quadp (p42, positive)

Percentage Accurate: 51.7% → 85.0%

Time: 12.3s

Alternatives: 8

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: 51.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e+59)
   (/ b (- a))
   (if (<= b 8.4e-35)
     (/ (- (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 <= -4.6e+59) {
		tmp = b / -a;
	} else if (b <= 8.4e-35) {
		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 <= (-4.6d+59)) then
        tmp = b / -a
    else if (b <= 8.4d-35) 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 <= -4.6e+59) {
		tmp = b / -a;
	} else if (b <= 8.4e-35) {
		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 <= -4.6e+59:
		tmp = b / -a
	elif b <= 8.4e-35:
		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 <= -4.6e+59)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 8.4e-35)
		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 <= -4.6e+59)
		tmp = b / -a;
	elseif (b <= 8.4e-35)
		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, -4.6e+59], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 8.4e-35], 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 < -4.60000000000000016e59

   1. Initial program 66.0%

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

      1. *-commutative 66.0%

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

   3. Simplified 66.0%

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

   5. Taylor expanded in b around -inf 93.6%

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

      1. associate-*r/ 93.6%

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

      2. mul-1-neg 93.6%

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

   7. Simplified 93.6%

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

   if -4.60000000000000016e59 < b < 8.3999999999999999e-35

   1. Initial program 82.9%

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

   if 8.3999999999999999e-35 < b

   1. Initial program 17.3%

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

      1. *-commutative 17.3%

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

   3. Simplified 17.3%

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

   5. Taylor expanded in b around inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-78}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-78)
   (/ b (- a))
   (if (<= b 7e-40) (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-78) {
		tmp = b / -a;
	} else if (b <= 7e-40) {
		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 <= (-2.9d-78)) then
        tmp = b / -a
    else if (b <= 7d-40) 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 <= -2.9e-78) {
		tmp = b / -a;
	} else if (b <= 7e-40) {
		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 <= -2.9e-78:
		tmp = b / -a
	elif b <= 7e-40:
		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 <= -2.9e-78)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 7e-40)
		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 <= -2.9e-78)
		tmp = b / -a;
	elseif (b <= 7e-40)
		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, -2.9e-78], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 7e-40], 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 < -2.9000000000000001e-78

   1. Initial program 74.5%

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

      1. *-commutative 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in b around -inf 86.8%

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

      1. associate-*r/ 86.8%

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

      2. mul-1-neg 86.8%

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

   7. Simplified 86.8%

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

   if -2.9000000000000001e-78 < b < 7.0000000000000003e-40

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in b around 0 78.7%

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

      1. associate-*r* 78.7%

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

      2. *-commutative 78.7%

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

      3. *-commutative 78.7%

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

   7. Simplified 78.7%

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

   if 7.0000000000000003e-40 < b

   1. Initial program 17.3%

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

      1. *-commutative 17.3%

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

   3. Simplified 17.3%

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

   5. Taylor expanded in b around inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-78}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-78}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.45e-78)
   (/ b (- a))
   (if (<= b 1.32e-37) (/ (sqrt (* c (* a -4.0))) (* a 2.0)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.45e-78) {
		tmp = b / -a;
	} else if (b <= 1.32e-37) {
		tmp = sqrt((c * (a * -4.0))) / (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 <= (-1.45d-78)) then
        tmp = b / -a
    else if (b <= 1.32d-37) then
        tmp = sqrt((c * (a * (-4.0d0)))) / (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 <= -1.45e-78) {
		tmp = b / -a;
	} else if (b <= 1.32e-37) {
		tmp = Math.sqrt((c * (a * -4.0))) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.45e-78:
		tmp = b / -a
	elif b <= 1.32e-37:
		tmp = math.sqrt((c * (a * -4.0))) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.45e-78)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.32e-37)
		tmp = Float64(sqrt(Float64(c * Float64(a * -4.0))) / 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 <= -1.45e-78)
		tmp = b / -a;
	elseif (b <= 1.32e-37)
		tmp = sqrt((c * (a * -4.0))) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.45e-78], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.32e-37], N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.45e-78

   1. Initial program 74.5%

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

      1. *-commutative 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in b around -inf 86.8%

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

      1. associate-*r/ 86.8%

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

      2. mul-1-neg 86.8%

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

   7. Simplified 86.8%

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

   if -1.45e-78 < b < 1.3200000000000001e-37

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

   3. Simplified 79.8%

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

      1. add-cube-cbrt 78.9%

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

      2. pow3 78.8%

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

      3. *-commutative 78.8%

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

      4. associate-*l* 78.8%

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

   6. Applied egg-rr 78.8%

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

   7. Taylor expanded in a around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 77.1%

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

      5. distribute-lft-neg-in 77.1%

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

      6. metadata-eval 77.1%

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

      7. rem-cube-cbrt 77.7%

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

   9. Simplified 77.7%

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

       1. *-un-lft-identity 77.7%

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

       2. add-sqr-sqrt 77.2%

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

       3. *-un-lft-identity 77.2%

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

       4. times-frac 77.3%

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

       5. pow1/2 77.3%

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

       6. sqrt-pow1 77.3%

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

       7. associate-*r* 77.3%

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

       8. *-commutative 77.3%

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

       9. associate-*l* 77.3%

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

       10. metadata-eval 77.3%

           \[\leadsto \frac{{\left(c \cdot \left(a \cdot -4\right)\right)}^{\color{blue}{0.25}}}{1} \cdot \frac{\sqrt{\sqrt{a \cdot \left(c \cdot -4\right)}}}{a \cdot 2} \]

       11. pow1/2 77.3%

           \[\leadsto \frac{{\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25}}{1} \cdot \frac{\sqrt{\color{blue}{{\left(a \cdot \left(c \cdot -4\right)\right)}^{0.5}}}}{a \cdot 2} \]

       12. sqrt-pow1 77.3%

           \[\leadsto \frac{{\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25}}{1} \cdot \frac{\color{blue}{{\left(a \cdot \left(c \cdot -4\right)\right)}^{\left(\frac{0.5}{2}\right)}}}{a \cdot 2} \]

       13. associate-*r* 77.3%

           \[\leadsto \frac{{\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25}}{1} \cdot \frac{{\color{blue}{\left(\left(a \cdot c\right) \cdot -4\right)}}^{\left(\frac{0.5}{2}\right)}}{a \cdot 2} \]

       14. *-commutative 77.3%

           \[\leadsto \frac{{\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25}}{1} \cdot \frac{{\left(\color{blue}{\left(c \cdot a\right)} \cdot -4\right)}^{\left(\frac{0.5}{2}\right)}}{a \cdot 2} \]

       15. associate-*l* 77.3%

           \[\leadsto \frac{{\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25}}{1} \cdot \frac{{\color{blue}{\left(c \cdot \left(a \cdot -4\right)\right)}}^{\left(\frac{0.5}{2}\right)}}{a \cdot 2} \]

       16. metadata-eval 77.3%

           \[\leadsto \frac{{\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25}}{1} \cdot \frac{{\left(c \cdot \left(a \cdot -4\right)\right)}^{\color{blue}{0.25}}}{a \cdot 2} \]

   11. Applied egg-rr 77.3%

       \[\leadsto \color{blue}{\frac{{\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25}}{1} \cdot \frac{{\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25}}{a \cdot 2}} \]
   12. Step-by-step derivation

       1. /-rgt-identity 77.3%

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

       2. associate-*r/ 77.2%

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

       3. pow-sqr 77.7%

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

       4. metadata-eval 77.7%

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

       5. unpow1/2 77.7%

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

   13. Simplified 77.7%

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

   if 1.3200000000000001e-37 < b

   1. Initial program 17.3%

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

      1. *-commutative 17.3%

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

   3. Simplified 17.3%

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

   5. Taylor expanded in b around inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-78}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{c \cdot \left(a \cdot -4\right)} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.3e-78)
   (/ b (- a))
   (if (<= b 5e-37) (* (sqrt (* c (* a -4.0))) (/ 0.5 a)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-78) {
		tmp = b / -a;
	} else if (b <= 5e-37) {
		tmp = 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 <= (-4.3d-78)) then
        tmp = b / -a
    else if (b <= 5d-37) then
        tmp = 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 <= -4.3e-78) {
		tmp = b / -a;
	} else if (b <= 5e-37) {
		tmp = 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 <= -4.3e-78:
		tmp = b / -a
	elif b <= 5e-37:
		tmp = 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 <= -4.3e-78)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 5e-37)
		tmp = Float64(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 <= -4.3e-78)
		tmp = b / -a;
	elseif (b <= 5e-37)
		tmp = 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, -4.3e-78], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 5e-37], N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.29999999999999994e-78

   1. Initial program 74.5%

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

      1. *-commutative 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in b around -inf 86.8%

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

      1. associate-*r/ 86.8%

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

      2. mul-1-neg 86.8%

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

   7. Simplified 86.8%

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

   if -4.29999999999999994e-78 < b < 4.9999999999999997e-37

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

   3. Simplified 79.8%

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

      1. add-cube-cbrt 78.9%

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

      2. pow3 78.8%

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

      3. *-commutative 78.8%

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

      4. associate-*l* 78.8%

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

   6. Applied egg-rr 78.8%

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

   7. Taylor expanded in a around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 77.1%

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

      5. distribute-lft-neg-in 77.1%

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

      6. metadata-eval 77.1%

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

      7. rem-cube-cbrt 77.7%

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

   9. Simplified 77.7%

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

       1. *-commutative 77.7%

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

       2. associate-*r/ 77.5%

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

       3. add-sqr-sqrt 77.1%

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

       4. associate-*l* 77.1%

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

       5. pow1/2 77.1%

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

       6. sqrt-pow1 77.2%

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

       7. associate-*r* 77.2%

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

       8. *-commutative 77.2%

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

       9. associate-*l* 77.2%

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

       10. metadata-eval 77.2%

           \[\leadsto {\left(c \cdot \left(a \cdot -4\right)\right)}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{a \cdot \left(c \cdot -4\right)}} \cdot \frac{1}{a \cdot 2}\right) \]

       11. pow1/2 77.2%

           \[\leadsto {\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot \left(\sqrt{\color{blue}{{\left(a \cdot \left(c \cdot -4\right)\right)}^{0.5}}} \cdot \frac{1}{a \cdot 2}\right) \]

       12. sqrt-pow1 77.1%

           \[\leadsto {\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot \left(\color{blue}{{\left(a \cdot \left(c \cdot -4\right)\right)}^{\left(\frac{0.5}{2}\right)}} \cdot \frac{1}{a \cdot 2}\right) \]

       13. associate-*r* 77.1%

           \[\leadsto {\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot \left({\color{blue}{\left(\left(a \cdot c\right) \cdot -4\right)}}^{\left(\frac{0.5}{2}\right)} \cdot \frac{1}{a \cdot 2}\right) \]

       14. *-commutative 77.1%

           \[\leadsto {\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot \left({\left(\color{blue}{\left(c \cdot a\right)} \cdot -4\right)}^{\left(\frac{0.5}{2}\right)} \cdot \frac{1}{a \cdot 2}\right) \]

       15. associate-*l* 77.1%

           \[\leadsto {\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot \left({\color{blue}{\left(c \cdot \left(a \cdot -4\right)\right)}}^{\left(\frac{0.5}{2}\right)} \cdot \frac{1}{a \cdot 2}\right) \]

       16. metadata-eval 77.1%

           \[\leadsto {\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot \left({\left(c \cdot \left(a \cdot -4\right)\right)}^{\color{blue}{0.25}} \cdot \frac{1}{a \cdot 2}\right) \]

       17. *-commutative 77.1%

           \[\leadsto {\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot \left({\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot \frac{1}{\color{blue}{2 \cdot a}}\right) \]

       18. associate-/r* 77.1%

           \[\leadsto {\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot \left({\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\right) \]

       19. metadata-eval 77.1%

           \[\leadsto {\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot \left({\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot \frac{\color{blue}{0.5}}{a}\right) \]

   11. Applied egg-rr 77.1%

       \[\leadsto \color{blue}{{\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot \left({\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot \frac{0.5}{a}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 77.1%

          \[\leadsto \color{blue}{\left({\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot {\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25}\right) \cdot \frac{0.5}{a}} \]

       2. pow-sqr 77.5%

          \[\leadsto \color{blue}{{\left(c \cdot \left(a \cdot -4\right)\right)}^{\left(2 \cdot 0.25\right)}} \cdot \frac{0.5}{a} \]

       3. metadata-eval 77.5%

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

       4. unpow1/2 77.5%

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

   13. Simplified 77.5%

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

   if 4.9999999999999997e-37 < b

   1. Initial program 17.3%

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

      1. *-commutative 17.3%

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

   3. Simplified 17.3%

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

   5. Taylor expanded in b around inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{c \cdot \left(a \cdot -4\right)} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-119}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-40}:\\ \;\;\;\;-0.5 \cdot \left(-\sqrt{c \cdot \frac{-4}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-119)
   (/ b (- a))
   (if (<= b 6.2e-40) (* -0.5 (- (sqrt (* c (/ -4.0 a))))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-119) {
		tmp = b / -a;
	} else if (b <= 6.2e-40) {
		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 <= (-3d-119)) then
        tmp = b / -a
    else if (b <= 6.2d-40) 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 <= -3e-119) {
		tmp = b / -a;
	} else if (b <= 6.2e-40) {
		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 <= -3e-119:
		tmp = b / -a
	elif b <= 6.2e-40:
		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 <= -3e-119)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 6.2e-40)
		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 <= -3e-119)
		tmp = b / -a;
	elseif (b <= 6.2e-40)
		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, -3e-119], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 6.2e-40], 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 < -3.0000000000000002e-119

   1. Initial program 74.5%

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

      1. *-commutative 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in b around -inf 83.5%

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

      1. associate-*r/ 83.5%

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

      2. mul-1-neg 83.5%

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

   7. Simplified 83.5%

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

   if -3.0000000000000002e-119 < b < 6.20000000000000021e-40

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

   3. Simplified 80.0%

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

      1. add-cube-cbrt 79.0%

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

      2. pow3 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-*l* 79.0%

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

   6. Applied egg-rr 79.0%

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

   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 32.3%

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

      4. rem-cube-cbrt 32.6%

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

      5. associate-/l* 32.5%

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

   9. Simplified 32.5%

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

   if 6.20000000000000021e-40 < b

   1. Initial program 17.3%

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

      1. *-commutative 17.3%

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

   3. Simplified 17.3%

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

   5. Taylor expanded in b around inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-119}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-40}:\\ \;\;\;\;-0.5 \cdot \left(-\sqrt{c \cdot \frac{-4}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-301}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.5e-301) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.5e-301) {
		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 <= 6.5d-301) 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 <= 6.5e-301) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.5e-301:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.5e-301)
		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 <= 6.5e-301)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.5e-301], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.49999999999999991e-301

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in b around -inf 67.9%

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

      1. associate-*r/ 67.9%

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

      2. mul-1-neg 67.9%

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

   7. Simplified 67.9%

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

   if 6.49999999999999991e-301 < b

   1. Initial program 32.7%

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

      1. *-commutative 32.7%

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

   3. Simplified 32.7%

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

   5. Taylor expanded in b around inf 67.4%

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

      1. associate-*r/ 67.4%

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

      2. neg-mul-1 67.4%

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

   7. Simplified 67.4%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-301}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.6% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -5e-310) (/ b (- a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b / -a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b / -a
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b / (-a)), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.0%

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

      1. *-commutative 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in b around -inf 68.4%

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

      1. associate-*r/ 68.4%

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

      2. mul-1-neg 68.4%

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

   7. Simplified 68.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 33.2%

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

      1. *-commutative 33.2%

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

   3. Simplified 33.2%

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

      1. add-cube-cbrt 32.8%

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

      2. pow3 32.8%

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

      3. *-commutative 32.8%

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

      4. associate-*l* 32.8%

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

   6. Applied egg-rr 32.8%

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

      1. clear-num 32.8%

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

      2. inv-pow 32.8%

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

      3. neg-mul-1 32.8%

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

      4. fma-define 32.8%

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

      5. pow2 32.8%

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

      6. unpow3 32.8%

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

      7. add-cube-cbrt 33.1%

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

   8. Applied egg-rr 33.1%

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

      1. unpow-1 33.1%

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

      2. associate-/l* 33.0%

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

   10. Simplified 33.0%

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

   11. Taylor expanded in a around 0 17.6%

       \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
   12. Step-by-step derivation

       1. distribute-rgt1-in 17.6%

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

       2. metadata-eval 17.6%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{0} \cdot b}{a} \]

       3. mul0-lft 17.6%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{0}}{a} \]

       4. div0 17.6%

          \[\leadsto 0.5 \cdot \color{blue}{0} \]

       5. metadata-eval 17.6%

          \[\leadsto \color{blue}{0} \]

   13. Simplified 17.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 11.5% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 54.4%

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

   1. *-commutative 54.4%

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

3. Simplified 54.4%

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

   1. add-cube-cbrt 54.1%

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

   2. pow3 54.1%

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

   3. *-commutative 54.1%

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

   4. associate-*l* 54.1%

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

6. Applied egg-rr 54.1%

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

   1. clear-num 54.0%

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

   2. inv-pow 54.0%

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

   3. neg-mul-1 54.0%

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

   4. fma-define 54.0%

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

   5. pow2 54.0%

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

   6. unpow3 54.0%

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

   7. add-cube-cbrt 54.2%

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

8. Applied egg-rr 54.2%

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

   1. unpow-1 54.2%

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

   2. associate-/l* 54.1%

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

10. Simplified 54.1%

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

11. Taylor expanded in a around 0 10.4%

    \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
12. Step-by-step derivation

    1. distribute-rgt1-in 10.4%

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

    2. metadata-eval 10.4%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{0} \cdot b}{a} \]

    3. mul0-lft 10.4%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{0}}{a} \]

    4. div0 10.4%

       \[\leadsto 0.5 \cdot \color{blue}{0} \]

    5. metadata-eval 10.4%

       \[\leadsto \color{blue}{0} \]

13. Simplified 10.4%

    \[\leadsto \color{blue}{0} \]
14. 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 2024136 
(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)))