quadp (p42, positive)

Percentage Accurate: 52.0% → 85.8%

Time: 12.3s

Alternatives: 9

Speedup: 12.9×

• Could not converge on a ground truth

  1. reg0 = (bf "3.585666772309482314614910037525685081407e-186")
  2. reg1 = (bf "-5.072205930318953122803015023065015736745e162")
  3. reg2 = (bf "-5.102973879885833340676367724344234271415e213")

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: 52.0% 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: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+154)
   (/ b (- a))
   (if (<= b 4.3e-75)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+154) {
		tmp = b / -a;
	} else if (b <= 4.3e-75) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d+154)) then
        tmp = b / -a
    else if (b <= 4.3d-75) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+154) {
		tmp = b / -a;
	} else if (b <= 4.3e-75) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+154:
		tmp = b / -a
	elif b <= 4.3e-75:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+154)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 4.3e-75)
		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 <= -1e+154)
		tmp = b / -a;
	elseif (b <= 4.3e-75)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+154], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 4.3e-75], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000004e154

   1. Initial program 44.5%

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

      1. *-commutative 44.5%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in b around -inf 97.6%

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

      1. associate-*r/ 97.6%

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

      2. mul-1-neg 97.6%

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

   7. Simplified 97.6%

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

   if -1.00000000000000004e154 < b < 4.2999999999999999e-75

   1. Initial program 85.8%

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

   if 4.2999999999999999e-75 < b

   1. Initial program 17.5%

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

      1. *-commutative 17.5%

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

   3. Simplified 17.5%

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

   5. Taylor expanded in b around inf 83.4%

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

      1. associate-*r/ 83.4%

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

      2. neg-mul-1 83.4%

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

   7. Simplified 83.4%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-142}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} - \frac{1}{a}\right)\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e-142)
   (* b (- (/ c (pow b 2.0)) (/ 1.0 a)))
   (if (<= b 1.9e-75)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e-142) {
		tmp = b * ((c / pow(b, 2.0)) - (1.0 / a));
	} else if (b <= 1.9e-75) {
		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 <= (-7d-142)) then
        tmp = b * ((c / (b ** 2.0d0)) - (1.0d0 / a))
    else if (b <= 1.9d-75) 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 <= -7e-142) {
		tmp = b * ((c / Math.pow(b, 2.0)) - (1.0 / a));
	} else if (b <= 1.9e-75) {
		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 <= -7e-142:
		tmp = b * ((c / math.pow(b, 2.0)) - (1.0 / a))
	elif b <= 1.9e-75:
		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 <= -7e-142)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) - Float64(1.0 / a)));
	elseif (b <= 1.9e-75)
		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 <= -7e-142)
		tmp = b * ((c / (b ^ 2.0)) - (1.0 / a));
	elseif (b <= 1.9e-75)
		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, -7e-142], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.9e-75], 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 < -7.00000000000000029e-142

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in b around -inf 92.3%

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

      1. mul-1-neg 92.3%

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

      2. distribute-rgt-neg-in 92.3%

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

      3. +-commutative 92.3%

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

      4. mul-1-neg 92.3%

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

      5. unsub-neg 92.3%

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

   7. Simplified 92.3%

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

   if -7.00000000000000029e-142 < b < 1.89999999999999997e-75

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in b around 0 74.9%

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

      1. associate-*r* 74.9%

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

      2. *-commutative 74.9%

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

      3. *-commutative 74.9%

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

   7. Simplified 74.9%

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

   if 1.89999999999999997e-75 < b

   1. Initial program 17.5%

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

      1. *-commutative 17.5%

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

   3. Simplified 17.5%

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

   5. Taylor expanded in b around inf 83.4%

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

      1. associate-*r/ 83.4%

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

      2. neg-mul-1 83.4%

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

   7. Simplified 83.4%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-142}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} - \frac{1}{a}\right)\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-142}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} - \frac{1}{a}\right)\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e-142)
   (* b (- (/ c (pow b 2.0)) (/ 1.0 a)))
   (if (<= b 5.4e-76) (/ (sqrt (* a (* c -4.0))) (* a 2.0)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e-142) {
		tmp = b * ((c / pow(b, 2.0)) - (1.0 / a));
	} else if (b <= 5.4e-76) {
		tmp = sqrt((a * (c * -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 <= (-7d-142)) then
        tmp = b * ((c / (b ** 2.0d0)) - (1.0d0 / a))
    else if (b <= 5.4d-76) then
        tmp = sqrt((a * (c * (-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 <= -7e-142) {
		tmp = b * ((c / Math.pow(b, 2.0)) - (1.0 / a));
	} else if (b <= 5.4e-76) {
		tmp = Math.sqrt((a * (c * -4.0))) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7e-142:
		tmp = b * ((c / math.pow(b, 2.0)) - (1.0 / a))
	elif b <= 5.4e-76:
		tmp = math.sqrt((a * (c * -4.0))) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7e-142)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) - Float64(1.0 / a)));
	elseif (b <= 5.4e-76)
		tmp = Float64(sqrt(Float64(a * Float64(c * -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 <= -7e-142)
		tmp = b * ((c / (b ^ 2.0)) - (1.0 / a));
	elseif (b <= 5.4e-76)
		tmp = sqrt((a * (c * -4.0))) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7e-142], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.4e-76], N[(N[Sqrt[N[(a * N[(c * -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 < -7.00000000000000029e-142

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in b around -inf 92.3%

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

      1. mul-1-neg 92.3%

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

      2. distribute-rgt-neg-in 92.3%

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

      3. +-commutative 92.3%

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

      4. mul-1-neg 92.3%

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

      5. unsub-neg 92.3%

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

   7. Simplified 92.3%

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

   if -7.00000000000000029e-142 < b < 5.4000000000000001e-76

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

   3. Simplified 76.4%

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

      1. add-cube-cbrt 75.6%

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

      2. pow3 75.6%

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

      3. *-commutative 75.6%

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

      4. associate-*l* 75.6%

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

   6. Applied egg-rr 75.6%

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

   7. 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 73.8%

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

      5. distribute-lft-neg-in 73.8%

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

      6. metadata-eval 73.8%

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

      7. rem-cube-cbrt 74.3%

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

   9. Simplified 74.3%

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

   if 5.4000000000000001e-76 < b

   1. Initial program 17.5%

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

      1. *-commutative 17.5%

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

   3. Simplified 17.5%

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

   5. Taylor expanded in b around inf 83.4%

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

      1. associate-*r/ 83.4%

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

      2. neg-mul-1 83.4%

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

   7. Simplified 83.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-142}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} - \frac{1}{a}\right)\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-169}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} - \frac{1}{a}\right)\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-169)
   (* b (- (/ c (pow b 2.0)) (/ 1.0 a)))
   (if (<= b 7.6e-170) (sqrt (/ c (- a))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-169) {
		tmp = b * ((c / pow(b, 2.0)) - (1.0 / a));
	} else if (b <= 7.6e-170) {
		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 <= (-6d-169)) then
        tmp = b * ((c / (b ** 2.0d0)) - (1.0d0 / a))
    else if (b <= 7.6d-170) 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 <= -6e-169) {
		tmp = b * ((c / Math.pow(b, 2.0)) - (1.0 / a));
	} else if (b <= 7.6e-170) {
		tmp = Math.sqrt((c / -a));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e-169:
		tmp = b * ((c / math.pow(b, 2.0)) - (1.0 / a))
	elif b <= 7.6e-170:
		tmp = math.sqrt((c / -a))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e-169)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) - Float64(1.0 / a)));
	elseif (b <= 7.6e-170)
		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 <= -6e-169)
		tmp = b * ((c / (b ^ 2.0)) - (1.0 / a));
	elseif (b <= 7.6e-170)
		tmp = sqrt((c / -a));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e-169], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.6e-170], N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.9999999999999998e-169

   1. Initial program 78.0%

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

      1. *-commutative 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in b around -inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. distribute-rgt-neg-in 87.2%

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

      3. +-commutative 87.2%

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

      4. mul-1-neg 87.2%

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

      5. unsub-neg 87.2%

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

   7. Simplified 87.2%

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

   if -5.9999999999999998e-169 < b < 7.5999999999999995e-170

   1. Initial program 78.5%

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

      1. *-commutative 78.5%

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

   3. Simplified 78.5%

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

      1. add-cube-cbrt 77.7%

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

      2. pow3 77.7%

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

      3. *-commutative 77.7%

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

      4. associate-*l* 77.7%

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

   6. Applied egg-rr 77.7%

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

   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 39.3%

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

      4. rem-cube-cbrt 39.7%

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

      5. associate-/l* 39.6%

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

   9. Simplified 39.6%

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

       1. add-sqr-sqrt 39.4%

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

       2. sqrt-unprod 39.6%

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

       3. *-commutative 39.6%

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

       4. *-commutative 39.6%

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

       5. swap-sqr 39.6%

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

       6. mul-1-neg 39.6%

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

       7. mul-1-neg 39.6%

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

       8. sqr-neg 39.6%

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

       9. add-sqr-sqrt 39.6%

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

       10. metadata-eval 39.6%

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

   11. Applied egg-rr 39.6%

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

   12. Taylor expanded in c around 0 39.7%

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

       1. associate-*r/ 39.7%

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

       2. neg-mul-1 39.7%

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

   14. Simplified 39.7%

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

   if 7.5999999999999995e-170 < b

   1. Initial program 24.5%

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

      1. *-commutative 24.5%

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

   3. Simplified 24.5%

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

   5. Taylor expanded in b around inf 77.0%

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

      1. associate-*r/ 77.0%

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

      2. neg-mul-1 77.0%

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

   7. Simplified 77.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-169}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} - \frac{1}{a}\right)\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-155}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e-155)
   (/ b (- a))
   (if (<= b 8e-170) (sqrt (/ c (- a))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-155) {
		tmp = b / -a;
	} else if (b <= 8e-170) {
		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 <= (-1.35d-155)) then
        tmp = b / -a
    else if (b <= 8d-170) 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 <= -1.35e-155) {
		tmp = b / -a;
	} else if (b <= 8e-170) {
		tmp = Math.sqrt((c / -a));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.35e-155:
		tmp = b / -a
	elif b <= 8e-170:
		tmp = math.sqrt((c / -a))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e-155)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 8e-170)
		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 <= -1.35e-155)
		tmp = b / -a;
	elseif (b <= 8e-170)
		tmp = sqrt((c / -a));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e-155], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 8e-170], N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.34999999999999991e-155

   1. Initial program 77.1%

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

      1. *-commutative 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in b around -inf 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 -1.34999999999999991e-155 < b < 7.99999999999999987e-170

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

   3. Simplified 80.4%

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

      1. add-cube-cbrt 79.6%

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

      2. pow3 79.6%

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

      3. *-commutative 79.6%

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

      4. associate-*l* 79.6%

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

   6. Applied egg-rr 79.6%

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

   7. 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 36.2%

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

      4. rem-cube-cbrt 36.5%

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

      5. associate-/l* 36.5%

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

   9. Simplified 36.5%

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

       1. add-sqr-sqrt 36.3%

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

       2. sqrt-unprod 36.5%

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

       3. *-commutative 36.5%

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

       4. *-commutative 36.5%

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

       5. swap-sqr 36.5%

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

       6. mul-1-neg 36.5%

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

       7. mul-1-neg 36.5%

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

       8. sqr-neg 36.5%

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

       9. add-sqr-sqrt 36.5%

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

       10. metadata-eval 36.5%

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

   11. Applied egg-rr 36.5%

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

   12. Taylor expanded in c around 0 36.5%

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

       1. associate-*r/ 36.5%

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

       2. neg-mul-1 36.5%

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

   14. Simplified 36.5%

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

   if 7.99999999999999987e-170 < b

   1. Initial program 24.5%

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

      1. *-commutative 24.5%

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

   3. Simplified 24.5%

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

   5. Taylor expanded in b around inf 77.0%

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

      1. associate-*r/ 77.0%

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

      2. neg-mul-1 77.0%

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

   7. Simplified 77.0%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-155}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-280}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.5e-280) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.5e-280) {
		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 <= 3.5d-280) 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 <= 3.5e-280) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.5e-280:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.5e-280)
		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 <= 3.5e-280)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.5e-280], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5000000000000001e-280

   1. Initial program 78.7%

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

      1. *-commutative 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in b around -inf 70.9%

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

      1. associate-*r/ 70.9%

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

      2. mul-1-neg 70.9%

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

   7. Simplified 70.9%

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

   if 3.5000000000000001e-280 < b

   1. Initial program 29.7%

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

      1. *-commutative 29.7%

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

   3. Simplified 29.7%

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

   5. Taylor expanded in b around inf 70.3%

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

      1. associate-*r/ 70.3%

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

      2. neg-mul-1 70.3%

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

   7. Simplified 70.3%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-280}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 2e-5) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e-5) {
		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 <= 2d-5) 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 <= 2e-5) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2e-5:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2e-5)
		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 <= 2e-5)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2e-5], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.00000000000000016e-5

   1. Initial program 71.8%

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

      1. *-commutative 71.8%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in b around -inf 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 2.00000000000000016e-5 < b

   1. Initial program 14.3%

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

      1. *-commutative 14.3%

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

   3. Simplified 14.3%

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

   5. Taylor expanded in b around inf 91.5%

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

      1. associate-*r/ 91.5%

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

      2. neg-mul-1 91.5%

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

   7. Simplified 91.5%

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

      1. add-sqr-sqrt 48.7%

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

      2. sqrt-unprod 50.6%

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

      3. sqr-neg 50.6%

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

      4. sqrt-unprod 10.5%

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

      5. add-sqr-sqrt 25.1%

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

      6. *-un-lft-identity 25.1%

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

   9. Applied egg-rr 25.1%

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

       1. *-lft-identity 25.1%

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

   11. Simplified 25.1%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 10.5% 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 53.8%

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

   1. *-commutative 53.8%

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

3. Simplified 53.8%

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

5. Taylor expanded in b around inf 36.9%

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

   1. associate-*r/ 36.9%

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

   2. neg-mul-1 36.9%

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

7. Simplified 36.9%

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

   1. add-sqr-sqrt 19.2%

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

   2. sqrt-unprod 19.3%

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

   3. sqr-neg 19.3%

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

   4. sqrt-unprod 4.1%

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

   5. add-sqr-sqrt 9.7%

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

   6. *-un-lft-identity 9.7%

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

9. Applied egg-rr 9.7%

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

    1. *-lft-identity 9.7%

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

11. Simplified 9.7%

    \[\leadsto \color{blue}{\frac{c}{b}} \]
12. 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 53.8%

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

   1. *-commutative 53.8%

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

3. Simplified 53.8%

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

5. Taylor expanded in b around -inf 36.3%

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

   1. associate-*r/ 36.3%

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

   2. mul-1-neg 36.3%

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

7. Simplified 36.3%

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

   1. add-log-exp 12.0%

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

9. Applied egg-rr 12.0%

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

    1. *-un-lft-identity 12.0%

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

    2. log-prod 12.0%

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

    3. metadata-eval 12.0%

       \[\leadsto \color{blue}{0} + \log \left(e^{\frac{-b}{a}}\right) \]

    4. rem-log-exp 36.3%

       \[\leadsto 0 + \color{blue}{\frac{-b}{a}} \]

    5. add-sqr-sqrt 34.6%

       \[\leadsto 0 + \frac{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}{a} \]

    6. sqrt-unprod 28.6%

       \[\leadsto 0 + \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}{a} \]

    7. sqr-neg 28.6%

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

    8. sqrt-prod 1.8%

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

    9. add-sqr-sqrt 2.6%

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

11. Applied egg-rr 2.6%

    \[\leadsto \color{blue}{0 + \frac{b}{a}} \]
12. Step-by-step derivation

    1. +-lft-identity 2.6%

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

13. Simplified 2.6%

    \[\leadsto \color{blue}{\frac{b}{a}} \]
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 2024141 
(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)))