Quadratic roots, full range

Percentage Accurate: 51.9% → 85.6%

Time: 11.0s

Alternatives: 6

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: 51.9% 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: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot -0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+153)
   (/ (- b) a)
   (if (<= b 2e-54)
     (/ (* (- 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 <= -1e+153) {
		tmp = -b / a;
	} else if (b <= 2e-54) {
		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 <= (-1d+153)) then
        tmp = -b / a
    else if (b <= 2d-54) 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 <= -1e+153) {
		tmp = -b / a;
	} else if (b <= 2e-54) {
		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 <= -1e+153:
		tmp = -b / a
	elif b <= 2e-54:
		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 <= -1e+153)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2e-54)
		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 <= -1e+153)
		tmp = -b / a;
	elseif (b <= 2e-54)
		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, -1e+153], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2e-54], 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 < -1e153

   1. Initial program 38.9%

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

      1. neg-sub0 38.9%

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

      2. associate-+l- 38.9%

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

      3. sub0-neg 38.9%

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

      4. neg-mul-1 38.9%

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

      5. associate-*l/ 38.9%

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

      6. *-commutative 38.9%

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

      7. associate-/r* 38.9%

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

      8. /-rgt-identity 38.9%

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

      9. metadata-eval 38.9%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -1e153 < b < 2.0000000000000001e-54

   1. Initial program 84.4%

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

      1. neg-sub0 84.4%

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

      2. associate-+l- 84.4%

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

      3. sub0-neg 84.4%

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

      4. neg-mul-1 84.4%

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

      5. associate-*l/ 84.3%

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

      6. *-commutative 84.3%

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

      7. associate-/r* 84.3%

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

      8. /-rgt-identity 84.3%

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

      9. metadata-eval 84.3%

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

   3. Simplified 84.3%

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

      1. fma-udef 84.3%

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

      2. *-commutative 84.3%

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

      3. associate-*r* 84.3%

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

      4. metadata-eval 84.3%

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

      5. distribute-rgt-neg-in 84.3%

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

      6. *-commutative 84.3%

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

      7. distribute-lft-neg-in 84.3%

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

      8. +-commutative 84.3%

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

      9. sub-neg 84.3%

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

      10. *-commutative 84.3%

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

      11. associate-*l* 84.3%

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

   5. Applied egg-rr 84.3%

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

      1. *-commutative 84.3%

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

   7. Simplified 84.3%

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

      1. associate-*r/ 84.4%

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

   9. Applied egg-rr 84.4%

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

   if 2.0000000000000001e-54 < b

   1. Initial program 15.9%

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

      1. neg-sub0 15.9%

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

      2. associate-+l- 15.9%

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

      3. sub0-neg 15.9%

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

      4. neg-mul-1 15.9%

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

      5. associate-*l/ 15.9%

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

      6. *-commutative 15.9%

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

      7. associate-/r* 15.9%

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

      8. /-rgt-identity 15.9%

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

      9. metadata-eval 15.9%

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

   3. Simplified 16.0%

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

   4. Taylor expanded in b around inf 93.2%

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

      1. mul-1-neg 93.2%

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

      2. distribute-neg-frac 93.2%

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

   6. Simplified 93.2%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot -0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-b}{a}\\ t_1 := \frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{if}\;b \leq -1.32 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- b) a)) (t_1 (* (/ -0.5 a) (- b (sqrt (* a (* c -4.0)))))))
   (if (<= b -1.32e-43)
     t_0
     (if (<= b -6.2e-106)
       t_1
       (if (<= b -1.32e-129) t_0 (if (<= b 3.9e-55) t_1 (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = -b / a;
	double t_1 = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	double tmp;
	if (b <= -1.32e-43) {
		tmp = t_0;
	} else if (b <= -6.2e-106) {
		tmp = t_1;
	} else if (b <= -1.32e-129) {
		tmp = t_0;
	} else if (b <= 3.9e-55) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -b / a
    t_1 = ((-0.5d0) / a) * (b - sqrt((a * (c * (-4.0d0)))))
    if (b <= (-1.32d-43)) then
        tmp = t_0
    else if (b <= (-6.2d-106)) then
        tmp = t_1
    else if (b <= (-1.32d-129)) then
        tmp = t_0
    else if (b <= 3.9d-55) then
        tmp = t_1
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -b / a;
	double t_1 = (-0.5 / a) * (b - Math.sqrt((a * (c * -4.0))));
	double tmp;
	if (b <= -1.32e-43) {
		tmp = t_0;
	} else if (b <= -6.2e-106) {
		tmp = t_1;
	} else if (b <= -1.32e-129) {
		tmp = t_0;
	} else if (b <= 3.9e-55) {
		tmp = t_1;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -b / a
	t_1 = (-0.5 / a) * (b - math.sqrt((a * (c * -4.0))))
	tmp = 0
	if b <= -1.32e-43:
		tmp = t_0
	elif b <= -6.2e-106:
		tmp = t_1
	elif b <= -1.32e-129:
		tmp = t_0
	elif b <= 3.9e-55:
		tmp = t_1
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) / a)
	t_1 = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(a * Float64(c * -4.0)))))
	tmp = 0.0
	if (b <= -1.32e-43)
		tmp = t_0;
	elseif (b <= -6.2e-106)
		tmp = t_1;
	elseif (b <= -1.32e-129)
		tmp = t_0;
	elseif (b <= 3.9e-55)
		tmp = t_1;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -b / a;
	t_1 = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	tmp = 0.0;
	if (b <= -1.32e-43)
		tmp = t_0;
	elseif (b <= -6.2e-106)
		tmp = t_1;
	elseif (b <= -1.32e-129)
		tmp = t_0;
	elseif (b <= 3.9e-55)
		tmp = t_1;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) / a), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.32e-43], t$95$0, If[LessEqual[b, -6.2e-106], t$95$1, If[LessEqual[b, -1.32e-129], t$95$0, If[LessEqual[b, 3.9e-55], t$95$1, N[((-c) / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.32000000000000002e-43 or -6.19999999999999971e-106 < b < -1.31999999999999992e-129

   1. Initial program 66.3%

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

      1. neg-sub0 66.3%

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

      2. associate-+l- 66.3%

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

      3. sub0-neg 66.3%

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

      4. neg-mul-1 66.3%

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

      5. associate-*l/ 66.1%

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

      6. *-commutative 66.1%

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

      7. associate-/r* 66.1%

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

      8. /-rgt-identity 66.1%

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

      9. metadata-eval 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in b around -inf 86.3%

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

      1. associate-*r/ 86.3%

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

      2. mul-1-neg 86.3%

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

   6. Simplified 86.3%

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

   if -1.32000000000000002e-43 < b < -6.19999999999999971e-106 or -1.31999999999999992e-129 < b < 3.9e-55

   1. Initial program 80.6%

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

      1. neg-sub0 80.6%

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

      2. associate-+l- 80.6%

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

      3. sub0-neg 80.6%

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

      4. neg-mul-1 80.6%

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

      5. associate-*l/ 80.6%

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

      6. *-commutative 80.6%

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

      7. associate-/r* 80.6%

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

      8. /-rgt-identity 80.6%

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

      9. metadata-eval 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in a around inf 76.7%

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

      1. *-commutative 76.7%

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

      2. *-commutative 76.7%

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

      3. associate-*r* 76.7%

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

   6. Simplified 76.7%

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

   if 3.9e-55 < b

   1. Initial program 15.9%

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

      1. neg-sub0 15.9%

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

      2. associate-+l- 15.9%

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

      3. sub0-neg 15.9%

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

      4. neg-mul-1 15.9%

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

      5. associate-*l/ 15.9%

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

      6. *-commutative 15.9%

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

      7. associate-/r* 15.9%

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

      8. /-rgt-identity 15.9%

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

      9. metadata-eval 15.9%

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

   3. Simplified 16.0%

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

   4. Taylor expanded in b around inf 93.2%

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

      1. mul-1-neg 93.2%

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

      2. distribute-neg-frac 93.2%

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

   6. Simplified 93.2%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-43}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-55}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{+126}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-51}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.22e+126)
   (/ (- b) a)
   (if (<= b 1.95e-51)
     (* (- 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 <= -1.22e+126) {
		tmp = -b / a;
	} else if (b <= 1.95e-51) {
		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 <= (-1.22d+126)) then
        tmp = -b / a
    else if (b <= 1.95d-51) 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 <= -1.22e+126) {
		tmp = -b / a;
	} else if (b <= 1.95e-51) {
		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 <= -1.22e+126:
		tmp = -b / a
	elif b <= 1.95e-51:
		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 <= -1.22e+126)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.95e-51)
		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 <= -1.22e+126)
		tmp = -b / a;
	elseif (b <= 1.95e-51)
		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, -1.22e+126], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.95e-51], 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 < -1.21999999999999995e126

   1. Initial program 48.1%

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

      1. neg-sub0 48.1%

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

      2. associate-+l- 48.1%

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

      3. sub0-neg 48.1%

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

      4. neg-mul-1 48.1%

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

      5. associate-*l/ 48.1%

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

      6. *-commutative 48.1%

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

      7. associate-/r* 48.1%

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

      8. /-rgt-identity 48.1%

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

      9. metadata-eval 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -1.21999999999999995e126 < b < 1.9499999999999999e-51

   1. Initial program 83.4%

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

      1. neg-sub0 83.4%

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

      2. associate-+l- 83.4%

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

      3. sub0-neg 83.4%

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

      4. neg-mul-1 83.4%

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

      5. associate-*l/ 83.3%

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

      6. *-commutative 83.3%

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

      7. associate-/r* 83.3%

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

      8. /-rgt-identity 83.3%

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

      9. metadata-eval 83.3%

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

   3. Simplified 83.3%

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

      1. fma-udef 83.3%

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

      2. *-commutative 83.3%

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

      3. associate-*r* 83.3%

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

      4. metadata-eval 83.3%

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

      5. distribute-rgt-neg-in 83.3%

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

      6. *-commutative 83.3%

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

      7. distribute-lft-neg-in 83.3%

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

      8. +-commutative 83.3%

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

      9. sub-neg 83.3%

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

      10. *-commutative 83.3%

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

      11. associate-*l* 83.3%

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

   5. Applied egg-rr 83.3%

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

      1. *-commutative 83.3%

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

   7. Simplified 83.3%

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

   if 1.9499999999999999e-51 < b

   1. Initial program 15.9%

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

      1. neg-sub0 15.9%

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

      2. associate-+l- 15.9%

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

      3. sub0-neg 15.9%

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

      4. neg-mul-1 15.9%

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

      5. associate-*l/ 15.9%

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

      6. *-commutative 15.9%

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

      7. associate-/r* 15.9%

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

      8. /-rgt-identity 15.9%

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

      9. metadata-eval 15.9%

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

   3. Simplified 16.0%

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

   4. Taylor expanded in b around inf 93.2%

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

      1. mul-1-neg 93.2%

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

      2. distribute-neg-frac 93.2%

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

   6. Simplified 93.2%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{+126}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-51}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 7.4 \cdot 10^{-54}:\\ \;\;\;\;-0.5 \cdot \frac{-\sqrt{c \cdot \frac{a}{-0.25}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.32e-129)
   (/ (- b) a)
   (if (<= b 7.4e-54)
     (* -0.5 (/ (- (sqrt (* c (/ a -0.25)))) a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.32e-129) {
		tmp = -b / a;
	} else if (b <= 7.4e-54) {
		tmp = -0.5 * (-sqrt((c * (a / -0.25))) / 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 <= (-1.32d-129)) then
        tmp = -b / a
    else if (b <= 7.4d-54) then
        tmp = (-0.5d0) * (-sqrt((c * (a / (-0.25d0)))) / 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 <= -1.32e-129) {
		tmp = -b / a;
	} else if (b <= 7.4e-54) {
		tmp = -0.5 * (-Math.sqrt((c * (a / -0.25))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.32e-129:
		tmp = -b / a
	elif b <= 7.4e-54:
		tmp = -0.5 * (-math.sqrt((c * (a / -0.25))) / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.32e-129)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 7.4e-54)
		tmp = Float64(-0.5 * Float64(Float64(-sqrt(Float64(c * Float64(a / -0.25)))) / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.32e-129)
		tmp = -b / a;
	elseif (b <= 7.4e-54)
		tmp = -0.5 * (-sqrt((c * (a / -0.25))) / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.32e-129], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 7.4e-54], N[(-0.5 * N[((-N[Sqrt[N[(c * N[(a / -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.31999999999999992e-129

   1. Initial program 68.5%

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

      1. neg-sub0 68.5%

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

      2. associate-+l- 68.5%

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

      3. sub0-neg 68.5%

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

      4. neg-mul-1 68.5%

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

      5. associate-*l/ 68.3%

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

      6. *-commutative 68.3%

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

      7. associate-/r* 68.3%

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

      8. /-rgt-identity 68.3%

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

      9. metadata-eval 68.3%

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

   3. Simplified 68.3%

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

   4. Taylor expanded in b around -inf 79.3%

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

      1. associate-*r/ 79.3%

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

      2. mul-1-neg 79.3%

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

   6. Simplified 79.3%

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

   if -1.31999999999999992e-129 < b < 7.4000000000000006e-54

   1. Initial program 79.9%

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

      1. neg-sub0 79.9%

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

      2. associate-+l- 79.9%

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

      3. sub0-neg 79.9%

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

      4. neg-mul-1 79.9%

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

      5. associate-*l/ 79.8%

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

      6. *-commutative 79.8%

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

      7. associate-/r* 79.8%

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

      8. /-rgt-identity 79.8%

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

      9. metadata-eval 79.8%

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

   3. Simplified 79.8%

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

      1. fma-udef 79.8%

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

      2. *-commutative 79.8%

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

      3. associate-*r* 79.8%

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

      4. metadata-eval 79.8%

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

      5. distribute-rgt-neg-in 79.8%

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

      6. *-commutative 79.8%

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

      7. distribute-lft-neg-in 79.8%

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

      8. +-commutative 79.8%

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

      9. sub-neg 79.8%

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

      10. add-sqr-sqrt 79.4%

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

      11. pow2 79.4%

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

   5. Applied egg-rr 79.5%

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

   6. Taylor expanded in c around -inf 44.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b - {\left(e^{0.25 \cdot \left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}\right)}^{2}}{a}} \]
   7. Step-by-step derivation

      1. unpow2 44.6%

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

      2. exp-prod 43.5%

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

      3. exp-prod 42.7%

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

      4. pow-sqr 42.7%

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

      5. mul-1-neg 42.7%

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

      6. unsub-neg 42.7%

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

      7. *-commutative 42.7%

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

   8. Simplified 42.7%

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

   9. Taylor expanded in b around 0 44.0%

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

       1. metadata-eval 44.0%

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

       2. log-prod 44.0%

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

       3. +-commutative 44.0%

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

       4. log-prod 44.0%

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

       5. log-div 73.3%

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

       6. associate-*r* 73.3%

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

       7. exp-prod 69.5%

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

       8. associate-*r/ 69.5%

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

   11. Simplified 78.9%

       \[\leadsto -0.5 \cdot \color{blue}{\frac{-\sqrt{c \cdot \frac{a}{-0.25}}}{a}} \]

   if 7.4000000000000006e-54 < b

   1. Initial program 15.9%

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

      1. neg-sub0 15.9%

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

      2. associate-+l- 15.9%

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

      3. sub0-neg 15.9%

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

      4. neg-mul-1 15.9%

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

      5. associate-*l/ 15.9%

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

      6. *-commutative 15.9%

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

      7. associate-/r* 15.9%

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

      8. /-rgt-identity 15.9%

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

      9. metadata-eval 15.9%

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

   3. Simplified 16.0%

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

   4. Taylor expanded in b around inf 93.2%

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

      1. mul-1-neg 93.2%

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

      2. distribute-neg-frac 93.2%

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

   6. Simplified 93.2%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 7.4 \cdot 10^{-54}:\\ \;\;\;\;-0.5 \cdot \frac{-\sqrt{c \cdot \frac{a}{-0.25}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 67.6% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.6e-300) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.6e-300) {
		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 <= 7.6d-300) 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 <= 7.6e-300) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.6e-300:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.6e-300)
		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 <= 7.6e-300)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.6e-300], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.60000000000000026e-300

   1. Initial program 72.5%

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

      1. neg-sub0 72.5%

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

      2. associate-+l- 72.5%

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

      3. sub0-neg 72.5%

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

      4. neg-mul-1 72.5%

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

      5. associate-*l/ 72.3%

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

      6. *-commutative 72.3%

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

      7. associate-/r* 72.3%

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

      8. /-rgt-identity 72.3%

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

      9. metadata-eval 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in b around -inf 65.0%

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

      1. associate-*r/ 65.0%

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

      2. mul-1-neg 65.0%

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

   6. Simplified 65.0%

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

   if 7.60000000000000026e-300 < b

   1. Initial program 36.1%

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

      1. neg-sub0 36.1%

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

      2. associate-+l- 36.1%

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

      3. sub0-neg 36.1%

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

      4. neg-mul-1 36.1%

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

      5. associate-*l/ 36.1%

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

      6. *-commutative 36.1%

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

      7. associate-/r* 36.1%

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

      8. /-rgt-identity 36.1%

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

      9. metadata-eval 36.1%

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

   3. Simplified 36.1%

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

   4. Taylor expanded in b around inf 68.0%

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

      1. mul-1-neg 68.0%

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

      2. distribute-neg-frac 68.0%

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

   6. Simplified 68.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 34.7% accurate, 29.0× 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(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 55.3%

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

   1. neg-sub0 55.3%

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

   2. associate-+l- 55.3%

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

   3. sub0-neg 55.3%

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

   4. neg-mul-1 55.3%

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

   5. associate-*l/ 55.2%

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

   6. *-commutative 55.2%

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

   7. associate-/r* 55.2%

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

   8. /-rgt-identity 55.2%

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

   9. metadata-eval 55.2%

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

3. Simplified 55.2%

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

4. Taylor expanded in b around -inf 35.6%

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

   1. associate-*r/ 35.6%

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

   2. mul-1-neg 35.6%

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

6. Simplified 35.6%

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

7. Final simplification 35.6%

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

Reproduce

herbie shell --seed 2023230 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))