The quadratic formula (r1)

Percentage Accurate: 52.5% → 84.9%

Time: 15.9s

Alternatives: 10

Speedup: 12.9×

Specification

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.5% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.8e+31)
   (/ b (- a))
   (if (<= b 1.65e-96)
     (/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.8e+31) {
		tmp = b / -a;
	} else if (b <= 1.65e-96) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.8e+31)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.65e-96)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.8e+31], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.65e-96], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $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 < -4.79999999999999965e31

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in b around -inf 93.0%

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

      1. associate-*r/ 93.0%

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

      2. mul-1-neg 93.0%

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

   7. Simplified 93.0%

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

   if -4.79999999999999965e31 < b < 1.64999999999999995e-96

   1. Initial program 85.9%

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

      1. *-commutative 85.9%

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

   3. Simplified 85.9%

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

   if 1.64999999999999995e-96 < b

   1. Initial program 16.9%

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

      1. *-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in a around 0 82.5%

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

      1. associate-*r/ 82.5%

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

      2. mul-1-neg 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.8e+31)
   (/ b (- a))
   (if (<= b 9.8e-98)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.8e+31) {
		tmp = b / -a;
	} else if (b <= 9.8e-98) {
		tmp = (sqrt(((b * b) - (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 <= (-4.8d+31)) then
        tmp = b / -a
    else if (b <= 9.8d-98) then
        tmp = (sqrt(((b * b) - (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 <= -4.8e+31) {
		tmp = b / -a;
	} else if (b <= 9.8e-98) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.8e+31:
		tmp = b / -a
	elif b <= 9.8e-98:
		tmp = (math.sqrt(((b * b) - (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 <= -4.8e+31)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 9.8e-98)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - 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 <= -4.8e+31)
		tmp = b / -a;
	elseif (b <= 9.8e-98)
		tmp = (sqrt(((b * b) - (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, -4.8e+31], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 9.8e-98], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $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 < -4.79999999999999965e31

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in b around -inf 93.0%

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

      1. associate-*r/ 93.0%

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

      2. mul-1-neg 93.0%

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

   7. Simplified 93.0%

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

   if -4.79999999999999965e31 < b < 9.80000000000000028e-98

   1. Initial program 85.9%

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

   if 9.80000000000000028e-98 < b

   1. Initial program 16.9%

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

      1. *-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in a around 0 82.5%

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

      1. associate-*r/ 82.5%

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

      2. mul-1-neg 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-97)
   (/ b (- a))
   (if (<= b 4.2e-95)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-97) {
		tmp = b / -a;
	} else if (b <= 4.2e-95) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.9d-97)) then
        tmp = b / -a
    else if (b <= 4.2d-95) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-97) {
		tmp = b / -a;
	} else if (b <= 4.2e-95) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.9e-97:
		tmp = b / -a
	elif b <= 4.2e-95:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e-97)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 4.2e-95)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.9e-97)
		tmp = b / -a;
	elseif (b <= 4.2e-95)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e-97], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 4.2e-95], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9e-97

   1. Initial program 66.1%

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

      1. *-commutative 66.1%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in b around -inf 87.1%

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

      1. associate-*r/ 87.1%

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

      2. mul-1-neg 87.1%

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

   7. Simplified 87.1%

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

   if -1.9e-97 < b < 4.2e-95

   1. Initial program 81.7%

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

      1. *-commutative 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in a around inf 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*r* 74.1%

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

   7. Simplified 74.1%

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

   if 4.2e-95 < b

   1. Initial program 16.9%

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

      1. *-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in a around 0 82.5%

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

      1. associate-*r/ 82.5%

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

      2. mul-1-neg 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-96}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-97)
   (/ b (- a))
   (if (<= b 5.1e-96)
     (* (/ 0.5 a) (- (sqrt (* -4.0 (* a c))) b))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-97) {
		tmp = b / -a;
	} else if (b <= 5.1e-96) {
		tmp = (0.5 / a) * (sqrt((-4.0 * (a * c))) - b);
	} 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 <= (-1.9d-97)) then
        tmp = b / -a
    else if (b <= 5.1d-96) then
        tmp = (0.5d0 / a) * (sqrt(((-4.0d0) * (a * c))) - b)
    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 <= -1.9e-97) {
		tmp = b / -a;
	} else if (b <= 5.1e-96) {
		tmp = (0.5 / a) * (Math.sqrt((-4.0 * (a * c))) - b);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.9e-97:
		tmp = b / -a
	elif b <= 5.1e-96:
		tmp = (0.5 / a) * (math.sqrt((-4.0 * (a * c))) - b)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e-97)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 5.1e-96)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.9e-97)
		tmp = b / -a;
	elseif (b <= 5.1e-96)
		tmp = (0.5 / a) * (sqrt((-4.0 * (a * c))) - b);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e-97], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 5.1e-96], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9e-97

   1. Initial program 66.1%

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

      1. *-commutative 66.1%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in b around -inf 87.1%

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

      1. associate-*r/ 87.1%

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

      2. mul-1-neg 87.1%

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

   7. Simplified 87.1%

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

   if -1.9e-97 < b < 5.09999999999999973e-96

   1. Initial program 81.7%

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

      1. *-commutative 81.7%

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

   3. Simplified 81.7%

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

      1. div-sub 81.7%

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

      2. sub-neg 81.7%

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

      3. div-inv 81.5%

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

      4. pow2 81.5%

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

      5. *-commutative 81.5%

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

      6. associate-/r* 81.5%

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

      7. metadata-eval 81.5%

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

      8. div-inv 81.5%

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

      9. *-commutative 81.5%

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

      10. associate-/r* 81.5%

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

      11. metadata-eval 81.5%

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

   6. Applied egg-rr 81.5%

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

      1. sub-neg 81.5%

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

      2. distribute-rgt-out-- 81.5%

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

   8. Simplified 81.5%

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

   9. Taylor expanded in a around inf 74.0%

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

   if 5.09999999999999973e-96 < b

   1. Initial program 16.9%

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

      1. *-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in a around 0 82.5%

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

      1. associate-*r/ 82.5%

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

      2. mul-1-neg 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-96}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-177}:\\ \;\;\;\;-0.5 \cdot \left(-\sqrt{c \cdot \frac{-4}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.6e-159)
   (/ b (- a))
   (if (<= b 3.5e-177) (* -0.5 (- (sqrt (* c (/ -4.0 a))))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-159) {
		tmp = b / -a;
	} else if (b <= 3.5e-177) {
		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 <= (-2.6d-159)) then
        tmp = b / -a
    else if (b <= 3.5d-177) 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 <= -2.6e-159) {
		tmp = b / -a;
	} else if (b <= 3.5e-177) {
		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 <= -2.6e-159:
		tmp = b / -a
	elif b <= 3.5e-177:
		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 <= -2.6e-159)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 3.5e-177)
		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 <= -2.6e-159)
		tmp = b / -a;
	elseif (b <= 3.5e-177)
		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, -2.6e-159], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 3.5e-177], 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 < -2.5999999999999998e-159

   1. Initial program 69.1%

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

      1. *-commutative 69.1%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 82.3%

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

      1. associate-*r/ 82.3%

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

      2. mul-1-neg 82.3%

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

   7. Simplified 82.3%

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

   if -2.5999999999999998e-159 < b < 3.5000000000000002e-177

   1. Initial program 76.5%

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

      1. *-commutative 76.5%

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

   3. Simplified 76.5%

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

      1. add-cube-cbrt 75.7%

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

      2. pow3 75.7%

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

   6. Applied egg-rr 75.7%

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

   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. rem-cube-cbrt 0.0%

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

      2. associate-/l* 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 46.3%

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

   9. Simplified 46.3%

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

   if 3.5000000000000002e-177 < b

   1. Initial program 25.9%

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

      1. *-commutative 25.9%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in a around 0 74.3%

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

      1. associate-*r/ 74.3%

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

      2. mul-1-neg 74.3%

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

   7. Simplified 74.3%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-177}:\\ \;\;\;\;-0.5 \cdot \left(-\sqrt{c \cdot \frac{-4}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = b / -a
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		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 <= -1e-310)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.4%

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

      1. *-commutative 70.4%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in b around -inf 72.9%

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

      1. associate-*r/ 72.9%

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

      2. mul-1-neg 72.9%

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

   7. Simplified 72.9%

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

   if -9.999999999999969e-311 < b

   1. Initial program 33.7%

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

      1. *-commutative 33.7%

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

   3. Simplified 33.7%

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

   5. Taylor expanded in a around 0 62.9%

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

      1. associate-*r/ 62.9%

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

      2. mul-1-neg 62.9%

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

   7. Simplified 62.9%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 4.4e+86) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e+86) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.4d+86) then
        tmp = b / -a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e+86) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.4e+86:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.4e+86)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.4e+86)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.4e+86], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.40000000000000006e86

   1. Initial program 65.2%

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

      1. *-commutative 65.2%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in b around -inf 50.3%

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

      1. associate-*r/ 50.3%

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

      2. mul-1-neg 50.3%

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

   7. Simplified 50.3%

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

   if 4.40000000000000006e86 < b

   1. Initial program 6.1%

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

      1. *-commutative 6.1%

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

   3. Simplified 6.1%

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

   5. Taylor expanded in a around 0 96.3%

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

      1. associate-*r/ 96.3%

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

      2. mul-1-neg 96.3%

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

   7. Simplified 96.3%

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

      1. add-sqr-sqrt 44.1%

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

      2. sqrt-unprod 56.5%

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

      3. sqr-neg 56.5%

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

      4. sqrt-unprod 22.1%

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

      5. add-sqr-sqrt 31.3%

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

      6. div-inv 31.3%

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

   9. Applied egg-rr 31.3%

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

       1. associate-*r/ 31.3%

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

       2. *-rgt-identity 31.3%

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

   11. Simplified 31.3%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 10.7% 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 53.6%

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

   1. *-commutative 53.6%

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

3. Simplified 53.7%

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

5. Taylor expanded in a around 0 30.0%

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

   1. associate-*r/ 30.0%

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

   2. mul-1-neg 30.0%

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

7. Simplified 30.0%

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

   1. add-sqr-sqrt 13.1%

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

   2. sqrt-unprod 15.9%

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

   3. sqr-neg 15.9%

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

   4. sqrt-unprod 5.4%

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

   5. add-sqr-sqrt 8.4%

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

   6. div-inv 8.4%

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

9. Applied egg-rr 8.4%

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

    1. associate-*r/ 8.4%

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

    2. *-rgt-identity 8.4%

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

11. Simplified 8.4%

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

Alternative 9: 5.3% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return a / 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 = a / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

   1. *-commutative 53.6%

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

3. Simplified 53.7%

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

5. Taylor expanded in b around -inf 40.9%

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

   1. associate-*r/ 40.9%

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

   2. mul-1-neg 40.9%

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

7. Simplified 40.9%

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

   1. neg-sub0 40.9%

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

   2. flip3-- 13.1%

      \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}}}{a} \]

   3. metadata-eval 13.1%

      \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}}{a} \]

   4. metadata-eval 13.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)}}{a} \]

   5. pow2 13.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)}}{a} \]

9. Applied egg-rr 13.1%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}}}{a} \]
10. Step-by-step derivation

    1. sub0-neg 13.1%

       \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)}}{a} \]

    2. +-lft-identity 13.1%

       \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}}}{a} \]

    3. mul0-lft 13.1%

       \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}}}{a} \]

    4. +-rgt-identity 13.1%

       \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2}}}}{a} \]

11. Simplified 13.1%

    \[\leadsto \frac{\color{blue}{\frac{-{b}^{3}}{{b}^{2}}}}{a} \]
12. Step-by-step derivation

    1. div-inv 13.1%

       \[\leadsto \color{blue}{\frac{-{b}^{3}}{{b}^{2}} \cdot \frac{1}{a}} \]

    2. add-sqr-sqrt 12.4%

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

    3. sqrt-unprod 13.1%

       \[\leadsto \color{blue}{\sqrt{\frac{-{b}^{3}}{{b}^{2}} \cdot \frac{-{b}^{3}}{{b}^{2}}}} \cdot \frac{1}{a} \]

    4. distribute-frac-neg 13.1%

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

    5. pow-div 13.1%

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

    6. metadata-eval 13.1%

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

    7. pow1 13.1%

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

    8. distribute-frac-neg 13.1%

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

    9. pow-div 29.1%

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

    10. metadata-eval 29.1%

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

    11. pow1 29.1%

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

    12. sqr-neg 29.1%

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

    13. unpow2 29.1%

        \[\leadsto \sqrt{\color{blue}{{b}^{2}}} \cdot \frac{1}{a} \]

    14. sqrt-pow1 2.3%

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

    15. metadata-eval 2.3%

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

    16. pow1 2.3%

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

13. Applied egg-rr 2.3%

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

    1. associate-*r/ 2.3%

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

    2. *-rgt-identity 2.3%

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

15. Simplified 2.3%

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

17. Applied egg-rr 2.2%

    \[\leadsto \color{blue}{e^{\log a - \log b}} \]
18. Step-by-step derivation

    1. exp-diff 2.2%

       \[\leadsto \color{blue}{\frac{e^{\log a}}{e^{\log b}}} \]

    2. rem-exp-log 4.2%

       \[\leadsto \frac{\color{blue}{a}}{e^{\log b}} \]

    3. rem-exp-log 5.0%

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

19. Simplified 5.0%

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

Alternative 10: 2.4% accurate, 38.7× speedup

Math

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

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 53.6%

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

   1. *-commutative 53.6%

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

3. Simplified 53.7%

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

5. Taylor expanded in b around -inf 40.9%

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

   1. associate-*r/ 40.9%

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

   2. mul-1-neg 40.9%

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

7. Simplified 40.9%

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

   1. neg-sub0 40.9%

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

   2. flip3-- 13.1%

      \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}}}{a} \]

   3. metadata-eval 13.1%

      \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}}{a} \]

   4. metadata-eval 13.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)}}{a} \]

   5. pow2 13.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)}}{a} \]

9. Applied egg-rr 13.1%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}}}{a} \]
10. Step-by-step derivation

    1. sub0-neg 13.1%

       \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)}}{a} \]

    2. +-lft-identity 13.1%

       \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}}}{a} \]

    3. mul0-lft 13.1%

       \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}}}{a} \]

    4. +-rgt-identity 13.1%

       \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2}}}}{a} \]

11. Simplified 13.1%

    \[\leadsto \frac{\color{blue}{\frac{-{b}^{3}}{{b}^{2}}}}{a} \]
12. Step-by-step derivation

    1. div-inv 13.1%

       \[\leadsto \color{blue}{\frac{-{b}^{3}}{{b}^{2}} \cdot \frac{1}{a}} \]

    2. add-sqr-sqrt 12.4%

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

    3. sqrt-unprod 13.1%

       \[\leadsto \color{blue}{\sqrt{\frac{-{b}^{3}}{{b}^{2}} \cdot \frac{-{b}^{3}}{{b}^{2}}}} \cdot \frac{1}{a} \]

    4. distribute-frac-neg 13.1%

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

    5. pow-div 13.1%

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

    6. metadata-eval 13.1%

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

    7. pow1 13.1%

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

    8. distribute-frac-neg 13.1%

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

    9. pow-div 29.1%

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

    10. metadata-eval 29.1%

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

    11. pow1 29.1%

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

    12. sqr-neg 29.1%

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

    13. unpow2 29.1%

        \[\leadsto \sqrt{\color{blue}{{b}^{2}}} \cdot \frac{1}{a} \]

    14. sqrt-pow1 2.3%

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

    15. metadata-eval 2.3%

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

    16. pow1 2.3%

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

13. Applied egg-rr 2.3%

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

    1. associate-*r/ 2.3%

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

    2. *-rgt-identity 2.3%

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

15. Simplified 2.3%

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

17. Applied egg-rr 2.5%

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

    1. *-commutative 2.5%

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

19. Simplified 2.5%

    \[\leadsto \color{blue}{b \cdot a} \]
20. Add Preprocessing

Developer Target 1: 70.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - t\_0}{2 \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (< b 0.0)
     (/ (+ (- b) t_0) (* 2.0 a))
     (/ c (* a (/ (- (- b) t_0) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b < 0.0d0) then
        tmp = (-b + t_0) / (2.0d0 * a)
    else
        tmp = c / (a * ((-b - t_0) / (2.0d0 * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b < 0.0:
		tmp = (-b + t_0) / (2.0 * a)
	else:
		tmp = c / (a * ((-b - t_0) / (2.0 * a)))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	else
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b < 0.0)
		tmp = (-b + t_0) / (2.0 * a);
	else
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c / N[(a * N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024152 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (- (* b b) (* (* 4 a) c)))) (let ((r1 (/ (+ (- b) (sqrt d)) (* 2 a)))) (let ((r2 (/ (- (- b) (sqrt d)) (* 2 a)))) (if (< b 0) r1 (/ c (* a r2)))))))

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