The quadratic formula (r2)

Percentage Accurate: 51.7% → 84.5%

Time: 15.7s

Alternatives: 8

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -37000000000000:\\ \;\;\;\;\frac{c}{0 - b}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+122}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -37000000000000.0)
   (/ c (- 0.0 b))
   (if (<= b 3.8e+122)
     (/ (+ b (sqrt (+ (* b b) (* a (* c -4.0))))) (* a -2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -37000000000000.0) {
		tmp = c / (0.0 - b);
	} else if (b <= 3.8e+122) {
		tmp = (b + sqrt(((b * b) + (a * (c * -4.0))))) / (a * -2.0);
	} else {
		tmp = (c / b) - (b / 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) :: tmp
    if (b <= (-37000000000000.0d0)) then
        tmp = c / (0.0d0 - b)
    else if (b <= 3.8d+122) then
        tmp = (b + sqrt(((b * b) + (a * (c * (-4.0d0)))))) / (a * (-2.0d0))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -37000000000000.0) {
		tmp = c / (0.0 - b);
	} else if (b <= 3.8e+122) {
		tmp = (b + Math.sqrt(((b * b) + (a * (c * -4.0))))) / (a * -2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -37000000000000.0:
		tmp = c / (0.0 - b)
	elif b <= 3.8e+122:
		tmp = (b + math.sqrt(((b * b) + (a * (c * -4.0))))) / (a * -2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -37000000000000.0)
		tmp = Float64(c / Float64(0.0 - b));
	elseif (b <= 3.8e+122)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0))))) / Float64(a * -2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -37000000000000.0)
		tmp = c / (0.0 - b);
	elseif (b <= 3.8e+122)
		tmp = (b + sqrt(((b * b) + (a * (c * -4.0))))) / (a * -2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -37000000000000.0], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e+122], N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7e13

   1. Initial program 5.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 5.5%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 93.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 93.1%

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

   if -3.7e13 < b < 3.7999999999999998e122

   1. Initial program 74.2%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 74.2%

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

   if 3.7999999999999998e122 < b

   1. Initial program 53.6%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 53.6%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{\color{blue}{b}}{a}\right)\right) \]

      6. /-lowering-/.f64 97.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 97.8%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -37000000000000:\\ \;\;\;\;\frac{c}{0 - b}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+122}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.36 \cdot 10^{+14}:\\ \;\;\;\;\frac{c}{0 - b}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+123}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.36e+14)
   (/ c (- 0.0 b))
   (if (<= b 7.5e+123)
     (* (+ b (sqrt (+ (* b b) (* a (* c -4.0))))) (/ -0.5 a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.36e+14) {
		tmp = c / (0.0 - b);
	} else if (b <= 7.5e+123) {
		tmp = (b + sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a);
	} else {
		tmp = (c / b) - (b / 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) :: tmp
    if (b <= (-1.36d+14)) then
        tmp = c / (0.0d0 - b)
    else if (b <= 7.5d+123) then
        tmp = (b + sqrt(((b * b) + (a * (c * (-4.0d0)))))) * ((-0.5d0) / a)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.36e+14) {
		tmp = c / (0.0 - b);
	} else if (b <= 7.5e+123) {
		tmp = (b + Math.sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.36e+14:
		tmp = c / (0.0 - b)
	elif b <= 7.5e+123:
		tmp = (b + math.sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.36e+14)
		tmp = Float64(c / Float64(0.0 - b));
	elseif (b <= 7.5e+123)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0))))) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.36e+14)
		tmp = c / (0.0 - b);
	elseif (b <= 7.5e+123)
		tmp = (b + sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.36e+14], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.5e+123], N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.36e14

   1. Initial program 5.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 5.5%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 93.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 93.1%

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

   if -1.36e14 < b < 7.4999999999999999e123

   1. Initial program 74.2%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 74.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot -2}\right), \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{-2 \cdot a}\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{-2}}{a}\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-2}\right), a\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \left(\sqrt{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 74.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 74.0%

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

   if 7.4999999999999999e123 < b

   1. Initial program 53.6%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 53.6%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{\color{blue}{b}}{a}\right)\right) \]

      6. /-lowering-/.f64 97.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 97.8%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.36 \cdot 10^{+14}:\\ \;\;\;\;\frac{c}{0 - b}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+123}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.85 \cdot 10^{-78}:\\ \;\;\;\;\frac{c}{0 - b}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{\frac{a}{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.85e-78)
   (/ c (- 0.0 b))
   (if (<= b 3.1e-36)
     (/ (+ b (sqrt (* a (* c -4.0)))) (/ a -0.5))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.85e-78) {
		tmp = c / (0.0 - b);
	} else if (b <= 3.1e-36) {
		tmp = (b + sqrt((a * (c * -4.0)))) / (a / -0.5);
	} else {
		tmp = (c / b) - (b / 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) :: tmp
    if (b <= (-3.85d-78)) then
        tmp = c / (0.0d0 - b)
    else if (b <= 3.1d-36) then
        tmp = (b + sqrt((a * (c * (-4.0d0))))) / (a / (-0.5d0))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.85e-78) {
		tmp = c / (0.0 - b);
	} else if (b <= 3.1e-36) {
		tmp = (b + Math.sqrt((a * (c * -4.0)))) / (a / -0.5);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.85e-78:
		tmp = c / (0.0 - b)
	elif b <= 3.1e-36:
		tmp = (b + math.sqrt((a * (c * -4.0)))) / (a / -0.5)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.85e-78)
		tmp = Float64(c / Float64(0.0 - b));
	elseif (b <= 3.1e-36)
		tmp = Float64(Float64(b + sqrt(Float64(a * Float64(c * -4.0)))) / Float64(a / -0.5));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.85e-78)
		tmp = c / (0.0 - b);
	elseif (b <= 3.1e-36)
		tmp = (b + sqrt((a * (c * -4.0)))) / (a / -0.5);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.85e-78], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.1e-36], N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a / -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.85e-78

   1. Initial program 16.6%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 16.6%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 81.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 81.8%

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

   if -3.85e-78 < b < 3.0999999999999999e-36

   1. Initial program 69.4%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 69.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot -2}\right), \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{-2 \cdot a}\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{-2}}{a}\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-2}\right), a\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \left(\sqrt{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 69.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 69.3%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(-4 \cdot a\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-4 \cdot a\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 61.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right) \]

   9. Simplified 61.0%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right), \color{blue}{\left(\frac{a}{\frac{-1}{2}}\right)}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\sqrt{c \cdot \left(a \cdot -4\right)}\right)\right), \left(\frac{\color{blue}{a}}{\frac{-1}{2}}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\sqrt{\left(a \cdot -4\right) \cdot c}\right)\right), \left(\frac{a}{\frac{-1}{2}}\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\sqrt{a \cdot \left(-4 \cdot c\right)}\right)\right), \left(\frac{a}{\frac{-1}{2}}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\sqrt{a \cdot \left(c \cdot -4\right)}\right)\right), \left(\frac{a}{\frac{-1}{2}}\right)\right) \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(a \cdot \left(c \cdot -4\right)\right)\right)\right), \left(\frac{a}{\frac{-1}{2}}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), \left(\frac{a}{\frac{-1}{2}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), \left(\frac{a}{\frac{-1}{2}}\right)\right) \]

       12. /-lowering-/.f64 62.2%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), \mathsf{/.f64}\left(a, \color{blue}{\frac{-1}{2}}\right)\right) \]

   11. Applied egg-rr 62.2%

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

   if 3.0999999999999999e-36 < b

   1. Initial program 75.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 75.5%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{\color{blue}{b}}{a}\right)\right) \]

      6. /-lowering-/.f64 91.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 91.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.85 \cdot 10^{-78}:\\ \;\;\;\;\frac{c}{0 - b}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{\frac{a}{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{c}{0 - b}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e-78)
   (/ c (- 0.0 b))
   (if (<= b 2.8e-40)
     (/ -0.5 (/ a (+ b (sqrt (* -4.0 (* c a))))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-78) {
		tmp = c / (0.0 - b);
	} else if (b <= 2.8e-40) {
		tmp = -0.5 / (a / (b + sqrt((-4.0 * (c * a)))));
	} else {
		tmp = (c / b) - (b / 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) :: tmp
    if (b <= (-3.5d-78)) then
        tmp = c / (0.0d0 - b)
    else if (b <= 2.8d-40) then
        tmp = (-0.5d0) / (a / (b + sqrt(((-4.0d0) * (c * a)))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-78) {
		tmp = c / (0.0 - b);
	} else if (b <= 2.8e-40) {
		tmp = -0.5 / (a / (b + Math.sqrt((-4.0 * (c * a)))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.5e-78:
		tmp = c / (0.0 - b)
	elif b <= 2.8e-40:
		tmp = -0.5 / (a / (b + math.sqrt((-4.0 * (c * a)))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e-78)
		tmp = Float64(c / Float64(0.0 - b));
	elseif (b <= 2.8e-40)
		tmp = Float64(-0.5 / Float64(a / Float64(b + sqrt(Float64(-4.0 * Float64(c * a))))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.5e-78)
		tmp = c / (0.0 - b);
	elseif (b <= 2.8e-40)
		tmp = -0.5 / (a / (b + sqrt((-4.0 * (c * a)))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.5e-78], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.8e-40], N[(-0.5 / N[(a / N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.4999999999999999e-78

   1. Initial program 16.6%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 16.6%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 81.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 81.8%

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

   if -3.4999999999999999e-78 < b < 2.8e-40

   1. Initial program 69.4%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 69.4%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{-2}\right), \color{blue}{\left(\frac{a}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right)}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\color{blue}{a}}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \left(\sqrt{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right)\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right)\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 69.3%

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 69.3%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a \cdot c\right), -4\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 62.1%

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right) \]

   9. Simplified 62.1%

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

   if 2.8e-40 < b

   1. Initial program 75.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 75.5%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{\color{blue}{b}}{a}\right)\right) \]

      6. /-lowering-/.f64 91.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 91.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{c}{0 - b}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{-79}:\\ \;\;\;\;\frac{c}{0 - b}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.65e-79)
   (/ c (- 0.0 b))
   (if (<= b 5.5e-41)
     (* (/ -0.5 a) (+ b (sqrt (* c (* a -4.0)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.65e-79) {
		tmp = c / (0.0 - b);
	} else if (b <= 5.5e-41) {
		tmp = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
	} else {
		tmp = (c / b) - (b / 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) :: tmp
    if (b <= (-2.65d-79)) then
        tmp = c / (0.0d0 - b)
    else if (b <= 5.5d-41) then
        tmp = ((-0.5d0) / a) * (b + sqrt((c * (a * (-4.0d0)))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.65e-79) {
		tmp = c / (0.0 - b);
	} else if (b <= 5.5e-41) {
		tmp = (-0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.65e-79:
		tmp = c / (0.0 - b)
	elif b <= 5.5e-41:
		tmp = (-0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.65e-79)
		tmp = Float64(c / Float64(0.0 - b));
	elseif (b <= 5.5e-41)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.65e-79)
		tmp = c / (0.0 - b);
	elseif (b <= 5.5e-41)
		tmp = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.65e-79], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.5e-41], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.6499999999999999e-79

   1. Initial program 16.6%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 16.6%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 81.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 81.8%

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

   if -2.6499999999999999e-79 < b < 5.50000000000000022e-41

   1. Initial program 69.4%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 69.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot -2}\right), \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{-2 \cdot a}\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{-2}}{a}\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-2}\right), a\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \left(\sqrt{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 69.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 69.3%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(-4 \cdot a\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-4 \cdot a\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 61.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right) \]

   9. Simplified 61.0%

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

   if 5.50000000000000022e-41 < b

   1. Initial program 75.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 75.5%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{\color{blue}{b}}{a}\right)\right) \]

      6. /-lowering-/.f64 91.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 91.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{-79}:\\ \;\;\;\;\frac{c}{0 - b}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{0 - b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311) (/ c (- 0.0 b)) (- 0.0 (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = c / (0.0 - b);
	} else {
		tmp = 0.0 - (b / 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) :: tmp
    if (b <= (-5d-311)) then
        tmp = c / (0.0d0 - b)
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = c / (0.0 - b);
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = c / (0.0 - b)
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(c / Float64(0.0 - b));
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = c / (0.0 - b);
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 31.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 31.8%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 62.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 62.9%

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

   if -5.00000000000023e-311 < b

   1. Initial program 75.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 75.5%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 69.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 69.0%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 69.0%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 69.0%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{0 - b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+26) (/ c b) (- 0.0 (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+26) {
		tmp = c / b;
	} else {
		tmp = 0.0 - (b / 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) :: tmp
    if (b <= (-8.5d+26)) then
        tmp = c / b
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+26) {
		tmp = c / b;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e+26:
		tmp = c / b
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+26)
		tmp = Float64(c / b);
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e+26)
		tmp = c / b;
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+26], N[(c / b), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -8.5e26

   1. Initial program 5.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 5.5%

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

   5. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot b + \frac{a \cdot c}{b}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot c}{b} + -1 \cdot b\right), a\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot c}{b} + \left(\mathsf{neg}\left(b\right)\right)\right), a\right) \]

      4. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot c}{b} - b\right), a\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{a \cdot c}{b}\right), b\right), a\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), b\right), b\right), a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(c \cdot a\right), b\right), b\right), a\right) \]

      8. *-lowering-*.f64 2.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\right), b\right), a\right) \]

   7. Simplified 2.5%

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

   8. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 31.2%

         \[\leadsto \mathsf{/.f64}\left(c, \color{blue}{b}\right) \]

   10. Simplified 31.2%

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

   if -8.5e26 < b

   1. Initial program 69.3%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 69.3%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 45.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 45.5%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 45.5%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 45.5%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 10.6% 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 52.6%

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

   1. sub-neg N/A

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

   2. distribute-neg-out N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

3. Simplified 52.6%

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

5. Taylor expanded in a around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot b + \frac{a \cdot c}{b}\right), \color{blue}{a}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot c}{b} + -1 \cdot b\right), a\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot c}{b} + \left(\mathsf{neg}\left(b\right)\right)\right), a\right) \]

   4. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{a \cdot c}{b} - b\right), a\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{a \cdot c}{b}\right), b\right), a\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), b\right), b\right), a\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(c \cdot a\right), b\right), b\right), a\right) \]

   8. *-lowering-*.f64 32.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\right), b\right), a\right) \]

7. Simplified 32.8%

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

8. Taylor expanded in c around inf

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

   1. /-lowering-/.f64 10.2%

      \[\leadsto \mathsf{/.f64}\left(c, \color{blue}{b}\right) \]

10. Simplified 10.2%

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

Developer Target 1: 69.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 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 = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-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 = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-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 = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-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 = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a))));
	else
		tmp = 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 = c / (a * ((-b + t_0) / (2.0 * a)));
	else
		tmp = (-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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024155 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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