Quadratic roots, full range

Percentage Accurate: 52.9% → 86.5%

Time: 13.7s

Alternatives: 8

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: 52.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e+139)
   (/ (- b) a)
   (if (<= b 1.2e-50)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+139) {
		tmp = -b / a;
	} else if (b <= 1.2e-50) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - 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 <= (-2.8d+139)) then
        tmp = -b / a
    else if (b <= 1.2d-50) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - 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 <= -2.8e+139) {
		tmp = -b / a;
	} else if (b <= 1.2e-50) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e+139:
		tmp = -b / a
	elif b <= 1.2e-50:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e+139)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.2e-50)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.8e+139)
		tmp = -b / a;
	elseif (b <= 1.2e-50)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e+139], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.2e-50], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $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 < -2.7999999999999998e139

   1. Initial program 41.9%

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

      1. *-commutative 41.9%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in b around -inf 94.9%

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

      1. associate-*r/ 94.9%

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

      2. mul-1-neg 94.9%

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

   7. Simplified 94.9%

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

   if -2.7999999999999998e139 < b < 1.20000000000000001e-50

   1. Initial program 81.0%

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

   if 1.20000000000000001e-50 < b

   1. Initial program 16.1%

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

      1. *-commutative 16.1%

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

   3. Simplified 16.1%

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

   5. Taylor expanded in b around inf 82.5%

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

      1. mul-1-neg 82.5%

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

      2. distribute-neg-frac 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(c \cdot -4\right)}\\ t_1 := \frac{c}{b} - \frac{b}{a}\\ \mathbf{if}\;b \leq -1.18 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{t_0 - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \left(\frac{t_0}{a} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a (* c -4.0)))) (t_1 (- (/ c b) (/ b a))))
   (if (<= b -1.18e+80)
     t_1
     (if (<= b -7.2e+14)
       (/ (- t_0 b) (* a 2.0))
       (if (<= b -2.1e-8)
         t_1
         (if (<= b 2.8e-53) (* 0.5 (- (/ t_0 a) (/ b a))) (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (c * -4.0)));
	double t_1 = (c / b) - (b / a);
	double tmp;
	if (b <= -1.18e+80) {
		tmp = t_1;
	} else if (b <= -7.2e+14) {
		tmp = (t_0 - b) / (a * 2.0);
	} else if (b <= -2.1e-8) {
		tmp = t_1;
	} else if (b <= 2.8e-53) {
		tmp = 0.5 * ((t_0 / a) - (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((a * (c * (-4.0d0))))
    t_1 = (c / b) - (b / a)
    if (b <= (-1.18d+80)) then
        tmp = t_1
    else if (b <= (-7.2d+14)) then
        tmp = (t_0 - b) / (a * 2.0d0)
    else if (b <= (-2.1d-8)) then
        tmp = t_1
    else if (b <= 2.8d-53) then
        tmp = 0.5d0 * ((t_0 / a) - (b / a))
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((a * (c * -4.0)));
	double t_1 = (c / b) - (b / a);
	double tmp;
	if (b <= -1.18e+80) {
		tmp = t_1;
	} else if (b <= -7.2e+14) {
		tmp = (t_0 - b) / (a * 2.0);
	} else if (b <= -2.1e-8) {
		tmp = t_1;
	} else if (b <= 2.8e-53) {
		tmp = 0.5 * ((t_0 / a) - (b / a));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((a * (c * -4.0)))
	t_1 = (c / b) - (b / a)
	tmp = 0
	if b <= -1.18e+80:
		tmp = t_1
	elif b <= -7.2e+14:
		tmp = (t_0 - b) / (a * 2.0)
	elif b <= -2.1e-8:
		tmp = t_1
	elif b <= 2.8e-53:
		tmp = 0.5 * ((t_0 / a) - (b / a))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * Float64(c * -4.0)))
	t_1 = Float64(Float64(c / b) - Float64(b / a))
	tmp = 0.0
	if (b <= -1.18e+80)
		tmp = t_1;
	elseif (b <= -7.2e+14)
		tmp = Float64(Float64(t_0 - b) / Float64(a * 2.0));
	elseif (b <= -2.1e-8)
		tmp = t_1;
	elseif (b <= 2.8e-53)
		tmp = Float64(0.5 * Float64(Float64(t_0 / a) - Float64(b / a)));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt((a * (c * -4.0)));
	t_1 = (c / b) - (b / a);
	tmp = 0.0;
	if (b <= -1.18e+80)
		tmp = t_1;
	elseif (b <= -7.2e+14)
		tmp = (t_0 - b) / (a * 2.0);
	elseif (b <= -2.1e-8)
		tmp = t_1;
	elseif (b <= 2.8e-53)
		tmp = 0.5 * ((t_0 / a) - (b / a));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.18e+80], t$95$1, If[LessEqual[b, -7.2e+14], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -2.1e-8], t$95$1, If[LessEqual[b, 2.8e-53], N[(0.5 * N[(N[(t$95$0 / a), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.18e80 or -7.2e14 < b < -2.09999999999999994e-8

   1. Initial program 60.9%

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

      1. *-commutative 60.9%

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

   3. Simplified 60.9%

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

   5. Taylor expanded in b around -inf 94.8%

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

      1. +-commutative 94.8%

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

      2. mul-1-neg 94.8%

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

      3. unsub-neg 94.8%

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

   7. Simplified 94.8%

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

   if -1.18e80 < b < -7.2e14

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. prod-diff 99.8%

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

      2. *-commutative 99.8%

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

      3. fma-def 99.8%

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

      4. associate-+l+ 99.8%

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

      5. pow2 99.8%

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

      6. distribute-lft-neg-in 99.8%

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

      7. *-commutative 99.8%

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

      8. distribute-rgt-neg-in 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-*r* 99.8%

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

      11. *-commutative 99.8%

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

      12. *-commutative 99.8%

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

      13. fma-udef 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+r+ 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-*r* 99.8%

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

      5. associate-*r* 99.8%

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

      6. distribute-lft-out 99.8%

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

      7. metadata-eval 99.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot \color{blue}{-8} + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}{a \cdot 2} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \color{blue}{\left({b}^{2} + \left(c \cdot 4\right) \cdot a\right)}}}{a \cdot 2} \]

      9. unpow2 99.8%

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

      10. fma-udef 99.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \color{blue}{\mathsf{fma}\left(b, b, \left(c \cdot 4\right) \cdot a\right)}}}{a \cdot 2} \]

      11. associate-*l* 99.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(4 \cdot a\right)}\right)}}{a \cdot 2} \]

      12. *-commutative 99.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot 4\right)}\right)}}{a \cdot 2} \]

   8. Simplified 99.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}{a \cdot 2} \]

   9. Taylor expanded in b around 0 81.0%

      \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}}{a \cdot 2} \]
   10. Step-by-step derivation

       1. mul-1-neg 81.0%

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

       2. unsub-neg 81.0%

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

       3. distribute-rgt-out 81.0%

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

       4. metadata-eval 81.0%

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

       5. associate-*r* 81.0%

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

   11. Simplified 81.0%

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

   if -2.09999999999999994e-8 < b < 2.79999999999999985e-53

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

   3. Simplified 74.9%

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

      1. prod-diff 74.6%

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

      2. *-commutative 74.6%

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

      3. fma-def 74.6%

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

      4. associate-+l+ 74.6%

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

      5. pow2 74.6%

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

      6. distribute-lft-neg-in 74.6%

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

      7. *-commutative 74.6%

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

      8. distribute-rgt-neg-in 74.6%

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

      9. metadata-eval 74.6%

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

      10. associate-*r* 74.6%

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

      11. *-commutative 74.6%

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

      12. *-commutative 74.6%

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

      13. fma-udef 74.6%

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

   6. Applied egg-rr 74.6%

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

      1. +-commutative 74.6%

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

      2. associate-+r+ 74.6%

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

      3. associate-+l+ 74.6%

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

      4. associate-*r* 74.6%

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

      5. associate-*r* 74.6%

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

      6. distribute-lft-out 74.6%

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

      7. metadata-eval 74.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot \color{blue}{-8} + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}{a \cdot 2} \]

      8. +-commutative 74.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \color{blue}{\left({b}^{2} + \left(c \cdot 4\right) \cdot a\right)}}}{a \cdot 2} \]

      9. unpow2 74.6%

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

      10. fma-udef 74.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \color{blue}{\mathsf{fma}\left(b, b, \left(c \cdot 4\right) \cdot a\right)}}}{a \cdot 2} \]

      11. associate-*l* 74.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(4 \cdot a\right)}\right)}}{a \cdot 2} \]

      12. *-commutative 74.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot 4\right)}\right)}}{a \cdot 2} \]

   8. Simplified 74.6%

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

      1. associate-*l* 74.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{a \cdot \left(c \cdot -8\right)} + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}{a \cdot 2} \]

      2. fma-def 74.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}}{a \cdot 2} \]

   10. Applied egg-rr 74.6%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}}{a \cdot 2} \]

   11. Taylor expanded in b around 0 68.1%

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

       1. +-commutative 68.1%

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

       2. *-commutative 68.1%

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

       3. associate-*r/ 68.3%

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

       4. *-rgt-identity 68.3%

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

       5. distribute-rgt-out 68.6%

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

       6. metadata-eval 68.6%

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

       7. metadata-eval 68.6%

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

       8. cancel-sign-sub-inv 68.6%

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

       9. distribute-lft-out-- 68.6%

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

       10. associate-*l* 68.6%

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

   13. Simplified 68.6%

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

   if 2.79999999999999985e-53 < b

   1. Initial program 16.1%

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

      1. *-commutative 16.1%

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

   3. Simplified 16.1%

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

   5. Taylor expanded in b around inf 82.5%

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

      1. mul-1-neg 82.5%

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

      2. distribute-neg-frac 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+80}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \left(\frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b} - \frac{b}{a}\\ t_1 := \frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.18 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (/ c b) (/ b a)))
        (t_1 (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))))
   (if (<= b -1.18e+80)
     t_0
     (if (<= b -7.2e+14)
       t_1
       (if (<= b -2.1e-8) t_0 (if (<= b 1.9e-50) t_1 (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -1.18e+80) {
		tmp = t_0;
	} else if (b <= -7.2e+14) {
		tmp = t_1;
	} else if (b <= -2.1e-8) {
		tmp = t_0;
	} else if (b <= 1.9e-50) {
		tmp = t_1;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (c / b) - (b / a)
    t_1 = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    if (b <= (-1.18d+80)) then
        tmp = t_0
    else if (b <= (-7.2d+14)) then
        tmp = t_1
    else if (b <= (-2.1d-8)) then
        tmp = t_0
    else if (b <= 1.9d-50) then
        tmp = t_1
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -1.18e+80) {
		tmp = t_0;
	} else if (b <= -7.2e+14) {
		tmp = t_1;
	} else if (b <= -2.1e-8) {
		tmp = t_0;
	} else if (b <= 1.9e-50) {
		tmp = t_1;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (c / b) - (b / a)
	t_1 = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	tmp = 0
	if b <= -1.18e+80:
		tmp = t_0
	elif b <= -7.2e+14:
		tmp = t_1
	elif b <= -2.1e-8:
		tmp = t_0
	elif b <= 1.9e-50:
		tmp = t_1
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / b) - Float64(b / a))
	t_1 = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (b <= -1.18e+80)
		tmp = t_0;
	elseif (b <= -7.2e+14)
		tmp = t_1;
	elseif (b <= -2.1e-8)
		tmp = t_0;
	elseif (b <= 1.9e-50)
		tmp = t_1;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (c / b) - (b / a);
	t_1 = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	tmp = 0.0;
	if (b <= -1.18e+80)
		tmp = t_0;
	elseif (b <= -7.2e+14)
		tmp = t_1;
	elseif (b <= -2.1e-8)
		tmp = t_0;
	elseif (b <= 1.9e-50)
		tmp = t_1;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.18e+80], t$95$0, If[LessEqual[b, -7.2e+14], t$95$1, If[LessEqual[b, -2.1e-8], t$95$0, If[LessEqual[b, 1.9e-50], t$95$1, N[((-c) / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.18e80 or -7.2e14 < b < -2.09999999999999994e-8

   1. Initial program 60.9%

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

      1. *-commutative 60.9%

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

   3. Simplified 60.9%

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

   5. Taylor expanded in b around -inf 94.8%

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

      1. +-commutative 94.8%

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

      2. mul-1-neg 94.8%

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

      3. unsub-neg 94.8%

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

   7. Simplified 94.8%

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

   if -1.18e80 < b < -7.2e14 or -2.09999999999999994e-8 < b < 1.9e-50

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

   3. Simplified 78.2%

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

      1. prod-diff 78.0%

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

      2. *-commutative 78.0%

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

      3. fma-def 78.0%

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

      4. associate-+l+ 78.0%

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

      5. pow2 78.0%

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

      6. distribute-lft-neg-in 78.0%

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

      7. *-commutative 78.0%

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

      8. distribute-rgt-neg-in 78.0%

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

      9. metadata-eval 78.0%

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

      10. associate-*r* 78.0%

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

      11. *-commutative 78.0%

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

      12. *-commutative 78.0%

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

      13. fma-udef 78.0%

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

   6. Applied egg-rr 78.0%

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

      1. +-commutative 78.0%

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

      2. associate-+r+ 78.0%

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

      3. associate-+l+ 78.0%

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

      4. associate-*r* 78.0%

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

      5. associate-*r* 78.0%

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

      6. distribute-lft-out 78.0%

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

      7. metadata-eval 78.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot \color{blue}{-8} + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}{a \cdot 2} \]

      8. +-commutative 78.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \color{blue}{\left({b}^{2} + \left(c \cdot 4\right) \cdot a\right)}}}{a \cdot 2} \]

      9. unpow2 78.0%

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

      10. fma-udef 78.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \color{blue}{\mathsf{fma}\left(b, b, \left(c \cdot 4\right) \cdot a\right)}}}{a \cdot 2} \]

      11. associate-*l* 78.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(4 \cdot a\right)}\right)}}{a \cdot 2} \]

      12. *-commutative 78.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot 4\right)}\right)}}{a \cdot 2} \]

   8. Simplified 78.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}{a \cdot 2} \]

   9. Taylor expanded in b around 0 70.0%

      \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}}{a \cdot 2} \]
   10. Step-by-step derivation

       1. mul-1-neg 70.0%

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

       2. unsub-neg 70.0%

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

       3. distribute-rgt-out 70.2%

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

       4. metadata-eval 70.2%

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

       5. associate-*r* 70.2%

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

   11. Simplified 70.2%

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

   if 1.9e-50 < b

   1. Initial program 16.1%

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

      1. *-commutative 16.1%

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

   3. Simplified 16.1%

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

   5. Taylor expanded in b around inf 82.5%

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

      1. mul-1-neg 82.5%

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

      2. distribute-neg-frac 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+80}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b} - \frac{b}{a}\\ t_1 := 0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{if}\;b \leq -8.6 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5.5 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (/ c b) (/ b a))) (t_1 (* 0.5 (/ (sqrt (* a (* c -4.0))) a))))
   (if (<= b -8.6e+38)
     t_0
     (if (<= b -7.2e+14)
       t_1
       (if (<= b -5.5e-7) t_0 (if (<= b 8.4e-53) t_1 (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = 0.5 * (sqrt((a * (c * -4.0))) / a);
	double tmp;
	if (b <= -8.6e+38) {
		tmp = t_0;
	} else if (b <= -7.2e+14) {
		tmp = t_1;
	} else if (b <= -5.5e-7) {
		tmp = t_0;
	} else if (b <= 8.4e-53) {
		tmp = t_1;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (c / b) - (b / a)
    t_1 = 0.5d0 * (sqrt((a * (c * (-4.0d0)))) / a)
    if (b <= (-8.6d+38)) then
        tmp = t_0
    else if (b <= (-7.2d+14)) then
        tmp = t_1
    else if (b <= (-5.5d-7)) then
        tmp = t_0
    else if (b <= 8.4d-53) then
        tmp = t_1
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = 0.5 * (Math.sqrt((a * (c * -4.0))) / a);
	double tmp;
	if (b <= -8.6e+38) {
		tmp = t_0;
	} else if (b <= -7.2e+14) {
		tmp = t_1;
	} else if (b <= -5.5e-7) {
		tmp = t_0;
	} else if (b <= 8.4e-53) {
		tmp = t_1;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (c / b) - (b / a)
	t_1 = 0.5 * (math.sqrt((a * (c * -4.0))) / a)
	tmp = 0
	if b <= -8.6e+38:
		tmp = t_0
	elif b <= -7.2e+14:
		tmp = t_1
	elif b <= -5.5e-7:
		tmp = t_0
	elif b <= 8.4e-53:
		tmp = t_1
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / b) - Float64(b / a))
	t_1 = Float64(0.5 * Float64(sqrt(Float64(a * Float64(c * -4.0))) / a))
	tmp = 0.0
	if (b <= -8.6e+38)
		tmp = t_0;
	elseif (b <= -7.2e+14)
		tmp = t_1;
	elseif (b <= -5.5e-7)
		tmp = t_0;
	elseif (b <= 8.4e-53)
		tmp = t_1;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (c / b) - (b / a);
	t_1 = 0.5 * (sqrt((a * (c * -4.0))) / a);
	tmp = 0.0;
	if (b <= -8.6e+38)
		tmp = t_0;
	elseif (b <= -7.2e+14)
		tmp = t_1;
	elseif (b <= -5.5e-7)
		tmp = t_0;
	elseif (b <= 8.4e-53)
		tmp = t_1;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8.6e+38], t$95$0, If[LessEqual[b, -7.2e+14], t$95$1, If[LessEqual[b, -5.5e-7], t$95$0, If[LessEqual[b, 8.4e-53], t$95$1, N[((-c) / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.5999999999999994e38 or -7.2e14 < b < -5.5000000000000003e-7

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in b around -inf 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 -8.5999999999999994e38 < b < -7.2e14 or -5.5000000000000003e-7 < b < 8.3999999999999991e-53

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

   3. Simplified 77.2%

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

      1. prod-diff 76.9%

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

      2. *-commutative 76.9%

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

      3. fma-def 76.9%

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

      4. associate-+l+ 76.9%

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

      5. pow2 76.9%

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

      6. distribute-lft-neg-in 76.9%

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

      7. *-commutative 76.9%

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

      8. distribute-rgt-neg-in 76.9%

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

      9. metadata-eval 76.9%

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

      10. associate-*r* 76.9%

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

      11. *-commutative 76.9%

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

      12. *-commutative 76.9%

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

      13. fma-udef 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. +-commutative 76.9%

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

      2. associate-+r+ 76.9%

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

      3. associate-+l+ 76.9%

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

      4. associate-*r* 76.9%

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

      5. associate-*r* 76.9%

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

      6. distribute-lft-out 76.9%

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

      7. metadata-eval 76.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot \color{blue}{-8} + \left(\left(c \cdot 4\right) \cdot a + {b}^{2}\right)}}{a \cdot 2} \]

      8. +-commutative 76.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \color{blue}{\left({b}^{2} + \left(c \cdot 4\right) \cdot a\right)}}}{a \cdot 2} \]

      9. unpow2 76.9%

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

      10. fma-udef 76.9%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \color{blue}{\mathsf{fma}\left(b, b, \left(c \cdot 4\right) \cdot a\right)}}}{a \cdot 2} \]

      11. associate-*l* 76.9%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(4 \cdot a\right)}\right)}}{a \cdot 2} \]

      12. *-commutative 76.9%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot 4\right)}\right)}}{a \cdot 2} \]

   8. Simplified 76.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -8 + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}{a \cdot 2} \]

   9. Taylor expanded in b around 0 68.1%

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

       1. associate-*l/ 68.3%

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

       2. *-lft-identity 68.3%

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

       3. distribute-rgt-out 68.6%

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

       4. metadata-eval 68.6%

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

       5. associate-*r* 68.6%

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

   11. Simplified 68.6%

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

   if 8.3999999999999991e-53 < b

   1. Initial program 16.1%

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

      1. *-commutative 16.1%

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

   3. Simplified 16.1%

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

   5. Taylor expanded in b around inf 82.5%

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

      1. mul-1-neg 82.5%

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

      2. distribute-neg-frac 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{+38}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{elif}\;b \leq -5.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.6% 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(Float64(-c) / b);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.5%

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

      1. *-commutative 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in b around -inf 62.3%

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

      1. +-commutative 62.3%

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

      2. mul-1-neg 62.3%

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

      3. unsub-neg 62.3%

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

   7. Simplified 62.3%

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

   if -4.999999999999985e-310 < b

   1. Initial program 33.6%

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

      1. *-commutative 33.6%

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

   3. Simplified 33.6%

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

   5. Taylor expanded in b around inf 63.2%

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

      1. mul-1-neg 63.2%

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

      2. distribute-neg-frac 63.2%

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

   7. Simplified 63.2%

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

4. Final simplification 62.8%

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

Alternative 6: 45.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -5e-310) (/ (- b) a) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = -b / a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -b / a
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[((-b) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.5%

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

      1. *-commutative 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in b around -inf 62.0%

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

      1. associate-*r/ 62.0%

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

      2. mul-1-neg 62.0%

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

   7. Simplified 62.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 33.6%

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

      1. *-commutative 33.6%

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

   3. Simplified 33.6%

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

   6. Applied egg-rr 31.8%

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

      1. fma-def 30.2%

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

      2. distribute-neg-frac 30.2%

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

   8. Simplified 30.2%

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

   9. Taylor expanded in a around 0 16.1%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot b + 0.5 \cdot b}{a}} \]
   10. Step-by-step derivation

       1. distribute-rgt-out 16.1%

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

       2. metadata-eval 16.1%

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

       3. associate-*l/ 14.3%

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

       4. mul0-rgt 16.1%

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

   11. Simplified 16.1%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{-290}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 7e-290) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7e-290) {
		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 <= 7d-290) 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 <= 7e-290) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7e-290:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7e-290)
		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 <= 7e-290)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7e-290], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.99999999999999963e-290

   1. Initial program 74.3%

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

      1. *-commutative 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in b around -inf 60.5%

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

      1. associate-*r/ 60.5%

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

      2. mul-1-neg 60.5%

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

   7. Simplified 60.5%

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

   if 6.99999999999999963e-290 < b

   1. Initial program 32.9%

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

      1. *-commutative 32.9%

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

   3. Simplified 32.9%

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

   5. Taylor expanded in b around inf 64.5%

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

      1. mul-1-neg 64.5%

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

      2. distribute-neg-frac 64.5%

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

   7. Simplified 64.5%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{-290}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 11.1% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 51.2%

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

   1. *-commutative 51.2%

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

3. Simplified 51.2%

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

6. Applied egg-rr 50.1%

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

   1. fma-def 49.2%

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

   2. distribute-neg-frac 49.2%

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

8. Simplified 49.2%

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

9. Taylor expanded in a around 0 10.3%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot b + 0.5 \cdot b}{a}} \]
10. Step-by-step derivation

    1. distribute-rgt-out 10.3%

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

    2. metadata-eval 10.3%

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

    3. associate-*l/ 9.2%

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

    4. mul0-rgt 10.3%

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

11. Simplified 10.3%

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

12. Final simplification 10.3%

    \[\leadsto 0 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))