Quadratic roots, full range

Percentage Accurate: 51.6% → 85.3%

Time: 16.3s

Alternatives: 9

Speedup: 12.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot -4\right)\\ \mathbf{if}\;b \leq -2 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{a \cdot 2}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.55 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{t_0}{b + \mathsf{hypot}\left(\sqrt{t_0}, b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(\frac{b}{c} \cdot -0.5 + 0.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c -4.0))))
   (if (<= b -2e+152)
     (/ (fma b -2.0 (/ (* a 2.0) (/ b c))) (* a 2.0))
     (if (<= b 5.9e-139)
       (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
       (if (<= b 3.5e-80)
         (/ (- c) b)
         (if (<= b 3.55e-37)
           (/ (/ t_0 (+ b (hypot (sqrt t_0) b))) (* a 2.0))
           (/ 1.0 (* 2.0 (+ (* (/ b c) -0.5) (* 0.5 (/ a b)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * -4.0);
	double tmp;
	if (b <= -2e+152) {
		tmp = fma(b, -2.0, ((a * 2.0) / (b / c))) / (a * 2.0);
	} else if (b <= 5.9e-139) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else if (b <= 3.5e-80) {
		tmp = -c / b;
	} else if (b <= 3.55e-37) {
		tmp = (t_0 / (b + hypot(sqrt(t_0), b))) / (a * 2.0);
	} else {
		tmp = 1.0 / (2.0 * (((b / c) * -0.5) + (0.5 * (a / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * -4.0))
	tmp = 0.0
	if (b <= -2e+152)
		tmp = Float64(fma(b, -2.0, Float64(Float64(a * 2.0) / Float64(b / c))) / Float64(a * 2.0));
	elseif (b <= 5.9e-139)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	elseif (b <= 3.5e-80)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 3.55e-37)
		tmp = Float64(Float64(t_0 / Float64(b + hypot(sqrt(t_0), b))) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(2.0 * Float64(Float64(Float64(b / c) * -0.5) + Float64(0.5 * Float64(a / b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e+152], N[(N[(b * -2.0 + N[(N[(a * 2.0), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.9e-139], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.5e-80], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 3.55e-37], N[(N[(t$95$0 / N[(b + N[Sqrt[N[Sqrt[t$95$0], $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 * N[(N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -2.0000000000000001e152

   1. Initial program 40.0%

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

      1. *-commutative 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in b around -inf 90.9%

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

      1. *-commutative 90.9%

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

      2. fma-def 90.9%

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

      3. associate-/l* 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -2.0000000000000001e152 < b < 5.8999999999999998e-139

   1. Initial program 83.3%

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

   if 5.8999999999999998e-139 < b < 3.50000000000000015e-80

   1. Initial program 23.4%

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

      1. *-commutative 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in b around inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. distribute-neg-frac 81.2%

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

   7. Simplified 81.2%

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

   if 3.50000000000000015e-80 < b < 3.54999999999999989e-37

   1. Initial program 76.3%

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

      1. *-commutative 76.3%

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

   3. Simplified 76.3%

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

      1. prod-diff 76.3%

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

      2. *-commutative 76.3%

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

      3. fma-def 76.3%

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

      4. associate-+l+ 76.3%

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

      5. pow2 76.3%

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

      6. distribute-lft-neg-in 76.3%

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

      7. *-commutative 76.3%

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

      8. distribute-rgt-neg-in 76.3%

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

      9. metadata-eval 76.3%

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

      10. associate-*r* 76.3%

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

      11. *-commutative 76.3%

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

      12. *-commutative 76.3%

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

      13. fma-udef 76.3%

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

   6. Applied egg-rr 76.3%

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

      1. fma-def 76.3%

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

      2. fma-def 76.3%

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

      3. associate-*l* 76.3%

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

   8. Simplified 76.3%

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

      1. +-commutative 76.3%

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

      2. add-sqr-sqrt 76.1%

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

      3. unpow2 76.1%

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

      4. hypot-def 76.3%

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

      5. *-commutative 76.3%

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

   10. Applied egg-rr 76.3%

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

       1. +-commutative 76.3%

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

       2. flip-+ 75.4%

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

   12. Applied egg-rr 75.3%

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

   14. Simplified 98.9%

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

   if 3.54999999999999989e-37 < b

   1. Initial program 14.1%

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

      1. *-commutative 14.1%

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

   3. Simplified 14.1%

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

   6. Applied egg-rr 5.3%

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

      1. unpow-1 5.3%

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

      2. associate-/r/ 5.3%

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

   8. Applied egg-rr 5.3%

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

      1. div-inv 5.3%

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

      2. frac-2neg 5.3%

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

      3. metadata-eval 5.3%

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

      4. distribute-neg-in 5.3%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 13.3%

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

      7. sqr-neg 13.3%

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

      8. sqrt-prod 11.7%

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

      9. add-sqr-sqrt 14.2%

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

      10. sub-neg 14.2%

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

   10. Applied egg-rr 14.2%

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

   11. Taylor expanded in a around 0 91.5%

       \[\leadsto \frac{1}{\color{blue}{\left(-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)} \cdot 2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{a \cdot 2}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.55 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(\frac{b}{c} \cdot -0.5 + 0.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{a \cdot 2}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-138}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(\frac{b}{c} \cdot -0.5 + 0.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e+150)
   (/ (fma b -2.0 (/ (* a 2.0) (/ b c))) (* a 2.0))
   (if (<= b 2.25e-138)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ 1.0 (* 2.0 (+ (* (/ b c) -0.5) (* 0.5 (/ a b))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e+150) {
		tmp = fma(b, -2.0, ((a * 2.0) / (b / c))) / (a * 2.0);
	} else if (b <= 2.25e-138) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / (2.0 * (((b / c) * -0.5) + (0.5 * (a / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e+150)
		tmp = Float64(fma(b, -2.0, Float64(Float64(a * 2.0) / Float64(b / c))) / Float64(a * 2.0));
	elseif (b <= 2.25e-138)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(2.0 * Float64(Float64(Float64(b / c) * -0.5) + Float64(0.5 * Float64(a / b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e+150], N[(N[(b * -2.0 + N[(N[(a * 2.0), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.25e-138], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 * N[(N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.15000000000000001e150

   1. Initial program 40.0%

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

      1. *-commutative 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in b around -inf 90.9%

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

      1. *-commutative 90.9%

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

      2. fma-def 90.9%

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

      3. associate-/l* 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -1.15000000000000001e150 < b < 2.25000000000000004e-138

   1. Initial program 83.3%

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

   if 2.25000000000000004e-138 < b

   1. Initial program 20.2%

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

      1. *-commutative 20.2%

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

   3. Simplified 20.2%

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

   6. Applied egg-rr 12.2%

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

      1. unpow-1 12.2%

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

      2. associate-/r/ 12.2%

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

   8. Applied egg-rr 12.2%

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

      1. div-inv 12.2%

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

      2. frac-2neg 12.2%

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

      3. metadata-eval 12.2%

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

      4. distribute-neg-in 12.2%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 19.5%

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

      7. sqr-neg 19.5%

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

      8. sqrt-prod 18.1%

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

      9. add-sqr-sqrt 20.2%

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

      10. sub-neg 20.2%

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

   10. Applied egg-rr 20.2%

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

   11. Taylor expanded in a around 0 85.6%

       \[\leadsto \frac{1}{\color{blue}{\left(-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)} \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{a \cdot 2}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-138}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(\frac{b}{c} \cdot -0.5 + 0.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.52 \cdot 10^{-164}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(\frac{b}{c} \cdot -0.5 + 0.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.52e-164)
   (- (/ c b) (/ b a))
   (if (<= b 2.5e-138)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ 1.0 (* 2.0 (+ (* (/ b c) -0.5) (* 0.5 (/ a b))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.52e-164) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.5e-138) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / (2.0 * (((b / c) * -0.5) + (0.5 * (a / b))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.52d-164)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.5d-138) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = 1.0d0 / (2.0d0 * (((b / c) * (-0.5d0)) + (0.5d0 * (a / b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.52e-164) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.5e-138) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / (2.0 * (((b / c) * -0.5) + (0.5 * (a / b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.52e-164:
		tmp = (c / b) - (b / a)
	elif b <= 2.5e-138:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = 1.0 / (2.0 * (((b / c) * -0.5) + (0.5 * (a / b))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.52e-164)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.5e-138)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(2.0 * Float64(Float64(Float64(b / c) * -0.5) + Float64(0.5 * Float64(a / b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.52e-164)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.5e-138)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = 1.0 / (2.0 * (((b / c) * -0.5) + (0.5 * (a / b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.52e-164], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5e-138], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 * N[(N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.52000000000000007e-164

   1. Initial program 69.5%

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

      1. *-commutative 69.5%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in b around -inf 79.0%

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

      1. +-commutative 79.0%

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

      2. mul-1-neg 79.0%

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

      3. unsub-neg 79.0%

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

   7. Simplified 79.0%

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

   if -1.52000000000000007e-164 < b < 2.49999999999999994e-138

   1. Initial program 77.9%

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

      1. *-commutative 77.9%

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

   3. Simplified 77.9%

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

      1. prod-diff 77.6%

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

      2. *-commutative 77.6%

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

      3. fma-def 77.6%

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

      4. associate-+l+ 77.6%

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

      5. pow2 77.6%

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

      6. distribute-lft-neg-in 77.6%

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

      7. *-commutative 77.6%

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

      8. distribute-rgt-neg-in 77.6%

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

      9. metadata-eval 77.6%

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

      10. associate-*r* 77.6%

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

      11. *-commutative 77.6%

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

      12. *-commutative 77.6%

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

      13. fma-udef 77.6%

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

   6. Applied egg-rr 77.6%

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

      1. fma-def 77.6%

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

      2. fma-def 77.7%

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

      3. associate-*l* 77.7%

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

   8. Simplified 77.7%

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

   9. Taylor expanded in b around 0 77.6%

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

       1. mul-1-neg 77.6%

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

       2. unsub-neg 77.6%

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

       3. distribute-rgt-out 77.8%

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

       4. metadata-eval 77.8%

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

       5. associate-*r* 77.9%

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

       6. *-commutative 77.9%

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

   11. Simplified 77.9%

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

   if 2.49999999999999994e-138 < b

   1. Initial program 20.2%

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

      1. *-commutative 20.2%

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

   3. Simplified 20.2%

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

   6. Applied egg-rr 12.2%

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

      1. unpow-1 12.2%

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

      2. associate-/r/ 12.2%

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

   8. Applied egg-rr 12.2%

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

      1. div-inv 12.2%

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

      2. frac-2neg 12.2%

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

      3. metadata-eval 12.2%

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

      4. distribute-neg-in 12.2%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 19.5%

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

      7. sqr-neg 19.5%

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

      8. sqrt-prod 18.1%

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

      9. add-sqr-sqrt 20.2%

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

      10. sub-neg 20.2%

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

   10. Applied egg-rr 20.2%

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

   11. Taylor expanded in a around 0 85.6%

       \[\leadsto \frac{1}{\color{blue}{\left(-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)} \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.52 \cdot 10^{-164}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(\frac{b}{c} \cdot -0.5 + 0.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.25 \cdot 10^{-170}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-138}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(\frac{b}{c} \cdot -0.5 + 0.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.25e-170)
   (- (/ c b) (/ b a))
   (if (<= b 2.5e-138)
     (* 0.5 (/ (sqrt (* a (* c -4.0))) a))
     (/ 1.0 (* 2.0 (+ (* (/ b c) -0.5) (* 0.5 (/ a b))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.25e-170) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.5e-138) {
		tmp = 0.5 * (sqrt((a * (c * -4.0))) / a);
	} else {
		tmp = 1.0 / (2.0 * (((b / c) * -0.5) + (0.5 * (a / b))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.25d-170)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.5d-138) then
        tmp = 0.5d0 * (sqrt((a * (c * (-4.0d0)))) / a)
    else
        tmp = 1.0d0 / (2.0d0 * (((b / c) * (-0.5d0)) + (0.5d0 * (a / b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.25e-170) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.5e-138) {
		tmp = 0.5 * (Math.sqrt((a * (c * -4.0))) / a);
	} else {
		tmp = 1.0 / (2.0 * (((b / c) * -0.5) + (0.5 * (a / b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.25e-170:
		tmp = (c / b) - (b / a)
	elif b <= 2.5e-138:
		tmp = 0.5 * (math.sqrt((a * (c * -4.0))) / a)
	else:
		tmp = 1.0 / (2.0 * (((b / c) * -0.5) + (0.5 * (a / b))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.25e-170)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.5e-138)
		tmp = Float64(0.5 * Float64(sqrt(Float64(a * Float64(c * -4.0))) / a));
	else
		tmp = Float64(1.0 / Float64(2.0 * Float64(Float64(Float64(b / c) * -0.5) + Float64(0.5 * Float64(a / b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.25e-170)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.5e-138)
		tmp = 0.5 * (sqrt((a * (c * -4.0))) / a);
	else
		tmp = 1.0 / (2.0 * (((b / c) * -0.5) + (0.5 * (a / b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.25e-170], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5e-138], N[(0.5 * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 * N[(N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.25000000000000018e-170

   1. Initial program 69.3%

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

      1. *-commutative 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 78.6%

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

      1. +-commutative 78.6%

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

      2. mul-1-neg 78.6%

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

      3. unsub-neg 78.6%

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

   7. Simplified 78.6%

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

   if -3.25000000000000018e-170 < b < 2.49999999999999994e-138

   1. Initial program 78.6%

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

      1. *-commutative 78.6%

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

   3. Simplified 78.6%

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

      1. prod-diff 78.3%

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

      2. *-commutative 78.3%

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

      3. fma-def 78.3%

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

      4. associate-+l+ 78.3%

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

      5. pow2 78.3%

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

      6. distribute-lft-neg-in 78.3%

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

      7. *-commutative 78.3%

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

      8. distribute-rgt-neg-in 78.3%

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

      9. metadata-eval 78.3%

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

      10. associate-*r* 78.3%

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

      11. *-commutative 78.3%

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

      12. *-commutative 78.3%

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

      13. fma-udef 78.3%

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

   6. Applied egg-rr 78.3%

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

      1. fma-def 78.3%

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

      2. fma-def 78.4%

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

      3. associate-*l* 78.4%

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

   8. Simplified 78.4%

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

   9. Taylor expanded in b around 0 78.0%

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

       1. associate-*l/ 78.1%

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

       2. *-lft-identity 78.1%

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

       3. distribute-rgt-out 78.3%

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

       4. metadata-eval 78.3%

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

       5. associate-*r* 78.4%

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

       6. *-commutative 78.4%

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

   11. Simplified 78.4%

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

   if 2.49999999999999994e-138 < b

   1. Initial program 20.2%

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

      1. *-commutative 20.2%

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

   3. Simplified 20.2%

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

   6. Applied egg-rr 12.2%

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

      1. unpow-1 12.2%

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

      2. associate-/r/ 12.2%

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

   8. Applied egg-rr 12.2%

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

      1. div-inv 12.2%

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

      2. frac-2neg 12.2%

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

      3. metadata-eval 12.2%

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

      4. distribute-neg-in 12.2%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 19.5%

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

      7. sqr-neg 19.5%

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

      8. sqrt-prod 18.1%

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

      9. add-sqr-sqrt 20.2%

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

      10. sub-neg 20.2%

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

   10. Applied egg-rr 20.2%

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

   11. Taylor expanded in a around 0 85.6%

       \[\leadsto \frac{1}{\color{blue}{\left(-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)} \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.25 \cdot 10^{-170}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-138}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(\frac{b}{c} \cdot -0.5 + 0.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-290}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(\frac{b}{c} \cdot -0.5 + 0.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-290)
   (- (/ c b) (/ b a))
   (/ 1.0 (* 2.0 (+ (* (/ b c) -0.5) (* 0.5 (/ a b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-290) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = 1.0 / (2.0 * (((b / c) * -0.5) + (0.5 * (a / b))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.4d-290)) then
        tmp = (c / b) - (b / a)
    else
        tmp = 1.0d0 / (2.0d0 * (((b / c) * (-0.5d0)) + (0.5d0 * (a / b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-290) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = 1.0 / (2.0 * (((b / c) * -0.5) + (0.5 * (a / b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e-290:
		tmp = (c / b) - (b / a)
	else:
		tmp = 1.0 / (2.0 * (((b / c) * -0.5) + (0.5 * (a / b))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e-290)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(1.0 / Float64(2.0 * Float64(Float64(Float64(b / c) * -0.5) + Float64(0.5 * Float64(a / b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.4e-290)
		tmp = (c / b) - (b / a);
	else
		tmp = 1.0 / (2.0 * (((b / c) * -0.5) + (0.5 * (a / b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e-290], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 * N[(N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.4000000000000001e-290

   1. Initial program 71.6%

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

      1. *-commutative 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in b around -inf 66.1%

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

      1. +-commutative 66.1%

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

      2. mul-1-neg 66.1%

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

      3. unsub-neg 66.1%

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

   7. Simplified 66.1%

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

   if -2.4000000000000001e-290 < b

   1. Initial program 35.0%

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

      1. *-commutative 35.0%

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

   3. Simplified 35.0%

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

   6. Applied egg-rr 29.2%

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

      1. unpow-1 29.2%

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

      2. associate-/r/ 29.2%

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

   8. Applied egg-rr 29.2%

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

      1. div-inv 29.2%

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

      2. frac-2neg 29.2%

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

      3. metadata-eval 29.2%

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

      4. distribute-neg-in 29.2%

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

      5. add-sqr-sqrt 1.7%

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

      6. sqrt-unprod 34.5%

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

      7. sqr-neg 34.5%

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

      8. sqrt-prod 31.8%

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

      9. add-sqr-sqrt 35.0%

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

      10. sub-neg 35.0%

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

   10. Applied egg-rr 35.0%

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

   11. Taylor expanded in a around 0 66.2%

       \[\leadsto \frac{1}{\color{blue}{\left(-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}\right)} \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-290}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \left(\frac{b}{c} \cdot -0.5 + 0.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.4% accurate, 9.7× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 71.5%

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

      1. *-commutative 71.5%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in b around -inf 64.8%

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

      1. +-commutative 64.8%

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

      2. mul-1-neg 64.8%

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

      3. unsub-neg 64.8%

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

   7. Simplified 64.8%

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

   if -9.999999999999969e-311 < b

   1. Initial program 34.1%

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

      1. *-commutative 34.1%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in b around inf 67.7%

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

      1. mul-1-neg 67.7%

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

      2. distribute-neg-frac 67.7%

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

   7. Simplified 67.7%

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

4. Final simplification 66.1%

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

Alternative 7: 44.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-308}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -8e-308) (/ (- b) a) (/ 0.0 a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-308) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8d-308)) then
        tmp = -b / a
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-308) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8e-308:
		tmp = -b / a
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e-308)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8e-308)
		tmp = -b / a;
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e-308], N[((-b) / a), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -8.00000000000000026e-308

   1. Initial program 71.3%

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

      1. *-commutative 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in b around -inf 64.4%

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

      1. associate-*r/ 64.4%

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

      2. mul-1-neg 64.4%

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

   7. Simplified 64.4%

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

   if -8.00000000000000026e-308 < b

   1. Initial program 34.7%

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

      1. *-commutative 34.7%

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

   3. Simplified 34.7%

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

   6. Applied egg-rr 34.3%

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

      1. fma-def 29.7%

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

      2. distribute-neg-frac 29.7%

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

   8. Simplified 29.7%

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

   9. Taylor expanded in a around 0 19.3%

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

       1. distribute-rgt-out 19.3%

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

       2. metadata-eval 19.3%

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

       3. mul0-rgt 19.3%

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

   11. Simplified 19.3%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-308}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{-305}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.7e-305) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.7e-305) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.7d-305) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.7e-305) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.7e-305:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.7e-305)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.7e-305)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.7e-305], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.6999999999999996e-305

   1. Initial program 71.7%

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

      1. *-commutative 71.7%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in b around -inf 63.6%

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

      1. associate-*r/ 63.6%

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

      2. mul-1-neg 63.6%

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

   7. Simplified 63.6%

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

   if 4.6999999999999996e-305 < b

   1. Initial program 33.6%

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

      1. *-commutative 33.6%

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

   3. Simplified 33.6%

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

   5. Taylor expanded in b around inf 68.3%

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

      1. mul-1-neg 68.3%

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

      2. distribute-neg-frac 68.3%

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

   7. Simplified 68.3%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{-305}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 11.7% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. *-commutative 54.6%

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

3. Simplified 54.6%

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

6. Applied egg-rr 54.4%

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

   1. fma-def 52.3%

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

   2. distribute-neg-frac 52.3%

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

8. Simplified 52.3%

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

9. Taylor expanded in a around 0 10.4%

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

    1. distribute-rgt-out 10.4%

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

    2. metadata-eval 10.4%

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

    3. mul0-rgt 10.4%

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

11. Simplified 10.4%

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

12. Final simplification 10.4%

    \[\leadsto \frac{0}{a} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))