The quadratic formula (r1)

Percentage Accurate: 52.0% → 85.9%

Time: 11.2s

Alternatives: 9

Speedup: 19.1×

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: 52.0% 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: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+124}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-46}:\\ \;\;\;\;\frac{\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot -0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e+124)
   (- (/ c b) (/ b a))
   (if (<= b 1.15e-46)
     (/ (* (- b (sqrt (- (* b b) (* a (* c 4.0))))) -0.5) a)
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e+124) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.15e-46) {
		tmp = ((b - sqrt(((b * b) - (a * (c * 4.0))))) * -0.5) / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.6d+124)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.15d-46) then
        tmp = ((b - sqrt(((b * b) - (a * (c * 4.0d0))))) * (-0.5d0)) / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e+124) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.15e-46) {
		tmp = ((b - Math.sqrt(((b * b) - (a * (c * 4.0))))) * -0.5) / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.6e+124:
		tmp = (c / b) - (b / a)
	elif b <= 1.15e-46:
		tmp = ((b - math.sqrt(((b * b) - (a * (c * 4.0))))) * -0.5) / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e+124)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.15e-46)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 4.0))))) * -0.5) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.6e+124)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.15e-46)
		tmp = ((b - sqrt(((b * b) - (a * (c * 4.0))))) * -0.5) / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e+124], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.15e-46], N[(N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.59999999999999986e124

   1. Initial program 37.2%

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

      1. neg-sub0 37.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 37.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 37.2%

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

      4. neg-mul-1 37.2%

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

      5. associate-*l/ 37.2%

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

      6. *-commutative 37.2%

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

      7. associate-/r* 37.2%

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

      8. /-rgt-identity 37.2%

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

      9. metadata-eval 37.2%

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

   3. Simplified 37.3%

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

   4. Taylor expanded in b around -inf 96.2%

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

      1. mul-1-neg 96.2%

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

      2. unsub-neg 96.2%

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

   6. Simplified 96.2%

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

   if -3.59999999999999986e124 < b < 1.15e-46

   1. Initial program 79.2%

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

      1. neg-sub0 79.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 79.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 79.2%

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

      4. neg-mul-1 79.2%

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

      5. associate-*l/ 79.0%

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

      6. *-commutative 79.0%

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

      7. associate-/r* 79.0%

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

      8. /-rgt-identity 79.0%

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

      9. metadata-eval 79.0%

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

   3. Simplified 79.0%

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

      1. fma-udef 79.0%

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

      2. *-commutative 79.0%

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

      3. associate-*r* 79.0%

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

      4. metadata-eval 79.0%

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

      5. distribute-rgt-neg-in 79.0%

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

      6. *-commutative 79.0%

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

      7. distribute-lft-neg-in 79.0%

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

      8. +-commutative 79.0%

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

      9. sub-neg 79.0%

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

      10. *-commutative 79.0%

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

      11. associate-*l* 79.0%

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

   5. Applied egg-rr 79.0%

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

      1. *-commutative 79.0%

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

   7. Simplified 79.0%

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

      1. associate-*r/ 79.2%

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

   9. Applied egg-rr 79.2%

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

   if 1.15e-46 < b

   1. Initial program 14.8%

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

      1. neg-sub0 14.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 14.8%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 14.8%

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

      4. neg-mul-1 14.8%

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

      5. associate-*l/ 14.8%

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

      6. *-commutative 14.8%

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

      7. associate-/r* 14.8%

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

      8. /-rgt-identity 14.8%

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

      9. metadata-eval 14.8%

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

   3. Simplified 14.8%

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

   4. Taylor expanded in b around inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. distribute-neg-frac 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+124}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-46}:\\ \;\;\;\;\frac{\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot -0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{+74}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.25e+74)
   (- (/ c b) (/ b a))
   (if (<= b 1.3e-46)
     (* (- b (sqrt (- (* b b) (* a (* c 4.0))))) (/ -0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e+74) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.3e-46) {
		tmp = (b - sqrt(((b * b) - (a * (c * 4.0))))) * (-0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.25d+74)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.3d-46) then
        tmp = (b - sqrt(((b * b) - (a * (c * 4.0d0))))) * ((-0.5d0) / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e+74) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.3e-46) {
		tmp = (b - Math.sqrt(((b * b) - (a * (c * 4.0))))) * (-0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.25e+74:
		tmp = (c / b) - (b / a)
	elif b <= 1.3e-46:
		tmp = (b - math.sqrt(((b * b) - (a * (c * 4.0))))) * (-0.5 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.25e+74)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.3e-46)
		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);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.25e+74)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.3e-46)
		tmp = (b - sqrt(((b * b) - (a * (c * 4.0))))) * (-0.5 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.25e+74], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.3e-46], 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[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.24999999999999991e74

   1. Initial program 48.5%

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

      1. neg-sub0 48.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 48.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 48.5%

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

      4. neg-mul-1 48.5%

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

      5. associate-*l/ 48.5%

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

      6. *-commutative 48.5%

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

      7. associate-/r* 48.5%

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

      8. /-rgt-identity 48.5%

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

      9. metadata-eval 48.5%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in b around -inf 96.9%

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

      1. mul-1-neg 96.9%

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

      2. unsub-neg 96.9%

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

   6. Simplified 96.9%

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

   if -1.24999999999999991e74 < b < 1.3000000000000001e-46

   1. Initial program 77.2%

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

      1. neg-sub0 77.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 77.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 77.2%

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

      4. neg-mul-1 77.2%

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

      5. associate-*l/ 77.1%

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

      6. *-commutative 77.1%

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

      7. associate-/r* 77.1%

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

      8. /-rgt-identity 77.1%

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

      9. metadata-eval 77.1%

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

   3. Simplified 77.1%

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

      1. fma-udef 77.1%

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

      2. *-commutative 77.1%

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

      3. associate-*r* 77.1%

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

      4. metadata-eval 77.1%

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

      5. distribute-rgt-neg-in 77.1%

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

      6. *-commutative 77.1%

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

      7. distribute-lft-neg-in 77.1%

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

      8. +-commutative 77.1%

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

      9. sub-neg 77.1%

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

      10. *-commutative 77.1%

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

      11. associate-*l* 77.1%

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

   5. Applied egg-rr 77.1%

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

      1. *-commutative 77.1%

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

   7. Simplified 77.1%

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

   if 1.3000000000000001e-46 < b

   1. Initial program 14.8%

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

      1. neg-sub0 14.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 14.8%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 14.8%

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

      4. neg-mul-1 14.8%

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

      5. associate-*l/ 14.8%

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

      6. *-commutative 14.8%

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

      7. associate-/r* 14.8%

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

      8. /-rgt-identity 14.8%

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

      9. metadata-eval 14.8%

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

   3. Simplified 14.8%

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

   4. Taylor expanded in b around inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. distribute-neg-frac 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{+74}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{-116}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.6e-116)
   (- (/ c b) (/ b a))
   (if (<= b 1.5e-46)
     (* (/ -0.5 a) (- b (sqrt (* a (* c -4.0)))))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.6e-116) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.5e-46) {
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	} 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 <= (-8.6d-116)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.5d-46) then
        tmp = ((-0.5d0) / a) * (b - sqrt((a * (c * (-4.0d0)))))
    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 <= -8.6e-116) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.5e-46) {
		tmp = (-0.5 / a) * (b - Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.6e-116:
		tmp = (c / b) - (b / a)
	elif b <= 1.5e-46:
		tmp = (-0.5 / a) * (b - math.sqrt((a * (c * -4.0))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.6e-116)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.5e-46)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.6e-116)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.5e-46)
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.6e-116], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.5e-46], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.5999999999999994e-116

   1. Initial program 67.6%

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

      1. neg-sub0 67.6%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 67.6%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 67.6%

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

      4. neg-mul-1 67.6%

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

      5. associate-*l/ 67.4%

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

      6. *-commutative 67.4%

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

      7. associate-/r* 67.4%

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

      8. /-rgt-identity 67.4%

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

      9. metadata-eval 67.4%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in b around -inf 86.3%

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

      1. mul-1-neg 86.3%

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

      2. unsub-neg 86.3%

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

   6. Simplified 86.3%

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

   if -8.5999999999999994e-116 < b < 1.49999999999999994e-46

   1. Initial program 67.5%

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

      1. neg-sub0 67.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 67.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 67.5%

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

      4. neg-mul-1 67.5%

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

      5. associate-*l/ 67.5%

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

      6. *-commutative 67.5%

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

      7. associate-/r* 67.5%

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

      8. /-rgt-identity 67.5%

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

      9. metadata-eval 67.5%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in a around inf 65.5%

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

      1. *-commutative 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*r* 65.5%

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

   6. Simplified 65.5%

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

   if 1.49999999999999994e-46 < b

   1. Initial program 14.8%

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

      1. neg-sub0 14.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 14.8%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 14.8%

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

      4. neg-mul-1 14.8%

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

      5. associate-*l/ 14.8%

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

      6. *-commutative 14.8%

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

      7. associate-/r* 14.8%

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

      8. /-rgt-identity 14.8%

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

      9. metadata-eval 14.8%

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

   3. Simplified 14.8%

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

   4. Taylor expanded in b around inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. distribute-neg-frac 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{-116}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-123}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-123)
   (- (/ c b) (/ b a))
   (if (<= b 1.5e-46)
     (/ (* c -2.0) (+ b (sqrt (* a (* c -4.0)))))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-123) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.5e-46) {
		tmp = (c * -2.0) / (b + sqrt((a * (c * -4.0))));
	} 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 <= (-9d-123)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.5d-46) then
        tmp = (c * (-2.0d0)) / (b + sqrt((a * (c * (-4.0d0)))))
    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 <= -9e-123) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.5e-46) {
		tmp = (c * -2.0) / (b + Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9e-123:
		tmp = (c / b) - (b / a)
	elif b <= 1.5e-46:
		tmp = (c * -2.0) / (b + math.sqrt((a * (c * -4.0))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9e-123)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.5e-46)
		tmp = Float64(Float64(c * -2.0) / Float64(b + sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9e-123)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.5e-46)
		tmp = (c * -2.0) / (b + sqrt((a * (c * -4.0))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9e-123], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.5e-46], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.99999999999999986e-123

   1. Initial program 67.6%

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

      1. neg-sub0 67.6%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 67.6%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 67.6%

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

      4. neg-mul-1 67.6%

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

      5. associate-*l/ 67.4%

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

      6. *-commutative 67.4%

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

      7. associate-/r* 67.4%

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

      8. /-rgt-identity 67.4%

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

      9. metadata-eval 67.4%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in b around -inf 86.3%

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

      1. mul-1-neg 86.3%

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

      2. unsub-neg 86.3%

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

   6. Simplified 86.3%

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

   if -8.99999999999999986e-123 < b < 1.49999999999999994e-46

   1. Initial program 67.5%

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

      1. neg-sub0 67.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 67.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 67.5%

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

      4. neg-mul-1 67.5%

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

      5. associate-*l/ 67.5%

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

      6. *-commutative 67.5%

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

      7. associate-/r* 67.5%

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

      8. /-rgt-identity 67.5%

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

      9. metadata-eval 67.5%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in a around inf 65.5%

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

      1. *-commutative 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*r* 65.5%

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

   6. Simplified 65.5%

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

      1. flip-- 64.7%

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

      2. add-sqr-sqrt 64.6%

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

   8. Applied egg-rr 64.6%

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

      1. associate-*l/ 64.6%

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

   10. Applied egg-rr 64.6%

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

   11. Taylor expanded in b around 0 66.9%

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

       1. *-commutative 66.9%

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

   13. Simplified 66.9%

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

   if 1.49999999999999994e-46 < b

   1. Initial program 14.8%

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

      1. neg-sub0 14.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 14.8%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 14.8%

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

      4. neg-mul-1 14.8%

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

      5. associate-*l/ 14.8%

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

      6. *-commutative 14.8%

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

      7. associate-/r* 14.8%

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

      8. /-rgt-identity 14.8%

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

      9. metadata-eval 14.8%

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

   3. Simplified 14.8%

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

   4. Taylor expanded in b around inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. distribute-neg-frac 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-123}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 68.1% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 69.4%

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

      1. neg-sub0 69.4%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 69.4%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 69.4%

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

      4. neg-mul-1 69.4%

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

      5. associate-*l/ 69.3%

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

      6. *-commutative 69.3%

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

      7. associate-/r* 69.3%

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

      8. /-rgt-identity 69.3%

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

      9. metadata-eval 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in b around -inf 68.6%

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

      1. mul-1-neg 68.6%

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

      2. unsub-neg 68.6%

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

   6. Simplified 68.6%

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

   if -9.999999999999969e-311 < b

   1. Initial program 32.2%

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

      1. neg-sub0 32.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 32.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 32.2%

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

      4. neg-mul-1 32.2%

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

      5. associate-*l/ 32.2%

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

      6. *-commutative 32.2%

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

      7. associate-/r* 32.2%

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

      8. /-rgt-identity 32.2%

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

      9. metadata-eval 32.2%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in b around inf 62.5%

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

      1. mul-1-neg 62.5%

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

      2. distribute-neg-frac 62.5%

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

   6. Simplified 62.5%

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

4. Final simplification 65.7%

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

Alternative 6: 43.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.25 \cdot 10^{-46}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 2.25e-46) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.25e-46) {
		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 <= 2.25d-46) 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 <= 2.25e-46) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.25e-46:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.25e-46)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.25e-46)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.25e-46], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.25e-46

   1. Initial program 67.6%

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

      1. neg-sub0 67.6%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 67.6%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 67.6%

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

      4. neg-mul-1 67.6%

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

      5. associate-*l/ 67.5%

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

      6. *-commutative 67.5%

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

      7. associate-/r* 67.5%

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

      8. /-rgt-identity 67.5%

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

      9. metadata-eval 67.5%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in b around -inf 52.4%

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

      1. associate-*r/ 52.4%

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

      2. mul-1-neg 52.4%

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

   6. Simplified 52.4%

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

   if 2.25e-46 < b

   1. Initial program 14.8%

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

   2. Taylor expanded in b around -inf 2.3%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}}{2 \cdot a} \]

   3. Taylor expanded in c around inf 20.5%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.25 \cdot 10^{-46}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 7: 67.9% accurate, 19.1× 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(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / 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 69.4%

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

      1. neg-sub0 69.4%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 69.4%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 69.4%

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

      4. neg-mul-1 69.4%

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

      5. associate-*l/ 69.3%

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

      6. *-commutative 69.3%

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

      7. associate-/r* 69.3%

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

      8. /-rgt-identity 69.3%

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

      9. metadata-eval 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in b around -inf 68.1%

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

      1. associate-*r/ 68.1%

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

      2. mul-1-neg 68.1%

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

   6. Simplified 68.1%

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

   if -9.999999999999969e-311 < b

   1. Initial program 32.2%

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

      1. neg-sub0 32.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 32.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 32.2%

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

      4. neg-mul-1 32.2%

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

      5. associate-*l/ 32.2%

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

      6. *-commutative 32.2%

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

      7. associate-/r* 32.2%

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

      8. /-rgt-identity 32.2%

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

      9. metadata-eval 32.2%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in b around inf 62.5%

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

      1. mul-1-neg 62.5%

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

      2. distribute-neg-frac 62.5%

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

   6. Simplified 62.5%

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

4. Final simplification 65.5%

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

Alternative 8: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

   1. neg-sub0 52.1%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 52.1%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 52.1%

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

   4. neg-mul-1 52.1%

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

   5. associate-*l/ 52.0%

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

   6. *-commutative 52.0%

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

   7. associate-/r* 52.0%

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

   8. /-rgt-identity 52.0%

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

   9. metadata-eval 52.0%

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

3. Simplified 52.0%

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

   1. associate-*r/ 52.1%

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

   2. clear-num 52.0%

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

5. Applied egg-rr 52.0%

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

6. Taylor expanded in a around 0 30.3%

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

   1. mul-1-neg 30.3%

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

   2. unsub-neg 30.3%

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

8. Simplified 30.3%

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

9. Taylor expanded in a around inf 2.5%

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

10. Final simplification 2.5%

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

Alternative 9: 11.3% 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 52.1%

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

2. Taylor expanded in b around -inf 35.7%

   \[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}}{2 \cdot a} \]

3. Taylor expanded in c around inf 8.1%

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

4. Final simplification 8.1%

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

Developer target: 70.8% accurate, 1.0× 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 2023171 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

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