The quadratic formula (r2)

Percentage Accurate: 51.5% → 85.4%

Time: 15.5s

Alternatives: 8

Speedup: 12.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-65}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.4e-65)
   (/ c (- b))
   (if (<= b 2.9e+84)
     (/ (- (- b) (sqrt (- (* b b) (* (* c 4.0) a)))) (* a 2.0))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.4e-65) {
		tmp = c / -b;
	} else if (b <= 2.9e+84) {
		tmp = (-b - sqrt(((b * b) - ((c * 4.0) * a)))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.4d-65)) then
        tmp = c / -b
    else if (b <= 2.9d+84) then
        tmp = (-b - sqrt(((b * b) - ((c * 4.0d0) * a)))) / (a * 2.0d0)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.4e-65) {
		tmp = c / -b;
	} else if (b <= 2.9e+84) {
		tmp = (-b - Math.sqrt(((b * b) - ((c * 4.0) * a)))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.4e-65:
		tmp = c / -b
	elif b <= 2.9e+84:
		tmp = (-b - math.sqrt(((b * b) - ((c * 4.0) * a)))) / (a * 2.0)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.4e-65)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 2.9e+84)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(c * 4.0) * a)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.4e-65)
		tmp = c / -b;
	elseif (b <= 2.9e+84)
		tmp = (-b - sqrt(((b * b) - ((c * 4.0) * a)))) / (a * 2.0);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.4e-65], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 2.9e+84], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(c * 4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.3999999999999997e-65

   1. Initial program 18.2%

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

      1. div-sub 17.6%

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

      2. sub-neg 17.6%

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

      3. neg-mul-1 17.6%

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

      4. *-commutative 17.6%

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

      5. associate-/l* 17.0%

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

      6. distribute-neg-frac 17.0%

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

      7. neg-mul-1 17.0%

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

      8. *-commutative 17.0%

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

      9. associate-/l* 17.7%

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

      10. distribute-rgt-out 18.2%

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

      11. associate-/r* 18.2%

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

      12. metadata-eval 18.2%

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

      13. sub-neg 18.2%

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

      14. +-commutative 18.2%

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

   3. Simplified 18.2%

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

   5. Taylor expanded in b around -inf 85.9%

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

      1. mul-1-neg 85.9%

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

      2. distribute-neg-frac2 85.9%

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

   7. Simplified 85.9%

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

   if -5.3999999999999997e-65 < b < 2.89999999999999989e84

   1. Initial program 80.2%

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

      1. *-commutative 80.2%

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

      2. *-commutative 80.2%

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

      3. sqr-neg 80.2%

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

      4. *-commutative 80.2%

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

      5. sqr-neg 80.2%

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

      6. *-commutative 80.2%

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

      7. associate-*r* 80.3%

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

   3. Simplified 80.3%

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

   if 2.89999999999999989e84 < b

   1. Initial program 65.5%

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

      1. div-sub 65.5%

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

      2. sub-neg 65.5%

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

      3. neg-mul-1 65.5%

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

      4. *-commutative 65.5%

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

      5. associate-/l* 65.4%

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

      6. distribute-neg-frac 65.4%

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

      7. neg-mul-1 65.4%

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

      8. *-commutative 65.4%

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

      9. associate-/l* 65.4%

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

      10. distribute-rgt-out 65.4%

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

      11. associate-/r* 65.4%

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

      12. metadata-eval 65.4%

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

      13. sub-neg 65.4%

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

      14. +-commutative 65.4%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in a around 0 96.7%

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

      1. associate-*r/ 96.7%

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

      2. mul-1-neg 96.7%

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

   7. Simplified 96.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-65}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e-64)
   (/ c (- b))
   (if (<= b 5e+81)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-64) {
		tmp = c / -b;
	} else if (b <= 5e+81) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.5d-64)) then
        tmp = c / -b
    else if (b <= 5d+81) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (a * 2.0d0)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-64) {
		tmp = c / -b;
	} else if (b <= 5e+81) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.5e-64:
		tmp = c / -b
	elif b <= 5e+81:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e-64)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 5e+81)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.5e-64)
		tmp = c / -b;
	elseif (b <= 5e+81)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e-64], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 5e+81], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5e-64

   1. Initial program 18.2%

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

      1. div-sub 17.6%

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

      2. sub-neg 17.6%

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

      3. neg-mul-1 17.6%

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

      4. *-commutative 17.6%

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

      5. associate-/l* 17.0%

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

      6. distribute-neg-frac 17.0%

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

      7. neg-mul-1 17.0%

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

      8. *-commutative 17.0%

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

      9. associate-/l* 17.7%

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

      10. distribute-rgt-out 18.2%

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

      11. associate-/r* 18.2%

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

      12. metadata-eval 18.2%

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

      13. sub-neg 18.2%

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

      14. +-commutative 18.2%

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

   3. Simplified 18.2%

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

   5. Taylor expanded in b around -inf 85.9%

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

      1. mul-1-neg 85.9%

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

      2. distribute-neg-frac2 85.9%

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

   7. Simplified 85.9%

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

   if -1.5e-64 < b < 4.9999999999999998e81

   1. Initial program 80.2%

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

   if 4.9999999999999998e81 < b

   1. Initial program 65.5%

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

      1. div-sub 65.5%

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

      2. sub-neg 65.5%

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

      3. neg-mul-1 65.5%

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

      4. *-commutative 65.5%

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

      5. associate-/l* 65.4%

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

      6. distribute-neg-frac 65.4%

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

      7. neg-mul-1 65.4%

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

      8. *-commutative 65.4%

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

      9. associate-/l* 65.4%

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

      10. distribute-rgt-out 65.4%

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

      11. associate-/r* 65.4%

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

      12. metadata-eval 65.4%

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

      13. sub-neg 65.4%

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

      14. +-commutative 65.4%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in a around 0 96.7%

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

      1. associate-*r/ 96.7%

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

      2. mul-1-neg 96.7%

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

   7. Simplified 96.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-68}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{-90}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-68)
   (/ c (- b))
   (if (<= b 3.3e-90)
     (* (/ -0.5 a) (+ b (sqrt (* c (* a -4.0)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-68) {
		tmp = c / -b;
	} else if (b <= 3.3e-90) {
		tmp = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-9d-68)) then
        tmp = c / -b
    else if (b <= 3.3d-90) then
        tmp = ((-0.5d0) / a) * (b + sqrt((c * (a * (-4.0d0)))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-68) {
		tmp = c / -b;
	} else if (b <= 3.3e-90) {
		tmp = (-0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9e-68:
		tmp = c / -b
	elif b <= 3.3e-90:
		tmp = (-0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9e-68)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 3.3e-90)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9e-68)
		tmp = c / -b;
	elseif (b <= 3.3e-90)
		tmp = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9e-68], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 3.3e-90], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.99999999999999998e-68

   1. Initial program 18.2%

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

      1. div-sub 17.6%

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

      2. sub-neg 17.6%

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

      3. neg-mul-1 17.6%

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

      4. *-commutative 17.6%

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

      5. associate-/l* 17.0%

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

      6. distribute-neg-frac 17.0%

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

      7. neg-mul-1 17.0%

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

      8. *-commutative 17.0%

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

      9. associate-/l* 17.7%

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

      10. distribute-rgt-out 18.2%

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

      11. associate-/r* 18.2%

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

      12. metadata-eval 18.2%

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

      13. sub-neg 18.2%

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

      14. +-commutative 18.2%

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

   3. Simplified 18.2%

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

   5. Taylor expanded in b around -inf 85.9%

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

      1. mul-1-neg 85.9%

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

      2. distribute-neg-frac2 85.9%

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

   7. Simplified 85.9%

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

   if -8.99999999999999998e-68 < b < 3.3e-90

   1. Initial program 71.0%

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

      1. div-sub 71.0%

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

      2. sub-neg 71.0%

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

      3. neg-mul-1 71.0%

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

      4. *-commutative 71.0%

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

      5. associate-/l* 71.0%

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

      6. distribute-neg-frac 71.0%

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

      7. neg-mul-1 71.0%

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

      8. *-commutative 71.0%

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

      9. associate-/l* 70.8%

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

      10. distribute-rgt-out 70.9%

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

      11. associate-/r* 70.9%

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

      12. metadata-eval 70.9%

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

      13. sub-neg 70.9%

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

      14. +-commutative 70.9%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in b around inf 20.9%

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

      1. associate-/l* 16.7%

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

   7. Simplified 16.7%

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

   8. Taylor expanded in b around 0 68.6%

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

      1. associate-*r* 68.7%

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

      2. *-commutative 68.7%

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

   10. Simplified 68.7%

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

   if 3.3e-90 < b

   1. Initial program 78.2%

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

      1. div-sub 78.2%

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

      2. sub-neg 78.2%

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

      3. neg-mul-1 78.2%

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

      4. *-commutative 78.2%

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

      5. associate-/l* 78.1%

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

      6. distribute-neg-frac 78.1%

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

      7. neg-mul-1 78.1%

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

      8. *-commutative 78.1%

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

      9. associate-/l* 78.0%

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

      10. distribute-rgt-out 78.0%

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

      11. associate-/r* 78.0%

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

      12. metadata-eval 78.0%

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

      13. sub-neg 78.0%

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

      14. +-commutative 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in c around 0 87.5%

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

      1. +-commutative 87.5%

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

      2. mul-1-neg 87.5%

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

      3. unsub-neg 87.5%

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

   7. Simplified 87.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-68}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{-90}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ c (- b)) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = c / -b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = c / -b
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = c / -b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = c / -b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = c / -b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(c / (-b)), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 34.9%

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

      1. div-sub 34.5%

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

      2. sub-neg 34.5%

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

      3. neg-mul-1 34.5%

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

      4. *-commutative 34.5%

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

      5. associate-/l* 34.1%

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

      6. distribute-neg-frac 34.1%

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

      7. neg-mul-1 34.1%

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

      8. *-commutative 34.1%

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

      9. associate-/l* 34.5%

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

      10. distribute-rgt-out 34.9%

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

      11. associate-/r* 34.9%

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

      12. metadata-eval 34.9%

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

      13. sub-neg 34.9%

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

      14. +-commutative 34.9%

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

   3. Simplified 34.9%

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

   5. Taylor expanded in b around -inf 63.7%

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

      1. mul-1-neg 63.7%

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

      2. distribute-neg-frac2 63.7%

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

   7. Simplified 63.7%

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

   if -1.999999999999994e-310 < b

   1. Initial program 76.1%

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

      1. div-sub 76.1%

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

      2. sub-neg 76.1%

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

      3. neg-mul-1 76.1%

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

      4. *-commutative 76.1%

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

      5. associate-/l* 76.0%

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

      6. distribute-neg-frac 76.0%

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

      7. neg-mul-1 76.0%

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

      8. *-commutative 76.0%

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

      9. associate-/l* 75.9%

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

      10. distribute-rgt-out 75.9%

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

      11. associate-/r* 75.9%

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

      12. metadata-eval 75.9%

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

      13. sub-neg 75.9%

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

      14. +-commutative 75.9%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in c around 0 72.2%

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

      1. +-commutative 72.2%

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

      2. mul-1-neg 72.2%

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

      3. unsub-neg 72.2%

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

   7. Simplified 72.2%

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

Alternative 5: 67.7% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-307}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.3e-307) (/ c (- b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-307) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.3d-307)) then
        tmp = c / -b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-307) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.3e-307:
		tmp = c / -b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.3e-307)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.3e-307)
		tmp = c / -b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.3e-307], N[(c / (-b)), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.3e-307

   1. Initial program 34.4%

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

      1. div-sub 34.0%

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

      2. sub-neg 34.0%

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

      3. neg-mul-1 34.0%

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

      4. *-commutative 34.0%

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

      5. associate-/l* 33.6%

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

      6. distribute-neg-frac 33.6%

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

      7. neg-mul-1 33.6%

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

      8. *-commutative 33.6%

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

      9. associate-/l* 34.0%

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

      10. distribute-rgt-out 34.4%

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

      11. associate-/r* 34.4%

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

      12. metadata-eval 34.4%

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

      13. sub-neg 34.4%

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

      14. +-commutative 34.4%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in b around -inf 64.2%

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

      1. mul-1-neg 64.2%

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

      2. distribute-neg-frac2 64.2%

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

   7. Simplified 64.2%

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

   if -3.3e-307 < b

   1. Initial program 76.3%

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

      1. div-sub 76.3%

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

      2. sub-neg 76.3%

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

      3. neg-mul-1 76.3%

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

      4. *-commutative 76.3%

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

      5. associate-/l* 76.2%

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

      6. distribute-neg-frac 76.2%

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

      7. neg-mul-1 76.2%

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

      8. *-commutative 76.2%

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

      9. associate-/l* 76.1%

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

      10. distribute-rgt-out 76.1%

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

      11. associate-/r* 76.1%

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

      12. metadata-eval 76.1%

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

      13. sub-neg 76.1%

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

      14. +-commutative 76.1%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in a around 0 71.3%

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

      1. associate-*r/ 71.3%

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

      2. mul-1-neg 71.3%

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

   7. Simplified 71.3%

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

Alternative 6: 35.1% accurate, 29.0× 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 / Float64(-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 54.5%

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

   1. div-sub 54.3%

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

   2. sub-neg 54.3%

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

   3. neg-mul-1 54.3%

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

   4. *-commutative 54.3%

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

   5. associate-/l* 54.1%

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

   6. distribute-neg-frac 54.1%

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

   7. neg-mul-1 54.1%

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

   8. *-commutative 54.1%

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

   9. associate-/l* 54.2%

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

   10. distribute-rgt-out 54.4%

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

   11. associate-/r* 54.4%

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

   12. metadata-eval 54.4%

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

   13. sub-neg 54.4%

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

   14. +-commutative 54.4%

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

3. Simplified 54.5%

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

5. Taylor expanded in b around -inf 34.3%

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

   1. mul-1-neg 34.3%

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

   2. distribute-neg-frac2 34.3%

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

7. Simplified 34.3%

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

Alternative 7: 10.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

   1. div-sub 54.3%

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

   2. sub-neg 54.3%

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

   3. neg-mul-1 54.3%

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

   4. *-commutative 54.3%

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

   5. associate-/l* 54.1%

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

   6. distribute-neg-frac 54.1%

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

   7. neg-mul-1 54.1%

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

   8. *-commutative 54.1%

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

   9. associate-/l* 54.2%

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

   10. distribute-rgt-out 54.4%

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

   11. associate-/r* 54.4%

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

   12. metadata-eval 54.4%

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

   13. sub-neg 54.4%

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

   14. +-commutative 54.4%

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

3. Simplified 54.5%

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

5. Taylor expanded in a around 0 34.9%

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

6. Taylor expanded in b around 0 10.4%

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

Alternative 8: 2.6% 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 54.5%

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

   1. div-sub 54.3%

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

   2. sub-neg 54.3%

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

   3. neg-mul-1 54.3%

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

   4. *-commutative 54.3%

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

   5. associate-/l* 54.1%

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

   6. distribute-neg-frac 54.1%

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

   7. neg-mul-1 54.1%

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

   8. *-commutative 54.1%

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

   9. associate-/l* 54.2%

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

   10. distribute-rgt-out 54.4%

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

   11. associate-/r* 54.4%

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

   12. metadata-eval 54.4%

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

   13. sub-neg 54.4%

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

   14. +-commutative 54.4%

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

3. Simplified 54.5%

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

6. Applied egg-rr 33.5%

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

7. Taylor expanded in b around -inf 2.5%

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

Developer target: 69.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024108 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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