The quadratic formula (r1)

Percentage Accurate: 52.4% → 86.2%

Time: 9.8s

Alternatives: 9

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.4% 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: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+151}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}}{a}, \frac{-0.5 \cdot b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+151)
   (- (/ c b) (/ b a))
   (if (<= b 1.95e-65)
     (fma 0.5 (/ (sqrt (fma a (* -4.0 c) (* b b))) a) (/ (* -0.5 b) a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+151) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-65) {
		tmp = fma(0.5, (sqrt(fma(a, (-4.0 * c), (b * b))) / a), ((-0.5 * b) / a));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+151)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.95e-65)
		tmp = fma(0.5, Float64(sqrt(fma(a, Float64(-4.0 * c), Float64(b * b))) / a), Float64(Float64(-0.5 * b) / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+151], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-65], N[(0.5 * N[(N[Sqrt[N[(a * N[(-4.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] + N[(N[(-0.5 * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.00000000000000003e151

   1. Initial program 42.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{{b}^{2}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

      10. unpow2 N/A

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

      11. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{c}{b}}{b}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{c}{b}}{b}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{c}{b}}}{b}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      14. distribute-frac-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, b, \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, b, \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}}\right) \]

      16. lower-neg.f64 96.1

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

   5. Applied rewrites 96.1%

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

   7. Applied rewrites 96.1%

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

   if -2.00000000000000003e151 < b < 1.9500000000000002e-65

   1. Initial program 84.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 84.4%

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

   6. Applied rewrites 84.4%

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

   if 1.9500000000000002e-65 < b

   1. Initial program 13.6%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 84.2

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

   5. Applied rewrites 84.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+151}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}}{a}, \frac{-0.5 \cdot b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+151}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+151)
   (- (/ c b) (/ b a))
   (if (<= b 1.95e-65)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+151) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-65) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d+151)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.95d-65) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (2.0d0 * a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+151) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-65) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e+151:
		tmp = (c / b) - (b / a)
	elif b <= 1.95e-65:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (2.0 * a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+151)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.95e-65)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e+151)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.95e-65)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (2.0 * a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+151], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-65], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.00000000000000003e151

   1. Initial program 42.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{{b}^{2}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

      10. unpow2 N/A

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

      11. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{c}{b}}{b}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{c}{b}}{b}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{c}{b}}}{b}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      14. distribute-frac-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, b, \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, b, \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}}\right) \]

      16. lower-neg.f64 96.1

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

   5. Applied rewrites 96.1%

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

   7. Applied rewrites 96.1%

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

   if -2.00000000000000003e151 < b < 1.9500000000000002e-65

   1. Initial program 84.4%

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

   if 1.9500000000000002e-65 < b

   1. Initial program 13.6%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 84.2

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

   5. Applied rewrites 84.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+151}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-65}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.6e+149)
   (- (/ c b) (/ b a))
   (if (<= b 1.95e-65)
     (* (- (sqrt (fma (* -4.0 c) a (* b b))) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.6e+149) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-65) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.6e+149)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.95e-65)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.6e+149], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-65], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.6e149

   1. Initial program 42.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{{b}^{2}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

      10. unpow2 N/A

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

      11. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{c}{b}}{b}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{c}{b}}{b}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{c}{b}}}{b}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      14. distribute-frac-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, b, \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, b, \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}}\right) \]

      16. lower-neg.f64 96.1

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

   5. Applied rewrites 96.1%

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

   7. Applied rewrites 96.1%

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

   if -6.6e149 < b < 1.9500000000000002e-65

   1. Initial program 84.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 84.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 84.2

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

   4. Applied rewrites 84.2%

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

   if 1.9500000000000002e-65 < b

   1. Initial program 13.6%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 84.2

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

   5. Applied rewrites 84.2%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-65}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-42)
   (- (/ c b) (/ b a))
   (if (<= b 1.95e-65)
     (/ (- (sqrt (* (* c a) -4.0)) b) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-42) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-65) {
		tmp = (sqrt(((c * a) * -4.0)) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.5d-42)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.95d-65) then
        tmp = (sqrt(((c * a) * (-4.0d0))) - b) / (2.0d0 * a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-42) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-65) {
		tmp = (Math.sqrt(((c * a) * -4.0)) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e-42:
		tmp = (c / b) - (b / a)
	elif b <= 1.95e-65:
		tmp = (math.sqrt(((c * a) * -4.0)) - b) / (2.0 * a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-42)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.95e-65)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -4.0)) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e-42)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.95e-65)
		tmp = (sqrt(((c * a) * -4.0)) - b) / (2.0 * a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e-42], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-65], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.4999999999999996e-42

   1. Initial program 68.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{{b}^{2}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

      10. unpow2 N/A

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

      11. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{c}{b}}{b}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{c}{b}}{b}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{c}{b}}}{b}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      14. distribute-frac-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, b, \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, b, \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}}\right) \]

      16. lower-neg.f64 90.7

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 90.7%

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

   if -8.4999999999999996e-42 < b < 1.9500000000000002e-65

   1. Initial program 77.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 69.3

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

   7. Applied rewrites 69.3%

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

   if 1.9500000000000002e-65 < b

   1. Initial program 13.6%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 84.2

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

   5. Applied rewrites 84.2%

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

Alternative 5: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-65}:\\ \;\;\;\;\left(\sqrt{\left(c \cdot a\right) \cdot -4} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-42)
   (- (/ c b) (/ b a))
   (if (<= b 1.95e-65)
     (* (- (sqrt (* (* c a) -4.0)) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-42) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-65) {
		tmp = (sqrt(((c * a) * -4.0)) - b) * (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 <= (-8.5d-42)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.95d-65) then
        tmp = (sqrt(((c * a) * (-4.0d0))) - b) * (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 <= -8.5e-42) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-65) {
		tmp = (Math.sqrt(((c * a) * -4.0)) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e-42:
		tmp = (c / b) - (b / a)
	elif b <= 1.95e-65:
		tmp = (math.sqrt(((c * a) * -4.0)) - b) * (0.5 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-42)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.95e-65)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -4.0)) - b) * 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 <= -8.5e-42)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.95e-65)
		tmp = (sqrt(((c * a) * -4.0)) - b) * (0.5 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e-42], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-65], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.4999999999999996e-42

   1. Initial program 68.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{{b}^{2}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

      10. unpow2 N/A

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

      11. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{c}{b}}{b}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{c}{b}}{b}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{c}{b}}}{b}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      14. distribute-frac-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, b, \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, b, \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}}\right) \]

      16. lower-neg.f64 90.7

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 90.7%

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

   if -8.4999999999999996e-42 < b < 1.9500000000000002e-65

   1. Initial program 77.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 77.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 77.4

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.1

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

   7. Applied rewrites 69.1%

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

   if 1.9500000000000002e-65 < b

   1. Initial program 13.6%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 84.2

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

   5. Applied rewrites 84.2%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-65}:\\ \;\;\;\;\left(\sqrt{\left(c \cdot a\right) \cdot -4} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.8% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 72.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{{b}^{2}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right)} \]

      10. unpow2 N/A

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

      11. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{c}{b}}{b}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{c}{b}}{b}}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{c}{b}}}{b}, b, \mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      14. distribute-frac-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, b, \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, b, \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}}\right) \]

      16. lower-neg.f64 70.4

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 70.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 30.3%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 65.0

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

   5. Applied rewrites 65.0%

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

Alternative 7: 67.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-302}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.2e-302) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-302) {
		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 <= 5.2d-302) 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 <= 5.2e-302) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.2e-302:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.2e-302)
		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 <= 5.2e-302)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.2e-302], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.20000000000000022e-302

   1. Initial program 72.8%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 69.2

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

   5. Applied rewrites 69.2%

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

   if 5.20000000000000022e-302 < b

   1. Initial program 29.2%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 66.0

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

   5. Applied rewrites 66.0%

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

Alternative 8: 43.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-295}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -1.2e-295) (/ (- b) a) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e-295) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	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 <= (-1.2d-295)) then
        tmp = -b / a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e-295) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.2e-295:
		tmp = -b / a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e-295)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.2e-295)
		tmp = -b / a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e-295], N[((-b) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -1.1999999999999999e-295

   1. Initial program 71.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -1.1999999999999999e-295 < b

   1. Initial program 31.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 31.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 31.4

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

   4. Applied rewrites 31.4%

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

   6. Applied rewrites 28.5%

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

   7. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{b}{a}} \]
   8. Step-by-step derivation

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt 17.6

         \[\leadsto \color{blue}{0} \]

   9. Applied rewrites 17.6%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 11.3% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 51.7%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 51.6

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 51.6

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

4. Applied rewrites 51.6%

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

6. Applied rewrites 50.2%

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

7. Taylor expanded in c around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{b}{a}} \]
8. Step-by-step derivation

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 10.1

      \[\leadsto \color{blue}{0} \]

9. Applied rewrites 10.1%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer Target 1: 71.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024276 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

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

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