exp-log 77.3%

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

       4. *-commutative 77.3%

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

       5. associate-*l* 77.3%

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

   11. Applied egg-rr 77.3%

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

   if 3.69999999999999976e-97 < b

   1. Initial program 19.3%

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

      1. *-commutative 19.3%

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

   3. Simplified 19.3%

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

   5. Taylor expanded in b around inf 84.5%

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

      1. associate-*r/ 84.5%

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

      2. neg-mul-1 84.5%

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

   7. Simplified 84.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.35 \cdot 10^{-79}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-98}:\\ \;\;\;\;\left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e-80)
   (/ b (- a))
   (if (<= b 1.45e-98)
     (* (+ b (sqrt (* c (* a -4.0)))) (/ 0.5 a))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-80) {
		tmp = b / -a;
	} else if (b <= 1.45e-98) {
		tmp = (b + sqrt((c * (a * -4.0)))) * (0.5 / 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 <= (-9.5d-80)) then
        tmp = b / -a
    else if (b <= 1.45d-98) then
        tmp = (b + sqrt((c * (a * (-4.0d0))))) * (0.5d0 / 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 <= -9.5e-80) {
		tmp = b / -a;
	} else if (b <= 1.45e-98) {
		tmp = (b + Math.sqrt((c * (a * -4.0)))) * (0.5 / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.5e-80:
		tmp = b / -a
	elif b <= 1.45e-98:
		tmp = (b + math.sqrt((c * (a * -4.0)))) * (0.5 / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.5e-80)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.45e-98)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) * Float64(0.5 / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.5e-80)
		tmp = b / -a;
	elseif (b <= 1.45e-98)
		tmp = (b + sqrt((c * (a * -4.0)))) * (0.5 / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.5e-80], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.45e-98], N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.5000000000000003e-80

   1. Initial program 62.8%

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

      1. *-commutative 62.8%

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

   3. Simplified 62.8%

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

   5. Taylor expanded in b around -inf 90.6%

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

      1. associate-*r/ 90.6%

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

      2. mul-1-neg 90.6%

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

   7. Simplified 90.6%

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

   if -9.5000000000000003e-80 < b < 1.45e-98

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

   3. Simplified 79.7%

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

      1. add-exp-log 74.5%

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

      2. sub-neg 74.5%

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

      3. +-commutative 74.5%

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

      4. *-commutative 74.5%

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

      5. distribute-rgt-neg-in 74.5%

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

      6. fma-define 74.5%

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

      7. metadata-eval 74.5%

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

      8. pow2 74.5%

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

   6. Applied egg-rr 74.5%

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

   7. Taylor expanded in b around 0 72.4%

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

      1. *-commutative 72.4%

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

   9. Simplified 72.4%

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

       1. div-inv 72.4%

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

       2. add-sqr-sqrt 34.7%

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

       3. sqrt-unprod 71.7%

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

       4. sqr-neg 71.7%

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

       5. sqrt-unprod 37.8%

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

       6. add-sqr-sqrt 71.0%

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

       7. rem-exp-log 75.6%

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

       8. *-commutative 75.6%

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

       9. associate-*l* 75.6%

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

       10. *-commutative 75.6%

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

       11. associate-/r* 75.6%

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

       12. metadata-eval 75.6%

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

   11. Applied egg-rr 75.6%

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

   if 1.45e-98 < b

   1. Initial program 19.3%

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

      1. *-commutative 19.3%

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

   3. Simplified 19.3%

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

   5. Taylor expanded in b around inf 84.5%

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

      1. associate-*r/ 84.5%

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

      2. neg-mul-1 84.5%

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

   7. Simplified 84.5%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-98}:\\ \;\;\;\;\left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.1 \cdot 10^{-220}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-127}:\\ \;\;\;\;0.5 \cdot \sqrt{c \cdot \frac{-4}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.1e-220)
   (/ b (- a))
   (if (<= b 1.7e-127) (* 0.5 (sqrt (* c (/ -4.0 a)))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.1e-220) {
		tmp = b / -a;
	} else if (b <= 1.7e-127) {
		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 <= (-5.1d-220)) then
        tmp = b / -a
    else if (b <= 1.7d-127) 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 <= -5.1e-220) {
		tmp = b / -a;
	} else if (b <= 1.7e-127) {
		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 <= -5.1e-220:
		tmp = b / -a
	elif b <= 1.7e-127:
		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 <= -5.1e-220)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.7e-127)
		tmp = Float64(0.5 * 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 <= -5.1e-220)
		tmp = b / -a;
	elseif (b <= 1.7e-127)
		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, -5.1e-220], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.7e-127], 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 < -5.1000000000000001e-220

   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 78.1%

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

      1. associate-*r/ 78.1%

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

      2. mul-1-neg 78.1%

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

   7. Simplified 78.1%

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

   if -5.1000000000000001e-220 < b < 1.6999999999999999e-127

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

   3. Simplified 83.6%

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

      1. add-cube-cbrt 82.7%

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

      2. pow3 82.7%

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

      3. *-commutative 82.7%

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

      4. associate-*l* 82.7%

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

   6. Applied egg-rr 82.7%

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

   7. Taylor expanded in a around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 45.7%

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

      4. rem-cube-cbrt 46.1%

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

      5. associate-/l* 46.1%

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

   9. Simplified 46.1%

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

       1. mul-1-neg 46.1%

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

       2. distribute-rgt-neg-out 46.1%

          \[\leadsto \color{blue}{--0.5 \cdot \sqrt{c \cdot \frac{-4}{a}}} \]

       3. clear-num 46.1%

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

       4. un-div-inv 46.1%

          \[\leadsto --0.5 \cdot \sqrt{\color{blue}{\frac{c}{\frac{a}{-4}}}} \]

       5. div-inv 46.1%

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

       6. metadata-eval 46.1%

          \[\leadsto --0.5 \cdot \sqrt{\frac{c}{a \cdot \color{blue}{-0.25}}} \]

   11. Applied egg-rr 46.1%

       \[\leadsto \color{blue}{--0.5 \cdot \sqrt{\frac{c}{a \cdot -0.25}}} \]
   12. Step-by-step derivation

       1. distribute-lft-neg-in 46.1%

          \[\leadsto \color{blue}{\left(--0.5\right) \cdot \sqrt{\frac{c}{a \cdot -0.25}}} \]

       2. metadata-eval 46.1%

          \[\leadsto \color{blue}{0.5} \cdot \sqrt{\frac{c}{a \cdot -0.25}} \]

       3. *-rgt-identity 46.1%

          \[\leadsto 0.5 \cdot \sqrt{\frac{\color{blue}{c \cdot 1}}{a \cdot -0.25}} \]

       4. times-frac 46.0%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{c}{a} \cdot \frac{1}{-0.25}}} \]

       5. metadata-eval 46.0%

          \[\leadsto 0.5 \cdot \sqrt{\frac{c}{a} \cdot \color{blue}{-4}} \]

       6. *-commutative 46.0%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot \frac{c}{a}}} \]

       7. associate-*r/ 46.1%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{-4 \cdot c}{a}}} \]

       8. *-commutative 46.1%

          \[\leadsto 0.5 \cdot \sqrt{\frac{\color{blue}{c \cdot -4}}{a}} \]

       9. associate-/l* 46.1%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{c \cdot \frac{-4}{a}}} \]

   13. Simplified 46.1%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{c \cdot \frac{-4}{a}}} \]

   if 1.6999999999999999e-127 < b

   1. Initial program 21.1%

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

      1. *-commutative 21.1%

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

   3. Simplified 21.1%

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

   5. Taylor expanded in b around inf 80.8%

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

      1. associate-*r/ 80.8%

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

      2. neg-mul-1 80.8%

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

   7. Simplified 80.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.1 \cdot 10^{-220}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-127}:\\ \;\;\;\;0.5 \cdot \sqrt{c \cdot \frac{-4}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.1 \cdot 10^{-220}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.1e-220)
   (/ b (- a))
   (if (<= b 3.1e-132) (sqrt (/ c (- a))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.1e-220) {
		tmp = b / -a;
	} else if (b <= 3.1e-132) {
		tmp = sqrt((c / -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 <= (-5.1d-220)) then
        tmp = b / -a
    else if (b <= 3.1d-132) then
        tmp = sqrt((c / -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 <= -5.1e-220) {
		tmp = b / -a;
	} else if (b <= 3.1e-132) {
		tmp = Math.sqrt((c / -a));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.1e-220:
		tmp = b / -a
	elif b <= 3.1e-132:
		tmp = math.sqrt((c / -a))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.1e-220)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 3.1e-132)
		tmp = sqrt(Float64(c / Float64(-a)));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.1e-220)
		tmp = b / -a;
	elseif (b <= 3.1e-132)
		tmp = sqrt((c / -a));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.1e-220], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 3.1e-132], N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.1000000000000001e-220

   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 78.1%

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

      1. associate-*r/ 78.1%

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

      2. mul-1-neg 78.1%

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

   7. Simplified 78.1%

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

   if -5.1000000000000001e-220 < b < 3.10000000000000008e-132

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

   3. Simplified 83.6%

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

      1. add-cube-cbrt 82.7%

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

      2. pow3 82.7%

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

      3. *-commutative 82.7%

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

      4. associate-*l* 82.7%

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

   6. Applied egg-rr 82.7%

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

   7. Taylor expanded in a around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 45.7%

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

      4. rem-cube-cbrt 46.1%

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

      5. associate-/l* 46.1%

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

   9. Simplified 46.1%

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

       1. add-sqr-sqrt 45.8%

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

       2. sqrt-unprod 46.0%

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

       3. swap-sqr 46.0%

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

       4. metadata-eval 46.0%

          \[\leadsto \sqrt{\color{blue}{0.25} \cdot \left(\left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right) \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)\right)} \]

       5. mul-1-neg 46.0%

          \[\leadsto \sqrt{0.25 \cdot \left(\color{blue}{\left(-\sqrt{c \cdot \frac{-4}{a}}\right)} \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)\right)} \]

       6. mul-1-neg 46.0%

          \[\leadsto \sqrt{0.25 \cdot \left(\left(-\sqrt{c \cdot \frac{-4}{a}}\right) \cdot \color{blue}{\left(-\sqrt{c \cdot \frac{-4}{a}}\right)}\right)} \]

       7. sqr-neg 46.0%

          \[\leadsto \sqrt{0.25 \cdot \color{blue}{\left(\sqrt{c \cdot \frac{-4}{a}} \cdot \sqrt{c \cdot \frac{-4}{a}}\right)}} \]

       8. add-sqr-sqrt 46.0%

          \[\leadsto \sqrt{0.25 \cdot \color{blue}{\left(c \cdot \frac{-4}{a}\right)}} \]

       9. clear-num 46.0%

          \[\leadsto \sqrt{0.25 \cdot \left(c \cdot \color{blue}{\frac{1}{\frac{a}{-4}}}\right)} \]

       10. un-div-inv 46.0%

           \[\leadsto \sqrt{0.25 \cdot \color{blue}{\frac{c}{\frac{a}{-4}}}} \]

       11. div-inv 46.0%

           \[\leadsto \sqrt{0.25 \cdot \frac{c}{\color{blue}{a \cdot \frac{1}{-4}}}} \]

       12. metadata-eval 46.0%

           \[\leadsto \sqrt{0.25 \cdot \frac{c}{a \cdot \color{blue}{-0.25}}} \]

   11. Applied egg-rr 46.0%

       \[\leadsto \color{blue}{\sqrt{0.25 \cdot \frac{c}{a \cdot -0.25}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 46.0%

          \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot c}{a \cdot -0.25}}} \]

       2. *-commutative 46.0%

          \[\leadsto \sqrt{\frac{0.25 \cdot c}{\color{blue}{-0.25 \cdot a}}} \]

       3. times-frac 46.0%

          \[\leadsto \sqrt{\color{blue}{\frac{0.25}{-0.25} \cdot \frac{c}{a}}} \]

       4. metadata-eval 46.0%

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

       5. metadata-eval 46.0%

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

       6. times-frac 46.0%

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

       7. *-lft-identity 46.0%

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

       8. neg-mul-1 46.0%

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

   13. Simplified 46.0%

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

   if 3.10000000000000008e-132 < b

   1. Initial program 21.1%

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

      1. *-commutative 21.1%

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

   3. Simplified 21.1%

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

   5. Taylor expanded in b around inf 80.8%

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

      1. associate-*r/ 80.8%

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

      2. neg-mul-1 80.8%

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

   7. Simplified 80.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.1 \cdot 10^{-220}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{-279}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.5e-279) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.5e-279) {
		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 <= 8.5d-279) 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 <= 8.5e-279) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.5e-279:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.5e-279)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8.5e-279)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.5e-279], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.5000000000000002e-279

   1. Initial program 68.4%

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

      1. *-commutative 68.4%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in b around -inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. mul-1-neg 69.9%

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

   7. Simplified 69.9%

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

   if 8.5000000000000002e-279 < b

   1. Initial program 34.3%

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

      1. *-commutative 34.3%

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

   3. Simplified 34.3%

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

   5. Taylor expanded in b around inf 66.6%

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

      1. associate-*r/ 66.6%

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

      2. neg-mul-1 66.6%

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

   7. Simplified 66.6%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{-279}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 4.8e-35) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.8e-35) {
		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 <= 4.8d-35) 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 <= 4.8e-35) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.8e-35:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.8e-35)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.8e-35)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.8e-35], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.8000000000000003e-35

   1. Initial program 67.4%

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

      1. *-commutative 67.4%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in b around -inf 51.7%

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

      1. associate-*r/ 51.7%

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

      2. mul-1-neg 51.7%

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

   7. Simplified 51.7%

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

   if 4.8000000000000003e-35 < b

   1. Initial program 17.0%

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

      1. *-commutative 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in b around -inf 2.4%

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

      1. mul-1-neg 2.4%

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

      2. *-commutative 2.4%

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

      3. distribute-rgt-neg-in 2.4%

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

      4. +-commutative 2.4%

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

      5. mul-1-neg 2.4%

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

      6. unsub-neg 2.4%

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

   7. Simplified 2.4%

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

   8. Taylor expanded in a around inf 33.3%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 10.8% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

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

   1. *-commutative 51.1%

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

3. Simplified 51.1%

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

5. Taylor expanded in b around -inf 34.0%

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

   1. mul-1-neg 34.0%

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

   2. *-commutative 34.0%

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

   3. distribute-rgt-neg-in 34.0%

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

   4. +-commutative 34.0%

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

   5. mul-1-neg 34.0%

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

   6. unsub-neg 34.0%

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

7. Simplified 34.0%

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

8. Taylor expanded in a around inf 12.7%

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

Alternative 9: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = b / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

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

   1. *-commutative 51.1%

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

3. Simplified 51.1%

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

5. Taylor expanded in b around -inf 35.7%

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

   1. associate-*r/ 35.7%

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

   2. mul-1-neg 35.7%

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

7. Simplified 35.7%

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

   1. add-sqr-sqrt 34.2%

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

   2. sqrt-unprod 24.0%

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

   3. sqr-neg 24.0%

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

   4. sqrt-prod 1.8%

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

   5. add-sqr-sqrt 2.4%

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

   6. *-un-lft-identity 2.4%

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

9. Applied egg-rr 2.4%

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

    1. *-lft-identity 2.4%

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

11. Simplified 2.4%

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

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024169 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

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

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