Quadratic roots, full range

Percentage Accurate: 61.1% → 83.5%

Time: 13.5s

Alternatives: 12

Speedup: 1.0×

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: 61.1% 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: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -24000000:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{-1}{\frac{\frac{-b}{a}}{c}}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -24000000.0)
   (- (/ c b) (/ b a))
   (if (<= b 6.5e+86)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ (* -2.0 (/ -1.0 (/ (/ (- b) a) c))) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -24000000.0) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.5e+86) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-2.0 * (-1.0 / ((-b / a) / c))) / (a * 2.0);
	}
	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 <= (-24000000.0d0)) then
        tmp = (c / b) - (b / a)
    else if (b <= 6.5d+86) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = ((-2.0d0) * ((-1.0d0) / ((-b / a) / c))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -24000000.0) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.5e+86) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-2.0 * (-1.0 / ((-b / a) / c))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -24000000.0:
		tmp = (c / b) - (b / a)
	elif b <= 6.5e+86:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (-2.0 * (-1.0 / ((-b / a) / c))) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -24000000.0)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6.5e+86)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(-1.0 / Float64(Float64(Float64(-b) / a) / c))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -24000000.0)
		tmp = (c / b) - (b / a);
	elseif (b <= 6.5e+86)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (-2.0 * (-1.0 / ((-b / a) / c))) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -24000000.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e+86], 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[(N[(-2.0 * N[(-1.0 / N[(N[((-b) / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4e7

   1. Initial program 64.8%

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

      1. *-commutative 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in b around -inf 94.5%

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

      1. +-commutative 94.5%

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

      2. mul-1-neg 94.5%

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

      3. unsub-neg 94.5%

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

   7. Simplified 94.5%

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

   if -2.4e7 < b < 6.49999999999999996e86

   1. Initial program 74.6%

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

   if 6.49999999999999996e86 < b

   1. Initial program 24.8%

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

      1. *-commutative 24.8%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in b around inf 67.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. *-commutative 67.5%

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

      2. associate-/l* 75.6%

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

   7. Simplified 75.6%

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

      1. *-un-lft-identity 75.6%

         \[\leadsto \frac{-2 \cdot \frac{\color{blue}{1 \cdot c}}{\frac{b}{a}}}{a \cdot 2} \]

      2. div-inv 75.6%

         \[\leadsto \frac{-2 \cdot \frac{1 \cdot c}{\color{blue}{b \cdot \frac{1}{a}}}}{a \cdot 2} \]

      3. times-frac 67.4%

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

   9. Applied egg-rr 67.4%

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

       1. frac-2neg 67.4%

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

       2. metadata-eval 67.4%

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

       3. clear-num 67.3%

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

       4. frac-times 67.4%

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

       5. metadata-eval 67.4%

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

   11. Applied egg-rr 67.4%

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

       1. associate-*r/ 75.6%

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

       2. associate-*r/ 75.7%

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

       3. *-rgt-identity 75.7%

          \[\leadsto \frac{-2 \cdot \frac{-1}{\frac{\frac{\color{blue}{-b}}{a}}{c}}}{a \cdot 2} \]

   13. Simplified 75.7%

       \[\leadsto \frac{-2 \cdot \color{blue}{\frac{-1}{\frac{\frac{-b}{a}}{c}}}}{a \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -24000000:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{-1}{\frac{\frac{-b}{a}}{c}}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-54}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 0.78:\\ \;\;\;\;0.5 \cdot \frac{{\left(c \cdot \left(a \cdot -4\right)\right)}^{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{-1}{\frac{\frac{-b}{a}}{c}}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.15e-54)
   (- (/ c b) (/ b a))
   (if (<= b 0.78)
     (* 0.5 (/ (pow (* c (* a -4.0)) 0.5) a))
     (/ (* -2.0 (/ -1.0 (/ (/ (- b) a) c))) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.15e-54) {
		tmp = (c / b) - (b / a);
	} else if (b <= 0.78) {
		tmp = 0.5 * (pow((c * (a * -4.0)), 0.5) / a);
	} else {
		tmp = (-2.0 * (-1.0 / ((-b / a) / c))) / (a * 2.0);
	}
	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 <= (-2.15d-54)) then
        tmp = (c / b) - (b / a)
    else if (b <= 0.78d0) then
        tmp = 0.5d0 * (((c * (a * (-4.0d0))) ** 0.5d0) / a)
    else
        tmp = ((-2.0d0) * ((-1.0d0) / ((-b / a) / c))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.15e-54) {
		tmp = (c / b) - (b / a);
	} else if (b <= 0.78) {
		tmp = 0.5 * (Math.pow((c * (a * -4.0)), 0.5) / a);
	} else {
		tmp = (-2.0 * (-1.0 / ((-b / a) / c))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.15e-54:
		tmp = (c / b) - (b / a)
	elif b <= 0.78:
		tmp = 0.5 * (math.pow((c * (a * -4.0)), 0.5) / a)
	else:
		tmp = (-2.0 * (-1.0 / ((-b / a) / c))) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.15e-54)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 0.78)
		tmp = Float64(0.5 * Float64((Float64(c * Float64(a * -4.0)) ^ 0.5) / a));
	else
		tmp = Float64(Float64(-2.0 * Float64(-1.0 / Float64(Float64(Float64(-b) / a) / c))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.15e-54)
		tmp = (c / b) - (b / a);
	elseif (b <= 0.78)
		tmp = 0.5 * (((c * (a * -4.0)) ^ 0.5) / a);
	else
		tmp = (-2.0 * (-1.0 / ((-b / a) / c))) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.15e-54], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.78], N[(0.5 * N[(N[Power[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(-1.0 / N[(N[((-b) / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.15e-54

   1. Initial program 70.7%

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

      1. *-commutative 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in b around -inf 92.2%

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

      1. +-commutative 92.2%

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

      2. mul-1-neg 92.2%

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

      3. unsub-neg 92.2%

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

   7. Simplified 92.2%

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

   if -2.15e-54 < b < 0.78000000000000003

   1. Initial program 71.6%

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

      1. *-commutative 71.6%

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

   3. Simplified 71.6%

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

      1. prod-diff 71.0%

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

      2. *-commutative 71.0%

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

      3. fma-def 71.0%

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

      4. associate-+l+ 71.0%

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

      5. pow2 71.0%

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

      6. distribute-lft-neg-in 71.0%

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

      7. *-commutative 71.0%

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

      8. distribute-rgt-neg-in 71.0%

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

      9. metadata-eval 71.0%

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

      10. associate-*r* 71.0%

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

      11. *-commutative 71.0%

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

      12. *-commutative 71.0%

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

      13. fma-udef 71.0%

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

   6. Applied egg-rr 71.0%

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

      1. fma-def 71.0%

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

      2. fma-def 71.0%

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

      3. *-commutative 71.0%

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

   8. Simplified 71.0%

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

   9. Taylor expanded in b around 0 62.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{a} \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 62.4%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}}{a}} \]

       2. distribute-rgt-out 64.1%

          \[\leadsto 0.5 \cdot \frac{1 \cdot \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 + 4\right)}}}{a} \]

       3. metadata-eval 64.1%

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

       4. associate-*r* 64.1%

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

       5. *-lft-identity 64.1%

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

       6. *-commutative 64.1%

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

   11. Simplified 64.1%

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

       1. div-inv 64.1%

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

       2. add-sqr-sqrt 63.8%

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

       3. associate-*l* 63.7%

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

       4. pow1/2 63.7%

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

       5. sqrt-pow1 63.8%

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

       6. *-commutative 63.8%

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

       7. *-commutative 63.8%

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

       8. associate-*l* 63.8%

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

       9. metadata-eval 63.8%

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

       10. pow1/2 63.8%

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

       11. sqrt-pow1 63.8%

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

       12. *-commutative 63.8%

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

       13. *-commutative 63.8%

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

       14. associate-*l* 63.8%

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

       15. metadata-eval 63.8%

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

   13. Applied egg-rr 63.8%

       \[\leadsto 0.5 \cdot \color{blue}{\left({\left(c \cdot \left(-4 \cdot a\right)\right)}^{0.25} \cdot \left({\left(c \cdot \left(-4 \cdot a\right)\right)}^{0.25} \cdot \frac{1}{a}\right)\right)} \]
   14. Step-by-step derivation

       1. associate-*r/ 63.8%

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

       2. *-rgt-identity 63.8%

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

       3. associate-*r/ 63.9%

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

       4. pow-sqr 64.1%

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

       5. *-commutative 64.1%

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

       6. metadata-eval 64.1%

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

   15. Simplified 64.1%

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

   if 0.78000000000000003 < b

   1. Initial program 39.1%

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

      1. *-commutative 39.1%

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

   3. Simplified 39.1%

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

   5. Taylor expanded in b around inf 61.8%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. *-commutative 61.8%

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

      2. associate-/l* 67.5%

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

   7. Simplified 67.5%

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

      1. *-un-lft-identity 67.5%

         \[\leadsto \frac{-2 \cdot \frac{\color{blue}{1 \cdot c}}{\frac{b}{a}}}{a \cdot 2} \]

      2. div-inv 67.5%

         \[\leadsto \frac{-2 \cdot \frac{1 \cdot c}{\color{blue}{b \cdot \frac{1}{a}}}}{a \cdot 2} \]

      3. times-frac 61.8%

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

   9. Applied egg-rr 61.8%

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

       1. frac-2neg 61.8%

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

       2. metadata-eval 61.8%

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

       3. clear-num 61.7%

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

       4. frac-times 61.8%

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

       5. metadata-eval 61.8%

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

   11. Applied egg-rr 61.8%

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

       1. associate-*r/ 67.5%

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

       2. associate-*r/ 67.5%

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

       3. *-rgt-identity 67.5%

          \[\leadsto \frac{-2 \cdot \frac{-1}{\frac{\frac{\color{blue}{-b}}{a}}{c}}}{a \cdot 2} \]

   13. Simplified 67.5%

       \[\leadsto \frac{-2 \cdot \color{blue}{\frac{-1}{\frac{\frac{-b}{a}}{c}}}}{a \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-54}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 0.78:\\ \;\;\;\;0.5 \cdot \frac{{\left(c \cdot \left(a \cdot -4\right)\right)}^{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{-1}{\frac{\frac{-b}{a}}{c}}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 480000:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{-1}{\frac{\frac{-b}{a}}{c}}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e-57)
   (- (/ c b) (/ b a))
   (if (<= b 480000.0)
     (* 0.5 (/ (sqrt (* a (* c -4.0))) a))
     (/ (* -2.0 (/ -1.0 (/ (/ (- b) a) c))) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-57) {
		tmp = (c / b) - (b / a);
	} else if (b <= 480000.0) {
		tmp = 0.5 * (sqrt((a * (c * -4.0))) / a);
	} else {
		tmp = (-2.0 * (-1.0 / ((-b / a) / c))) / (a * 2.0);
	}
	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 <= (-1.5d-57)) then
        tmp = (c / b) - (b / a)
    else if (b <= 480000.0d0) then
        tmp = 0.5d0 * (sqrt((a * (c * (-4.0d0)))) / a)
    else
        tmp = ((-2.0d0) * ((-1.0d0) / ((-b / a) / c))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-57) {
		tmp = (c / b) - (b / a);
	} else if (b <= 480000.0) {
		tmp = 0.5 * (Math.sqrt((a * (c * -4.0))) / a);
	} else {
		tmp = (-2.0 * (-1.0 / ((-b / a) / c))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.5e-57:
		tmp = (c / b) - (b / a)
	elif b <= 480000.0:
		tmp = 0.5 * (math.sqrt((a * (c * -4.0))) / a)
	else:
		tmp = (-2.0 * (-1.0 / ((-b / a) / c))) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e-57)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 480000.0)
		tmp = Float64(0.5 * Float64(sqrt(Float64(a * Float64(c * -4.0))) / a));
	else
		tmp = Float64(Float64(-2.0 * Float64(-1.0 / Float64(Float64(Float64(-b) / a) / c))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.5e-57)
		tmp = (c / b) - (b / a);
	elseif (b <= 480000.0)
		tmp = 0.5 * (sqrt((a * (c * -4.0))) / a);
	else
		tmp = (-2.0 * (-1.0 / ((-b / a) / c))) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e-57], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 480000.0], N[(0.5 * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(-1.0 / N[(N[((-b) / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5e-57

   1. Initial program 70.7%

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

      1. *-commutative 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in b around -inf 92.2%

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

      1. +-commutative 92.2%

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

      2. mul-1-neg 92.2%

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

      3. unsub-neg 92.2%

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

   7. Simplified 92.2%

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

   if -1.5e-57 < b < 4.8e5

   1. Initial program 71.6%

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

      1. *-commutative 71.6%

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

   3. Simplified 71.6%

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

      1. prod-diff 71.0%

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

      2. *-commutative 71.0%

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

      3. fma-def 71.0%

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

      4. associate-+l+ 71.0%

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

      5. pow2 71.0%

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

      6. distribute-lft-neg-in 71.0%

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

      7. *-commutative 71.0%

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

      8. distribute-rgt-neg-in 71.0%

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

      9. metadata-eval 71.0%

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

      10. associate-*r* 71.0%

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

      11. *-commutative 71.0%

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

      12. *-commutative 71.0%

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

      13. fma-udef 71.0%

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

   6. Applied egg-rr 71.0%

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

      1. fma-def 71.0%

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

      2. fma-def 71.0%

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

      3. *-commutative 71.0%

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

   8. Simplified 71.0%

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

   9. Taylor expanded in b around 0 62.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{a} \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 62.4%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}}{a}} \]

       2. distribute-rgt-out 64.1%

          \[\leadsto 0.5 \cdot \frac{1 \cdot \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 + 4\right)}}}{a} \]

       3. metadata-eval 64.1%

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

       4. associate-*r* 64.1%

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

       5. *-lft-identity 64.1%

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

       6. *-commutative 64.1%

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

   11. Simplified 64.1%

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

   if 4.8e5 < b

   1. Initial program 39.1%

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

      1. *-commutative 39.1%

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

   3. Simplified 39.1%

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

   5. Taylor expanded in b around inf 61.8%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. *-commutative 61.8%

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

      2. associate-/l* 67.5%

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

   7. Simplified 67.5%

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

      1. *-un-lft-identity 67.5%

         \[\leadsto \frac{-2 \cdot \frac{\color{blue}{1 \cdot c}}{\frac{b}{a}}}{a \cdot 2} \]

      2. div-inv 67.5%

         \[\leadsto \frac{-2 \cdot \frac{1 \cdot c}{\color{blue}{b \cdot \frac{1}{a}}}}{a \cdot 2} \]

      3. times-frac 61.8%

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

   9. Applied egg-rr 61.8%

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

       1. frac-2neg 61.8%

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

       2. metadata-eval 61.8%

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

       3. clear-num 61.7%

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

       4. frac-times 61.8%

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

       5. metadata-eval 61.8%

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

   11. Applied egg-rr 61.8%

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

       1. associate-*r/ 67.5%

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

       2. associate-*r/ 67.5%

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

       3. *-rgt-identity 67.5%

          \[\leadsto \frac{-2 \cdot \frac{-1}{\frac{\frac{\color{blue}{-b}}{a}}{c}}}{a \cdot 2} \]

   13. Simplified 67.5%

       \[\leadsto \frac{-2 \cdot \color{blue}{\frac{-1}{\frac{\frac{-b}{a}}{c}}}}{a \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 480000:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{-1}{\frac{\frac{-b}{a}}{c}}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-b}{a}\\ \mathbf{if}\;b \leq 4 \cdot 10^{-309}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{\left(b + -2 \cdot \frac{c \cdot a}{b}\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{-1}{\frac{t\_0}{c}}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- b) a)))
   (if (<= b 4e-309)
     t_0
     (if (<= b 3.2e+86)
       (/ (- (+ b (* -2.0 (/ (* c a) b))) b) (* a 2.0))
       (/ (* -2.0 (/ -1.0 (/ t_0 c))) (* a 2.0))))))

C

double code(double a, double b, double c) {
	double t_0 = -b / a;
	double tmp;
	if (b <= 4e-309) {
		tmp = t_0;
	} else if (b <= 3.2e+86) {
		tmp = ((b + (-2.0 * ((c * a) / b))) - b) / (a * 2.0);
	} else {
		tmp = (-2.0 * (-1.0 / (t_0 / c))) / (a * 2.0);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -b / a
    if (b <= 4d-309) then
        tmp = t_0
    else if (b <= 3.2d+86) then
        tmp = ((b + ((-2.0d0) * ((c * a) / b))) - b) / (a * 2.0d0)
    else
        tmp = ((-2.0d0) * ((-1.0d0) / (t_0 / c))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -b / a;
	double tmp;
	if (b <= 4e-309) {
		tmp = t_0;
	} else if (b <= 3.2e+86) {
		tmp = ((b + (-2.0 * ((c * a) / b))) - b) / (a * 2.0);
	} else {
		tmp = (-2.0 * (-1.0 / (t_0 / c))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -b / a
	tmp = 0
	if b <= 4e-309:
		tmp = t_0
	elif b <= 3.2e+86:
		tmp = ((b + (-2.0 * ((c * a) / b))) - b) / (a * 2.0)
	else:
		tmp = (-2.0 * (-1.0 / (t_0 / c))) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) / a)
	tmp = 0.0
	if (b <= 4e-309)
		tmp = t_0;
	elseif (b <= 3.2e+86)
		tmp = Float64(Float64(Float64(b + Float64(-2.0 * Float64(Float64(c * a) / b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(-1.0 / Float64(t_0 / c))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -b / a;
	tmp = 0.0;
	if (b <= 4e-309)
		tmp = t_0;
	elseif (b <= 3.2e+86)
		tmp = ((b + (-2.0 * ((c * a) / b))) - b) / (a * 2.0);
	else
		tmp = (-2.0 * (-1.0 / (t_0 / c))) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) / a), $MachinePrecision]}, If[LessEqual[b, 4e-309], t$95$0, If[LessEqual[b, 3.2e+86], N[(N[(N[(b + N[(-2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(-1.0 / N[(t$95$0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.9999999999999977e-309

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. mul-1-neg 67.1%

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

   7. Simplified 67.1%

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

   if 3.9999999999999977e-309 < b < 3.2e86

   1. Initial program 68.4%

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

      1. *-commutative 68.4%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in b around inf 33.2%

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

   if 3.2e86 < b

   1. Initial program 24.8%

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

      1. *-commutative 24.8%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in b around inf 67.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. *-commutative 67.5%

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

      2. associate-/l* 75.6%

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

   7. Simplified 75.6%

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

      1. *-un-lft-identity 75.6%

         \[\leadsto \frac{-2 \cdot \frac{\color{blue}{1 \cdot c}}{\frac{b}{a}}}{a \cdot 2} \]

      2. div-inv 75.6%

         \[\leadsto \frac{-2 \cdot \frac{1 \cdot c}{\color{blue}{b \cdot \frac{1}{a}}}}{a \cdot 2} \]

      3. times-frac 67.4%

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

   9. Applied egg-rr 67.4%

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

       1. frac-2neg 67.4%

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

       2. metadata-eval 67.4%

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

       3. clear-num 67.3%

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

       4. frac-times 67.4%

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

       5. metadata-eval 67.4%

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

   11. Applied egg-rr 67.4%

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

       1. associate-*r/ 75.6%

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

       2. associate-*r/ 75.7%

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

       3. *-rgt-identity 75.7%

          \[\leadsto \frac{-2 \cdot \frac{-1}{\frac{\frac{\color{blue}{-b}}{a}}{c}}}{a \cdot 2} \]

   13. Simplified 75.7%

       \[\leadsto \frac{-2 \cdot \color{blue}{\frac{-1}{\frac{\frac{-b}{a}}{c}}}}{a \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{\left(b + -2 \cdot \frac{c \cdot a}{b}\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{-1}{\frac{\frac{-b}{a}}{c}}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-b}{a}\\ \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{-1}{\frac{t\_0}{c}}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- b) a)))
   (if (<= b -5e-310) t_0 (/ (* -2.0 (/ -1.0 (/ t_0 c))) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = -b / a;
	double tmp;
	if (b <= -5e-310) {
		tmp = t_0;
	} else {
		tmp = (-2.0 * (-1.0 / (t_0 / c))) / (a * 2.0);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -b / a
    if (b <= (-5d-310)) then
        tmp = t_0
    else
        tmp = ((-2.0d0) * ((-1.0d0) / (t_0 / c))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -b / a;
	double tmp;
	if (b <= -5e-310) {
		tmp = t_0;
	} else {
		tmp = (-2.0 * (-1.0 / (t_0 / c))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -b / a
	tmp = 0
	if b <= -5e-310:
		tmp = t_0
	else:
		tmp = (-2.0 * (-1.0 / (t_0 / c))) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) / a)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = t_0;
	else
		tmp = Float64(Float64(-2.0 * Float64(-1.0 / Float64(t_0 / c))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -b / a;
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = t_0;
	else
		tmp = (-2.0 * (-1.0 / (t_0 / c))) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) / a), $MachinePrecision]}, If[LessEqual[b, -5e-310], t$95$0, N[(N[(-2.0 * N[(-1.0 / N[(t$95$0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. mul-1-neg 67.1%

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

   7. Simplified 67.1%

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

   if -4.999999999999985e-310 < b

   1. Initial program 48.4%

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

      1. *-commutative 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in b around inf 45.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. *-commutative 45.5%

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

      2. associate-/l* 49.4%

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

   7. Simplified 49.4%

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

      1. *-un-lft-identity 49.4%

         \[\leadsto \frac{-2 \cdot \frac{\color{blue}{1 \cdot c}}{\frac{b}{a}}}{a \cdot 2} \]

      2. div-inv 49.4%

         \[\leadsto \frac{-2 \cdot \frac{1 \cdot c}{\color{blue}{b \cdot \frac{1}{a}}}}{a \cdot 2} \]

      3. times-frac 45.5%

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

   9. Applied egg-rr 45.5%

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

       1. frac-2neg 45.5%

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

       2. metadata-eval 45.5%

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

       3. clear-num 45.1%

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

       4. frac-times 45.2%

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

       5. metadata-eval 45.2%

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

   11. Applied egg-rr 45.2%

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

       1. associate-*r/ 49.3%

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

       2. associate-*r/ 49.4%

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

       3. *-rgt-identity 49.4%

          \[\leadsto \frac{-2 \cdot \frac{-1}{\frac{\frac{\color{blue}{-b}}{a}}{c}}}{a \cdot 2} \]

   13. Simplified 49.4%

       \[\leadsto \frac{-2 \cdot \color{blue}{\frac{-1}{\frac{\frac{-b}{a}}{c}}}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{-1}{\frac{\frac{-b}{a}}{c}}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-308}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{b}{a}}}{2} \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.6e-308) (/ (- b) a) (* (/ (/ c (/ b a)) 2.0) (/ -2.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6e-308) {
		tmp = -b / a;
	} else {
		tmp = ((c / (b / a)) / 2.0) * (-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) :: tmp
    if (b <= 1.6d-308) then
        tmp = -b / a
    else
        tmp = ((c / (b / a)) / 2.0d0) * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6e-308) {
		tmp = -b / a;
	} else {
		tmp = ((c / (b / a)) / 2.0) * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.6e-308:
		tmp = -b / a
	else:
		tmp = ((c / (b / a)) / 2.0) * (-2.0 / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.6e-308)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(Float64(c / Float64(b / a)) / 2.0) * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.6e-308)
		tmp = -b / a;
	else
		tmp = ((c / (b / a)) / 2.0) * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.6e-308], N[((-b) / a), $MachinePrecision], N[(N[(N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001e-308

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. mul-1-neg 67.1%

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

   7. Simplified 67.1%

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

   if 1.6000000000000001e-308 < b

   1. Initial program 48.4%

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

      1. *-commutative 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in b around inf 45.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. *-commutative 45.5%

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

      2. associate-/l* 49.4%

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

   7. Simplified 49.4%

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

      1. *-un-lft-identity 49.4%

         \[\leadsto \frac{-2 \cdot \frac{\color{blue}{1 \cdot c}}{\frac{b}{a}}}{a \cdot 2} \]

      2. div-inv 49.4%

         \[\leadsto \frac{-2 \cdot \frac{1 \cdot c}{\color{blue}{b \cdot \frac{1}{a}}}}{a \cdot 2} \]

      3. times-frac 45.5%

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

   9. Applied egg-rr 45.5%

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

       1. *-commutative 45.5%

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

       2. *-commutative 45.5%

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

       3. times-frac 45.5%

          \[\leadsto \color{blue}{\frac{\frac{1}{b} \cdot \frac{c}{\frac{1}{a}}}{2} \cdot \frac{-2}{a}} \]

       4. frac-times 49.4%

          \[\leadsto \frac{\color{blue}{\frac{1 \cdot c}{b \cdot \frac{1}{a}}}}{2} \cdot \frac{-2}{a} \]

       5. *-un-lft-identity 49.4%

          \[\leadsto \frac{\frac{\color{blue}{c}}{b \cdot \frac{1}{a}}}{2} \cdot \frac{-2}{a} \]

       6. div-inv 49.4%

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

   11. Applied egg-rr 49.4%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-308}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{b}{a}}}{2} \cdot \frac{-2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.9% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. mul-1-neg 67.1%

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

   7. Simplified 67.1%

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

   if -4.999999999999985e-310 < b

   1. Initial program 48.4%

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

      1. *-commutative 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in b around inf 45.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. *-commutative 45.5%

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

      2. associate-/l* 49.4%

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

   7. Simplified 49.4%

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

      1. *-un-lft-identity 49.4%

         \[\leadsto \frac{-2 \cdot \frac{\color{blue}{1 \cdot c}}{\frac{b}{a}}}{a \cdot 2} \]

      2. div-inv 49.4%

         \[\leadsto \frac{-2 \cdot \frac{1 \cdot c}{\color{blue}{b \cdot \frac{1}{a}}}}{a \cdot 2} \]

      3. times-frac 45.5%

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

   9. Applied egg-rr 45.5%

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

       1. *-commutative 45.5%

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

       2. times-frac 45.5%

          \[\leadsto \color{blue}{\frac{\frac{1}{b} \cdot \frac{c}{\frac{1}{a}}}{a} \cdot \frac{-2}{2}} \]

       3. frac-times 49.4%

          \[\leadsto \frac{\color{blue}{\frac{1 \cdot c}{b \cdot \frac{1}{a}}}}{a} \cdot \frac{-2}{2} \]

       4. *-un-lft-identity 49.4%

          \[\leadsto \frac{\frac{\color{blue}{c}}{b \cdot \frac{1}{a}}}{a} \cdot \frac{-2}{2} \]

       5. div-inv 49.4%

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

       6. metadata-eval 49.4%

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

   11. Applied egg-rr 49.4%

       \[\leadsto \color{blue}{\frac{\frac{c}{\frac{b}{a}}}{a} \cdot -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.2%

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

Alternative 8: 56.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{-300}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.4e-300) (/ (- b) a) (/ (- b b) (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.4e-300) {
		tmp = -b / a;
	} else {
		tmp = (b - b) / (a * 2.0);
	}
	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 <= (-7.4d-300)) then
        tmp = -b / a
    else
        tmp = (b - b) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.4e-300) {
		tmp = -b / a;
	} else {
		tmp = (b - b) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.4e-300:
		tmp = -b / a
	else:
		tmp = (b - b) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.4e-300)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(b - b) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.4e-300)
		tmp = -b / a;
	else
		tmp = (b - b) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.4e-300], N[((-b) / a), $MachinePrecision], N[(N[(b - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.4000000000000003e-300

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in b around -inf 68.1%

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

      1. associate-*r/ 68.1%

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

      2. mul-1-neg 68.1%

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

   7. Simplified 68.1%

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

   if -7.4000000000000003e-300 < b

   1. Initial program 49.2%

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

      1. *-commutative 49.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in b around inf 46.3%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{b}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{-300}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.0% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 3.4e+15) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.4e+15) {
		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 <= 3.4d+15) 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 <= 3.4e+15) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.4e+15:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.4e+15)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.4e+15)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.4e+15], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.4e15

   1. Initial program 69.6%

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

      1. *-commutative 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in b around -inf 49.6%

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

      1. associate-*r/ 49.6%

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

      2. mul-1-neg 49.6%

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

   7. Simplified 49.6%

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

   if 3.4e15 < b

   1. Initial program 40.9%

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

      1. *-commutative 40.9%

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

   3. Simplified 40.9%

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

   6. Applied egg-rr 6.5%

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

   7. Taylor expanded in b around -inf 29.7%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.9% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. mul-1-neg 67.1%

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

   7. Simplified 67.1%

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

   if -4.999999999999985e-310 < b

   1. Initial program 48.4%

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

      1. *-commutative 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in b around inf 39.7%

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

      1. mul-1-neg 39.7%

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

      2. distribute-neg-frac 39.7%

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

   7. Simplified 39.7%

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

4. Final simplification 53.3%

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

Alternative 11: 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 60.5%

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

   1. *-commutative 60.5%

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

3. Simplified 60.5%

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

6. Applied egg-rr 29.3%

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

7. Taylor expanded in a around 0 2.6%

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

8. Final simplification 2.6%

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

Alternative 12: 10.8% 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 60.5%

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

   1. *-commutative 60.5%

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

3. Simplified 60.5%

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

6. Applied egg-rr 29.3%

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

7. Taylor expanded in b around -inf 11.5%

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

8. Final simplification 11.5%

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

Reproduce

herbie shell --seed 2024031 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))