jeff quadratic root 1

Percentage Accurate: 71.7% → 89.5%

Time: 10.6s

Alternatives: 9

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_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 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (-b - t_0) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + t_0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_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 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (-b - t_0) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + t_0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\\ t_1 := \frac{c}{-b}\\ \mathbf{if}\;b \leq -1.22 \cdot 10^{+156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+121}:\\ \;\;\;\;\frac{t\_0 + b}{a} \cdot -0.5\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+279}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\mathsf{fma}\left(-2, \frac{c \cdot a}{b}, b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+300}:\\ \;\;\;\;c \cdot \frac{-b}{c \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma -4.0 (* c a) (* b b)))) (t_1 (/ c (- b))))
   (if (<= b -1.22e+156)
     t_1
     (if (<= b -1.2e-281)
       (/ (+ c c) (- t_0 b))
       (if (<= b 5e+121)
         (* (/ (+ t_0 b) a) -0.5)
         (if (<= b 3.1e+279)
           (if (>= b 0.0)
             (fma (/ b a) -1.0 (/ c b))
             (/ (+ c c) (- (fma -2.0 (/ (* c a) b) b) b)))
           (if (<= b 1.45e+300) (* c (/ (- b) (* c a))) t_1)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(-4.0, (c * a), (b * b)));
	double t_1 = c / -b;
	double tmp;
	if (b <= -1.22e+156) {
		tmp = t_1;
	} else if (b <= -1.2e-281) {
		tmp = (c + c) / (t_0 - b);
	} else if (b <= 5e+121) {
		tmp = ((t_0 + b) / a) * -0.5;
	} else if (b <= 3.1e+279) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = fma((b / a), -1.0, (c / b));
		} else {
			tmp_1 = (c + c) / (fma(-2.0, ((c * a) / b), b) - b);
		}
		tmp = tmp_1;
	} else if (b <= 1.45e+300) {
		tmp = c * (-b / (c * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))
	t_1 = Float64(c / Float64(-b))
	tmp = 0.0
	if (b <= -1.22e+156)
		tmp = t_1;
	elseif (b <= -1.2e-281)
		tmp = Float64(Float64(c + c) / Float64(t_0 - b));
	elseif (b <= 5e+121)
		tmp = Float64(Float64(Float64(t_0 + b) / a) * -0.5);
	elseif (b <= 3.1e+279)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = fma(Float64(b / a), -1.0, Float64(c / b));
		else
			tmp_1 = Float64(Float64(c + c) / Float64(fma(-2.0, Float64(Float64(c * a) / b), b) - b));
		end
		tmp = tmp_1;
	elseif (b <= 1.45e+300)
		tmp = Float64(c * Float64(Float64(-b) / Float64(c * a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c / (-b)), $MachinePrecision]}, If[LessEqual[b, -1.22e+156], t$95$1, If[LessEqual[b, -1.2e-281], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+121], N[(N[(N[(t$95$0 + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[b, 3.1e+279], If[GreaterEqual[b, 0.0], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[(-2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.45e+300], N[(c * N[((-b) / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.21999999999999991e156 or 1.44999999999999993e300 < b

   1. Initial program 40.8%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 38.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 38.5%

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

   8. Taylor expanded in a around -inf

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 91.2%

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

   if -1.21999999999999991e156 < b < -1.2e-281

   1. Initial program 89.8%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 89.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 89.8%

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

   9. Applied rewrites 89.8%

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

   if -1.2e-281 < b < 5.00000000000000007e121

   1. Initial program 80.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 80.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right)\\ \end{array} \]

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}{a}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      4. if-same N/A

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

      5. *-commutative N/A

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

   7. Applied rewrites 80.4%

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

   if 5.00000000000000007e121 < b < 3.09999999999999984e279

   1. Initial program 44.5%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   4. Step-by-step derivation

   5. Applied rewrites 44.5%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ } \end{array}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a} + \color{blue}{\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, \color{blue}{-1}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]
   9. Step-by-step derivation

   10. Applied rewrites 97.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{\color{blue}{c}}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]

   11. Taylor expanded in a around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(b + -2 \cdot \frac{a \cdot c}{b}\right) - b}\\ \end{array} \]
   12. Step-by-step derivation

   13. Applied rewrites 97.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\mathsf{fma}\left(-2, \frac{c \cdot a}{b}, b\right) - b}\\ \end{array} \]

   if 3.09999999999999984e279 < b < 1.44999999999999993e300

   1. Initial program 44.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 0.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 1.9%

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

   9. Applied rewrites 1.9%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 100.0%

       \[\leadsto c \cdot \frac{-b}{\color{blue}{c \cdot a}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{-b}\\ t_1 := c \cdot \frac{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)}{c}\\ \mathbf{if}\;b \leq -1.22 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-235}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{+308}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (- b))) (t_1 (* c (/ (fma (/ b a) -1.0 (/ c b)) c))))
   (if (<= b -1.22e+156)
     t_0
     (if (<= b 3.6e-235)
       (/ (+ c c) (- (sqrt (fma -4.0 (* c a) (* b b))) b))
       (if (<= b 7e-159)
         t_1
         (if (<= b 1.8e+148)
           (if (>= b 0.0)
             (/ (+ b (sqrt (* b b))) (* 2.0 (- a)))
             (/ (* 2.0 c) (+ (- b) (- b))))
           (if (<= b 1.25e+308) t_1 t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = c / -b;
	double t_1 = c * (fma((b / a), -1.0, (c / b)) / c);
	double tmp;
	if (b <= -1.22e+156) {
		tmp = t_0;
	} else if (b <= 3.6e-235) {
		tmp = (c + c) / (sqrt(fma(-4.0, (c * a), (b * b))) - b);
	} else if (b <= 7e-159) {
		tmp = t_1;
	} else if (b <= 1.8e+148) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (b + sqrt((b * b))) / (2.0 * -a);
		} else {
			tmp_1 = (2.0 * c) / (-b + -b);
		}
		tmp = tmp_1;
	} else if (b <= 1.25e+308) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / Float64(-b))
	t_1 = Float64(c * Float64(fma(Float64(b / a), -1.0, Float64(c / b)) / c))
	tmp = 0.0
	if (b <= -1.22e+156)
		tmp = t_0;
	elseif (b <= 3.6e-235)
		tmp = Float64(Float64(c + c) / Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b));
	elseif (b <= 7e-159)
		tmp = t_1;
	elseif (b <= 1.8e+148)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = Float64(Float64(b + sqrt(Float64(b * b))) / Float64(2.0 * Float64(-a)));
		else
			tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + Float64(-b)));
		end
		tmp = tmp_1;
	elseif (b <= 1.25e+308)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / (-b)), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.22e+156], t$95$0, If[LessEqual[b, 3.6e-235], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e-159], t$95$1, If[LessEqual[b, 1.8e+148], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + (-b)), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.25e+308], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.21999999999999991e156 or 1.25e308 < b

   1. Initial program 41.7%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 39.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 39.3%

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

   8. Taylor expanded in a around -inf

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 93.4%

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

   if -1.21999999999999991e156 < b < 3.59999999999999999e-235

   1. Initial program 88.6%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 88.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 88.7%

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

   9. Applied rewrites 88.7%

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

   if 3.59999999999999999e-235 < b < 7.00000000000000005e-159 or 1.80000000000000003e148 < b < 1.25e308

   1. Initial program 36.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 9.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 13.2%

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

   9. Applied rewrites 13.2%

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

   10. Taylor expanded in c around 0

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

   12. Applied rewrites 76.3%

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

   if 7.00000000000000005e-159 < b < 1.80000000000000003e148

   1. Initial program 86.7%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + -1 \cdot b}}\\ \end{array} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(\mathsf{neg}\left(b\right)\right)}}\\ \end{array} \]

      2. lower-neg.f64 86.7

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(-b\right)}}\\ \end{array} \]

   5. Applied rewrites 86.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(-b\right)}}\\ \end{array} \]

   6. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]

      2. lower-*.f64 63.2

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]

   8. Applied rewrites 63.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{+156}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-235}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-159}:\\ \;\;\;\;c \cdot \frac{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)}{c}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{+308}:\\ \;\;\;\;c \cdot \frac{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.22 \cdot 10^{+156}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-235}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-167} \lor \neg \left(b \leq 8.5 \cdot 10^{+149}\right):\\ \;\;\;\;c \cdot \frac{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + b}{a} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma -4.0 (* c a) (* b b)))))
   (if (<= b -1.22e+156)
     (/ c (- b))
     (if (<= b 3.6e-235)
       (/ (+ c c) (- t_0 b))
       (if (or (<= b 1.8e-167) (not (<= b 8.5e+149)))
         (* c (/ (fma (/ b a) -1.0 (/ c b)) c))
         (* (/ (+ t_0 b) a) -0.5))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(-4.0, (c * a), (b * b)));
	double tmp;
	if (b <= -1.22e+156) {
		tmp = c / -b;
	} else if (b <= 3.6e-235) {
		tmp = (c + c) / (t_0 - b);
	} else if ((b <= 1.8e-167) || !(b <= 8.5e+149)) {
		tmp = c * (fma((b / a), -1.0, (c / b)) / c);
	} else {
		tmp = ((t_0 + b) / a) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))
	tmp = 0.0
	if (b <= -1.22e+156)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 3.6e-235)
		tmp = Float64(Float64(c + c) / Float64(t_0 - b));
	elseif ((b <= 1.8e-167) || !(b <= 8.5e+149))
		tmp = Float64(c * Float64(fma(Float64(b / a), -1.0, Float64(c / b)) / c));
	else
		tmp = Float64(Float64(Float64(t_0 + b) / a) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.22e+156], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 3.6e-235], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 1.8e-167], N[Not[LessEqual[b, 8.5e+149]], $MachinePrecision]], N[(c * N[(N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.21999999999999991e156

   1. Initial program 40.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 40.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 40.2%

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

   8. Taylor expanded in a around -inf

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 95.6%

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

   if -1.21999999999999991e156 < b < 3.59999999999999999e-235

   1. Initial program 88.6%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 88.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 88.7%

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

   9. Applied rewrites 88.7%

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

   if 3.59999999999999999e-235 < b < 1.8e-167 or 8.49999999999999956e149 < b

   1. Initial program 36.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 9.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 13.2%

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

   9. Applied rewrites 13.3%

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

   10. Taylor expanded in c around 0

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

   12. Applied rewrites 76.3%

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

   if 1.8e-167 < b < 8.49999999999999956e149

   1. Initial program 86.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 86.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right)\\ \end{array} \]

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}{a}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      4. if-same N/A

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

      5. *-commutative N/A

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

   7. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{a} \cdot -0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{+156}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-235}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-167} \lor \neg \left(b \leq 8.5 \cdot 10^{+149}\right):\\ \;\;\;\;c \cdot \frac{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{a} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{-b}\\ \mathbf{if}\;b \leq -1.22 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-147}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+300}:\\ \;\;\;\;c \cdot \frac{-b}{c \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (- b))))
   (if (<= b -1.22e+156)
     t_0
     (if (<= b 3.8e-147)
       (/ (+ c c) (- (sqrt (fma -4.0 (* c a) (* b b))) b))
       (if (<= b 1.8e+148)
         (if (>= b 0.0)
           (/ (+ b (sqrt (* b b))) (* 2.0 (- a)))
           (/ (* 2.0 c) (+ (- b) (- b))))
         (if (<= b 1.45e+300) (* c (/ (- b) (* c a))) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = c / -b;
	double tmp;
	if (b <= -1.22e+156) {
		tmp = t_0;
	} else if (b <= 3.8e-147) {
		tmp = (c + c) / (sqrt(fma(-4.0, (c * a), (b * b))) - b);
	} else if (b <= 1.8e+148) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (b + sqrt((b * b))) / (2.0 * -a);
		} else {
			tmp_1 = (2.0 * c) / (-b + -b);
		}
		tmp = tmp_1;
	} else if (b <= 1.45e+300) {
		tmp = c * (-b / (c * a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / Float64(-b))
	tmp = 0.0
	if (b <= -1.22e+156)
		tmp = t_0;
	elseif (b <= 3.8e-147)
		tmp = Float64(Float64(c + c) / Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b));
	elseif (b <= 1.8e+148)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = Float64(Float64(b + sqrt(Float64(b * b))) / Float64(2.0 * Float64(-a)));
		else
			tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + Float64(-b)));
		end
		tmp = tmp_1;
	elseif (b <= 1.45e+300)
		tmp = Float64(c * Float64(Float64(-b) / Float64(c * a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / (-b)), $MachinePrecision]}, If[LessEqual[b, -1.22e+156], t$95$0, If[LessEqual[b, 3.8e-147], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e+148], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + (-b)), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.45e+300], N[(c * N[((-b) / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.21999999999999991e156 or 1.44999999999999993e300 < b

   1. Initial program 40.8%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 38.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 38.5%

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

   8. Taylor expanded in a around -inf

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 91.2%

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

   if -1.21999999999999991e156 < b < 3.80000000000000028e-147

   1. Initial program 85.3%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 84.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 84.8%

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

   9. Applied rewrites 84.8%

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

   if 3.80000000000000028e-147 < b < 1.80000000000000003e148

   1. Initial program 86.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + -1 \cdot b}}\\ \end{array} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(\mathsf{neg}\left(b\right)\right)}}\\ \end{array} \]

      2. lower-neg.f64 86.4

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(-b\right)}}\\ \end{array} \]

   5. Applied rewrites 86.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(-b\right)}}\\ \end{array} \]

   6. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]

      2. lower-*.f64 64.2

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]

   8. Applied rewrites 64.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]

   if 1.80000000000000003e148 < b < 1.44999999999999993e300

   1. Initial program 35.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 3.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 7.6%

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

   9. Applied rewrites 7.7%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 71.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{+156}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-147}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+300}:\\ \;\;\;\;c \cdot \frac{-b}{c \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{-b}\\ \mathbf{if}\;b \leq -4 \cdot 10^{-135}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.4 \cdot 10^{-181}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+300}:\\ \;\;\;\;c \cdot \frac{-b}{c \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (- b))))
   (if (<= b -4e-135)
     t_0
     (if (<= b 7.4e-181)
       (- (sqrt (* (/ c a) -1.0)))
       (if (<= b 1.8e+148)
         (if (>= b 0.0)
           (/ (+ b (sqrt (* b b))) (* 2.0 (- a)))
           (/ (* 2.0 c) (+ (- b) (- b))))
         (if (<= b 1.45e+300) (* c (/ (- b) (* c a))) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = c / -b;
	double tmp;
	if (b <= -4e-135) {
		tmp = t_0;
	} else if (b <= 7.4e-181) {
		tmp = -sqrt(((c / a) * -1.0));
	} else if (b <= 1.8e+148) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (b + sqrt((b * b))) / (2.0 * -a);
		} else {
			tmp_1 = (2.0 * c) / (-b + -b);
		}
		tmp = tmp_1;
	} else if (b <= 1.45e+300) {
		tmp = c * (-b / (c * a));
	} else {
		tmp = t_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
    real(8) :: tmp_1
    t_0 = c / -b
    if (b <= (-4d-135)) then
        tmp = t_0
    else if (b <= 7.4d-181) then
        tmp = -sqrt(((c / a) * (-1.0d0)))
    else if (b <= 1.8d+148) then
        if (b >= 0.0d0) then
            tmp_1 = (b + sqrt((b * b))) / (2.0d0 * -a)
        else
            tmp_1 = (2.0d0 * c) / (-b + -b)
        end if
        tmp = tmp_1
    else if (b <= 1.45d+300) then
        tmp = c * (-b / (c * a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c / -b;
	double tmp;
	if (b <= -4e-135) {
		tmp = t_0;
	} else if (b <= 7.4e-181) {
		tmp = -Math.sqrt(((c / a) * -1.0));
	} else if (b <= 1.8e+148) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (b + Math.sqrt((b * b))) / (2.0 * -a);
		} else {
			tmp_1 = (2.0 * c) / (-b + -b);
		}
		tmp = tmp_1;
	} else if (b <= 1.45e+300) {
		tmp = c * (-b / (c * a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c / -b
	tmp = 0
	if b <= -4e-135:
		tmp = t_0
	elif b <= 7.4e-181:
		tmp = -math.sqrt(((c / a) * -1.0))
	elif b <= 1.8e+148:
		tmp_1 = 0
		if b >= 0.0:
			tmp_1 = (b + math.sqrt((b * b))) / (2.0 * -a)
		else:
			tmp_1 = (2.0 * c) / (-b + -b)
		tmp = tmp_1
	elif b <= 1.45e+300:
		tmp = c * (-b / (c * a))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c / Float64(-b))
	tmp = 0.0
	if (b <= -4e-135)
		tmp = t_0;
	elseif (b <= 7.4e-181)
		tmp = Float64(-sqrt(Float64(Float64(c / a) * -1.0)));
	elseif (b <= 1.8e+148)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = Float64(Float64(b + sqrt(Float64(b * b))) / Float64(2.0 * Float64(-a)));
		else
			tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + Float64(-b)));
		end
		tmp = tmp_1;
	elseif (b <= 1.45e+300)
		tmp = Float64(c * Float64(Float64(-b) / Float64(c * a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = c / -b;
	tmp = 0.0;
	if (b <= -4e-135)
		tmp = t_0;
	elseif (b <= 7.4e-181)
		tmp = -sqrt(((c / a) * -1.0));
	elseif (b <= 1.8e+148)
		tmp_2 = 0.0;
		if (b >= 0.0)
			tmp_2 = (b + sqrt((b * b))) / (2.0 * -a);
		else
			tmp_2 = (2.0 * c) / (-b + -b);
		end
		tmp = tmp_2;
	elseif (b <= 1.45e+300)
		tmp = c * (-b / (c * a));
	else
		tmp = t_0;
	end
	tmp_3 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / (-b)), $MachinePrecision]}, If[LessEqual[b, -4e-135], t$95$0, If[LessEqual[b, 7.4e-181], (-N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[b, 1.8e+148], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + (-b)), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.45e+300], N[(c * N[((-b) / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.0000000000000002e-135 or 1.44999999999999993e300 < b

   1. Initial program 73.6%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 72.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 72.7%

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

   8. Taylor expanded in a around -inf

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 79.1%

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

   if -4.0000000000000002e-135 < b < 7.39999999999999968e-181

   1. Initial program 76.6%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 75.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in a around -inf

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

   10. Applied rewrites 0.0%

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

   12. Applied rewrites 46.9%

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

   if 7.39999999999999968e-181 < b < 1.80000000000000003e148

   1. Initial program 83.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + -1 \cdot b}}\\ \end{array} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(\mathsf{neg}\left(b\right)\right)}}\\ \end{array} \]

      2. lower-neg.f64 83.4

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(-b\right)}}\\ \end{array} \]

   5. Applied rewrites 83.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(-b\right)}}\\ \end{array} \]

   6. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]

      2. lower-*.f64 59.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]

   8. Applied rewrites 59.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]

   if 1.80000000000000003e148 < b < 1.44999999999999993e300

   1. Initial program 35.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 3.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 7.6%

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

   9. Applied rewrites 7.7%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 71.9%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-135}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 7.4 \cdot 10^{-181}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+300}:\\ \;\;\;\;c \cdot \frac{-b}{c \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{-b}\\ \mathbf{if}\;b \leq -4 \cdot 10^{-135}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.4 \cdot 10^{-181}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{+292}:\\ \;\;\;\;c \cdot \frac{-b}{c \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (- b))))
   (if (<= b -4e-135)
     t_0
     (if (<= b 7.4e-181)
       (- (sqrt (* (/ c a) -1.0)))
       (if (<= b 2.1e+292) (* c (/ (- b) (* c a))) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = c / -b;
	double tmp;
	if (b <= -4e-135) {
		tmp = t_0;
	} else if (b <= 7.4e-181) {
		tmp = -sqrt(((c / a) * -1.0));
	} else if (b <= 2.1e+292) {
		tmp = c * (-b / (c * a));
	} else {
		tmp = t_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 = c / -b
    if (b <= (-4d-135)) then
        tmp = t_0
    else if (b <= 7.4d-181) then
        tmp = -sqrt(((c / a) * (-1.0d0)))
    else if (b <= 2.1d+292) then
        tmp = c * (-b / (c * a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c / -b;
	double tmp;
	if (b <= -4e-135) {
		tmp = t_0;
	} else if (b <= 7.4e-181) {
		tmp = -Math.sqrt(((c / a) * -1.0));
	} else if (b <= 2.1e+292) {
		tmp = c * (-b / (c * a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c / -b
	tmp = 0
	if b <= -4e-135:
		tmp = t_0
	elif b <= 7.4e-181:
		tmp = -math.sqrt(((c / a) * -1.0))
	elif b <= 2.1e+292:
		tmp = c * (-b / (c * a))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c / Float64(-b))
	tmp = 0.0
	if (b <= -4e-135)
		tmp = t_0;
	elseif (b <= 7.4e-181)
		tmp = Float64(-sqrt(Float64(Float64(c / a) * -1.0)));
	elseif (b <= 2.1e+292)
		tmp = Float64(c * Float64(Float64(-b) / Float64(c * a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c / -b;
	tmp = 0.0;
	if (b <= -4e-135)
		tmp = t_0;
	elseif (b <= 7.4e-181)
		tmp = -sqrt(((c / a) * -1.0));
	elseif (b <= 2.1e+292)
		tmp = c * (-b / (c * a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / (-b)), $MachinePrecision]}, If[LessEqual[b, -4e-135], t$95$0, If[LessEqual[b, 7.4e-181], (-N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[b, 2.1e+292], N[(c * N[((-b) / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.0000000000000002e-135 or 2.1000000000000002e292 < b

   1. Initial program 73.3%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 71.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in a around -inf

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 77.8%

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

   if -4.0000000000000002e-135 < b < 7.39999999999999968e-181

   1. Initial program 76.6%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 75.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in a around -inf

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

   10. Applied rewrites 0.0%

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

   12. Applied rewrites 46.9%

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

   if 7.39999999999999968e-181 < b < 2.1000000000000002e292

   1. Initial program 67.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 21.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 22.1%

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

   9. Applied rewrites 22.1%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 51.3%

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

Alternative 7: 59.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-308} \lor \neg \left(b \leq 2.1 \cdot 10^{+292}\right):\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-b}{c \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (or (<= b -1.1e-308) (not (<= b 2.1e+292)))
   (/ c (- b))
   (* c (/ (- b) (* c a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if ((b <= -1.1e-308) || !(b <= 2.1e+292)) {
		tmp = c / -b;
	} else {
		tmp = c * (-b / (c * 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 <= (-1.1d-308)) .or. (.not. (b <= 2.1d+292))) then
        tmp = c / -b
    else
        tmp = c * (-b / (c * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if ((b <= -1.1e-308) || !(b <= 2.1e+292)) {
		tmp = c / -b;
	} else {
		tmp = c * (-b / (c * a));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if (b <= -1.1e-308) or not (b <= 2.1e+292):
		tmp = c / -b
	else:
		tmp = c * (-b / (c * a))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if ((b <= -1.1e-308) || !(b <= 2.1e+292))
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(c * Float64(Float64(-b) / Float64(c * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if ((b <= -1.1e-308) || ~((b <= 2.1e+292)))
		tmp = c / -b;
	else
		tmp = c * (-b / (c * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[Or[LessEqual[b, -1.1e-308], N[Not[LessEqual[b, 2.1e+292]], $MachinePrecision]], N[(c / (-b)), $MachinePrecision], N[(c * N[((-b) / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001e-308 or 2.1000000000000002e292 < b

   1. Initial program 74.1%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 72.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 72.7%

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

   8. Taylor expanded in a around -inf

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 65.1%

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

   if -1.1000000000000001e-308 < b < 2.1000000000000002e292

   1. Initial program 68.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 29.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 30.1%

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

   9. Applied rewrites 30.1%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 43.7%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-308} \lor \neg \left(b \leq 2.1 \cdot 10^{+292}\right):\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-b}{c \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-264} \lor \neg \left(b \leq 7.6 \cdot 10^{+212}\right):\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot b}{0} \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (or (<= b -3e-264) (not (<= b 7.6e+212)))
   (/ c (- b))
   (* (/ (* c b) 0.0) 4.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if ((b <= -3e-264) || !(b <= 7.6e+212)) {
		tmp = c / -b;
	} else {
		tmp = ((c * b) / 0.0) * 4.0;
	}
	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 <= (-3d-264)) .or. (.not. (b <= 7.6d+212))) then
        tmp = c / -b
    else
        tmp = ((c * b) / 0.0d0) * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if ((b <= -3e-264) || !(b <= 7.6e+212)) {
		tmp = c / -b;
	} else {
		tmp = ((c * b) / 0.0) * 4.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if (b <= -3e-264) or not (b <= 7.6e+212):
		tmp = c / -b
	else:
		tmp = ((c * b) / 0.0) * 4.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if ((b <= -3e-264) || !(b <= 7.6e+212))
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(Float64(c * b) / 0.0) * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if ((b <= -3e-264) || ~((b <= 7.6e+212)))
		tmp = c / -b;
	else
		tmp = ((c * b) / 0.0) * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[Or[LessEqual[b, -3e-264], N[Not[LessEqual[b, 7.6e+212]], $MachinePrecision]], N[(c / (-b)), $MachinePrecision], N[(N[(N[(c * b), $MachinePrecision] / 0.0), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3e-264 or 7.59999999999999976e212 < b

   1. Initial program 68.6%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 63.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 63.9%

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

   8. Taylor expanded in a around -inf

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 59.5%

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

   if -3e-264 < b < 7.59999999999999976e212

   1. Initial program 76.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 37.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

      4. if-same N/A

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

      5. associate-*r/ N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

   7. Applied rewrites 38.1%

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

   9. Applied rewrites 34.8%

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

   10. Taylor expanded in a around 0

       \[\leadsto 4 \cdot \color{blue}{\frac{b \cdot c}{-1 \cdot {b}^{2} + {b}^{2}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 6.2%

       \[\leadsto \frac{c \cdot b}{0} \cdot \color{blue}{4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-264} \lor \neg \left(b \leq 7.6 \cdot 10^{+212}\right):\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot b}{0} \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.5%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
2. Add Preprocessing

4. Applied rewrites 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

5. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ } \end{array}} \]
6. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

   2. metadata-eval N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} + -1 \cdot b}\\ \end{array} \]

   3. +-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}\\ \end{array} \]

   4. if-same N/A

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

   5. associate-*r/ N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

7. Applied rewrites 53.9%

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

8. Taylor expanded in a around -inf

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

10. Applied rewrites 0.0%

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

11. Taylor expanded in b around -inf

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

13. Applied rewrites 37.3%

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

Reproduce

herbie shell --seed 2024338 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))