Quadratic roots, full range

Percentage Accurate: 52.6% → 85.9%

Time: 13.9s

Alternatives: 9

Speedup: 12.9×

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: 52.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+119}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e+119)
   (- (/ c b) (/ b a))
   (if (<= b 1.15e-110)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+119) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.15e-110) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d+119)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.15d-110) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+119) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.15e-110) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e+119:
		tmp = (c / b) - (b / a)
	elif b <= 1.15e-110:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e+119)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.15e-110)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e+119)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.15e-110)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e+119], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.15e-110], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.99999999999999978e119

   1. Initial program 38.0%

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

      1. *-commutative 38.0%

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

   3. Simplified 38.0%

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

   6. Applied egg-rr 38.1%

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

      1. sub-neg 38.1%

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

      2. distribute-rgt-out-- 38.1%

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

   8. Simplified 38.1%

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

   9. Taylor expanded in b around -inf 90.3%

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

       1. mul-1-neg 90.3%

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

       2. *-commutative 90.3%

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

       3. distribute-rgt-neg-in 90.3%

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

       4. +-commutative 90.3%

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

       5. mul-1-neg 90.3%

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

       6. unsub-neg 90.3%

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

   11. Simplified 90.3%

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

   12. Taylor expanded in a around inf 90.6%

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

       1. +-commutative 90.6%

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

       2. mul-1-neg 90.6%

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

       3. unsub-neg 90.6%

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

   14. Simplified 90.6%

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

   if -3.99999999999999978e119 < b < 1.1500000000000001e-110

   1. Initial program 78.9%

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

   if 1.1500000000000001e-110 < b

   1. Initial program 16.8%

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

      1. *-commutative 16.8%

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

   3. Simplified 16.8%

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

   5. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

      2. mul-1-neg 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+119}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.6e-99)
   (- (/ c b) (/ b a))
   (if (<= b 8.5e-110)
     (* (/ 1.0 (* a -2.0)) (- b (sqrt (* c (* a -4.0)))))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-99) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8.5e-110) {
		tmp = (1.0 / (a * -2.0)) * (b - sqrt((c * (a * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.6d-99)) then
        tmp = (c / b) - (b / a)
    else if (b <= 8.5d-110) then
        tmp = (1.0d0 / (a * (-2.0d0))) * (b - sqrt((c * (a * (-4.0d0)))))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-99) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8.5e-110) {
		tmp = (1.0 / (a * -2.0)) * (b - Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.6e-99:
		tmp = (c / b) - (b / a)
	elif b <= 8.5e-110:
		tmp = (1.0 / (a * -2.0)) * (b - math.sqrt((c * (a * -4.0))))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.6e-99)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 8.5e-110)
		tmp = Float64(Float64(1.0 / Float64(a * -2.0)) * Float64(b - sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.6e-99)
		tmp = (c / b) - (b / a);
	elseif (b <= 8.5e-110)
		tmp = (1.0 / (a * -2.0)) * (b - sqrt((c * (a * -4.0))));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.6e-99], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.5e-110], N[(N[(1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.60000000000000005e-99

   1. Initial program 68.8%

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

      1. *-commutative 68.8%

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

   3. Simplified 68.8%

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

   6. Applied egg-rr 68.7%

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

      1. sub-neg 68.7%

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

      2. distribute-rgt-out-- 68.7%

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

   8. Simplified 68.7%

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

   9. Taylor expanded in b around -inf 81.5%

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

       1. mul-1-neg 81.5%

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

       2. *-commutative 81.5%

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

       3. distribute-rgt-neg-in 81.5%

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

       4. +-commutative 81.5%

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

       5. mul-1-neg 81.5%

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

       6. unsub-neg 81.5%

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

   11. Simplified 81.5%

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

   12. Taylor expanded in a around inf 81.7%

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

       1. +-commutative 81.7%

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

       2. mul-1-neg 81.7%

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

       3. unsub-neg 81.7%

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

   14. Simplified 81.7%

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

   if -2.60000000000000005e-99 < b < 8.50000000000000029e-110

   1. Initial program 66.8%

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

      1. *-commutative 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in b around 0 62.2%

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

      1. *-commutative 62.2%

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

      2. *-commutative 62.2%

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

      3. associate-*r* 62.2%

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

   7. Simplified 62.2%

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

      1. frac-2neg 62.2%

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

      2. div-inv 62.3%

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

      3. distribute-neg-in 62.3%

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

      4. add-sqr-sqrt 26.1%

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

      5. sqrt-unprod 61.8%

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

      6. sqr-neg 61.8%

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

      7. sqrt-prod 36.5%

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

      8. add-sqr-sqrt 60.8%

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

      9. sub-neg 60.8%

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

      10. add-sqr-sqrt 24.4%

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

      11. sqrt-unprod 60.7%

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

      12. sqr-neg 60.7%

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

      13. sqrt-prod 36.2%

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

      14. add-sqr-sqrt 62.3%

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

      15. distribute-rgt-neg-in 62.3%

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

      16. metadata-eval 62.3%

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

   9. Applied egg-rr 62.3%

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

       1. *-commutative 62.3%

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

       2. *-commutative 62.3%

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

   11. Simplified 62.3%

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

   if 8.50000000000000029e-110 < b

   1. Initial program 16.8%

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

      1. *-commutative 16.8%

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

   3. Simplified 16.8%

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

   5. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

      2. mul-1-neg 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-108}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.8e-101)
   (- (/ c b) (/ b a))
   (if (<= b 6.8e-108)
     (* (/ 0.5 a) (+ b (sqrt (* c (* a -4.0)))))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.8e-101) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.8e-108) {
		tmp = (0.5 / a) * (b + sqrt((c * (a * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.8d-101)) then
        tmp = (c / b) - (b / a)
    else if (b <= 6.8d-108) then
        tmp = (0.5d0 / a) * (b + sqrt((c * (a * (-4.0d0)))))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.8e-101) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.8e-108) {
		tmp = (0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.8e-101:
		tmp = (c / b) - (b / a)
	elif b <= 6.8e-108:
		tmp = (0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.8e-101)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6.8e-108)
		tmp = Float64(Float64(0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.8e-101)
		tmp = (c / b) - (b / a);
	elseif (b <= 6.8e-108)
		tmp = (0.5 / a) * (b + sqrt((c * (a * -4.0))));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.8e-101], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e-108], N[(N[(0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.79999999999999978e-101

   1. Initial program 68.8%

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

      1. *-commutative 68.8%

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

   3. Simplified 68.8%

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

   6. Applied egg-rr 68.7%

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

      1. sub-neg 68.7%

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

      2. distribute-rgt-out-- 68.7%

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

   8. Simplified 68.7%

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

   9. Taylor expanded in b around -inf 81.5%

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

       1. mul-1-neg 81.5%

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

       2. *-commutative 81.5%

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

       3. distribute-rgt-neg-in 81.5%

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

       4. +-commutative 81.5%

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

       5. mul-1-neg 81.5%

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

       6. unsub-neg 81.5%

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

   11. Simplified 81.5%

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

   12. Taylor expanded in a around inf 81.7%

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

       1. +-commutative 81.7%

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

       2. mul-1-neg 81.7%

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

       3. unsub-neg 81.7%

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

   14. Simplified 81.7%

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

   if -6.79999999999999978e-101 < b < 6.80000000000000004e-108

   1. Initial program 66.8%

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

      1. *-commutative 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in b around 0 62.2%

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

      1. *-commutative 62.2%

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

      2. *-commutative 62.2%

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

      3. associate-*r* 62.2%

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

   7. Simplified 62.2%

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

      1. add-cbrt-cube 53.8%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt[3]{\left(\sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \sqrt{c \cdot \left(a \cdot -4\right)}}}}{a \cdot 2} \]

      2. pow1/3 50.4%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\left(\sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \sqrt{c \cdot \left(a \cdot -4\right)}\right)}^{0.3333333333333333}}}{a \cdot 2} \]

      3. pow3 50.4%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(\sqrt{c \cdot \left(a \cdot -4\right)}\right)}^{3}\right)}}^{0.3333333333333333}}{a \cdot 2} \]

      4. sqrt-pow2 50.4%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(c \cdot \left(a \cdot -4\right)\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333}}{a \cdot 2} \]

      5. metadata-eval 50.4%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(c \cdot \left(a \cdot -4\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}}{a \cdot 2} \]

   9. Applied egg-rr 50.4%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(c \cdot \left(a \cdot -4\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{a \cdot 2} \]
   10. Step-by-step derivation

       1. unpow1/3 53.9%

          \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt[3]{{\left(c \cdot \left(a \cdot -4\right)\right)}^{1.5}}}}{a \cdot 2} \]

   11. Simplified 53.9%

       \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt[3]{{\left(c \cdot \left(a \cdot -4\right)\right)}^{1.5}}}}{a \cdot 2} \]
   12. Step-by-step derivation

       1. *-un-lft-identity 53.9%

          \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(-b\right) + \sqrt[3]{{\left(c \cdot \left(a \cdot -4\right)\right)}^{1.5}}\right)}}{a \cdot 2} \]

       2. *-commutative 53.9%

          \[\leadsto \frac{1 \cdot \left(\left(-b\right) + \sqrt[3]{{\left(c \cdot \left(a \cdot -4\right)\right)}^{1.5}}\right)}{\color{blue}{2 \cdot a}} \]

       3. times-frac 53.9%

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt[3]{{\left(c \cdot \left(a \cdot -4\right)\right)}^{1.5}}}{a}} \]

       4. metadata-eval 53.9%

          \[\leadsto \color{blue}{0.5} \cdot \frac{\left(-b\right) + \sqrt[3]{{\left(c \cdot \left(a \cdot -4\right)\right)}^{1.5}}}{a} \]

       5. add-sqr-sqrt 22.5%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt[3]{{\left(c \cdot \left(a \cdot -4\right)\right)}^{1.5}}}{a} \]

       6. sqrt-unprod 53.5%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt[3]{{\left(c \cdot \left(a \cdot -4\right)\right)}^{1.5}}}{a} \]

       7. sqr-neg 53.5%

          \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{b \cdot b}} + \sqrt[3]{{\left(c \cdot \left(a \cdot -4\right)\right)}^{1.5}}}{a} \]

       8. sqrt-prod 31.8%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt[3]{{\left(c \cdot \left(a \cdot -4\right)\right)}^{1.5}}}{a} \]

       9. add-sqr-sqrt 52.6%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{b} + \sqrt[3]{{\left(c \cdot \left(a \cdot -4\right)\right)}^{1.5}}}{a} \]

       10. pow1/3 49.1%

           \[\leadsto 0.5 \cdot \frac{b + \color{blue}{{\left({\left(c \cdot \left(a \cdot -4\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{a} \]

       11. pow-pow 60.8%

           \[\leadsto 0.5 \cdot \frac{b + \color{blue}{{\left(c \cdot \left(a \cdot -4\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{a} \]

       12. metadata-eval 60.8%

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

       13. pow1/2 60.8%

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

   13. Applied egg-rr 60.8%

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

       1. *-commutative 60.8%

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

       2. associate-*l/ 60.8%

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

       3. associate-*r/ 60.8%

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

       4. *-commutative 60.8%

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

   15. Simplified 60.8%

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

   if 6.80000000000000004e-108 < b

   1. Initial program 16.8%

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

      1. *-commutative 16.8%

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

   3. Simplified 16.8%

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

   5. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

      2. mul-1-neg 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-108}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-101}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.4e-101)
   (- (/ c b) (/ b a))
   (if (<= b 8e-109) (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.4e-101) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8e-109) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.4d-101)) then
        tmp = (c / b) - (b / a)
    else if (b <= 8d-109) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.4e-101) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8e-109) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.4e-101:
		tmp = (c / b) - (b / a)
	elif b <= 8e-109:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.4e-101)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 8e-109)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.4e-101)
		tmp = (c / b) - (b / a);
	elseif (b <= 8e-109)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.4e-101], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-109], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.39999999999999957e-101

   1. Initial program 68.8%

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

      1. *-commutative 68.8%

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

   3. Simplified 68.8%

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

   6. Applied egg-rr 68.7%

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

      1. sub-neg 68.7%

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

      2. distribute-rgt-out-- 68.7%

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

   8. Simplified 68.7%

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

   9. Taylor expanded in b around -inf 81.5%

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

       1. mul-1-neg 81.5%

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

       2. *-commutative 81.5%

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

       3. distribute-rgt-neg-in 81.5%

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

       4. +-commutative 81.5%

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

       5. mul-1-neg 81.5%

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

       6. unsub-neg 81.5%

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

   11. Simplified 81.5%

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

   12. Taylor expanded in a around inf 81.7%

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

       1. +-commutative 81.7%

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

       2. mul-1-neg 81.7%

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

       3. unsub-neg 81.7%

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

   14. Simplified 81.7%

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

   if -6.39999999999999957e-101 < b < 7.9999999999999999e-109

   1. Initial program 66.8%

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

      1. *-commutative 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in b around 0 62.2%

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

      1. *-commutative 62.2%

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

      2. *-commutative 62.2%

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

      3. associate-*r* 62.2%

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

   7. Simplified 62.2%

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

      1. +-commutative 62.2%

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

      2. unsub-neg 62.2%

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

   9. Applied egg-rr 62.2%

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

   if 7.9999999999999999e-109 < b

   1. Initial program 16.8%

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

      1. *-commutative 16.8%

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

   3. Simplified 16.8%

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

   5. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

      2. mul-1-neg 90.0%

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

   7. Simplified 90.0%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-101}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.0% accurate, 9.7× 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(c / Float64(-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 67.3%

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

      1. *-commutative 67.3%

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

   3. Simplified 67.3%

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

   6. Applied egg-rr 67.3%

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

      1. sub-neg 67.3%

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

      2. distribute-rgt-out-- 67.3%

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

   8. Simplified 67.3%

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

   9. Taylor expanded in b around -inf 63.4%

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

       1. mul-1-neg 63.4%

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

       2. *-commutative 63.4%

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

       3. distribute-rgt-neg-in 63.4%

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

       4. +-commutative 63.4%

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

       5. mul-1-neg 63.4%

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

       6. unsub-neg 63.4%

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

   11. Simplified 63.4%

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

   12. Taylor expanded in a around inf 64.7%

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

       1. +-commutative 64.7%

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

       2. mul-1-neg 64.7%

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

       3. unsub-neg 64.7%

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

   14. Simplified 64.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 29.5%

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

      1. *-commutative 29.5%

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

   3. Simplified 29.5%

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

   5. Taylor expanded in b around inf 71.8%

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

      1. associate-*r/ 71.8%

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

      2. mul-1-neg 71.8%

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

   7. Simplified 71.8%

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

4. Final simplification 68.5%

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

Alternative 6: 44.7% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 2.5e-28) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5e-28) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.5d-28) then
        tmp = b / -a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5e-28) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.5e-28:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.5e-28)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.5e-28)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.5e-28], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5000000000000001e-28

   1. Initial program 65.8%

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

      1. *-commutative 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in b around -inf 47.9%

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

      1. mul-1-neg 47.9%

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

      2. distribute-neg-frac2 47.9%

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

   7. Simplified 47.9%

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

   if 2.5000000000000001e-28 < b

   1. Initial program 14.6%

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

      1. *-commutative 14.6%

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

   3. Simplified 14.6%

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

   6. Applied egg-rr 7.4%

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

      1. unpow-1 7.4%

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

      2. associate-/l* 7.4%

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

   8. Simplified 7.4%

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

   9. Taylor expanded in b around -inf 34.3%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.5e-308) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5e-308) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.5d-308) then
        tmp = b / -a
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5e-308) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.5e-308:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.5e-308)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.5e-308)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.5e-308], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.49999999999999977e-308

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in b around -inf 64.1%

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

      1. mul-1-neg 64.1%

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

      2. distribute-neg-frac2 64.1%

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

   7. Simplified 64.1%

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

   if 2.49999999999999977e-308 < b

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

   3. Simplified 29.0%

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

   5. Taylor expanded in b around inf 72.3%

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

      1. associate-*r/ 72.3%

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

      2. mul-1-neg 72.3%

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

   7. Simplified 72.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.8%

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

   1. *-commutative 46.8%

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

3. Simplified 46.8%

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

6. Applied egg-rr 25.5%

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

   1. unpow-1 25.5%

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

   2. associate-/l* 25.5%

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

8. Simplified 25.5%

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

9. Taylor expanded in a around 0 2.4%

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

10. Final simplification 2.4%

    \[\leadsto \frac{b}{a} \]
11. Add Preprocessing

Alternative 9: 11.2% 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 46.8%

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

   1. *-commutative 46.8%

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

3. Simplified 46.8%

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

6. Applied egg-rr 25.5%

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

   1. unpow-1 25.5%

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

   2. associate-/l* 25.5%

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

8. Simplified 25.5%

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

9. Taylor expanded in b around -inf 14.8%

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

10. Final simplification 14.8%

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

Reproduce

herbie shell --seed 2024076 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))