quadp (p42, positive)

Percentage Accurate: 53.0% → 84.3%

Time: 14.3s

Alternatives: 8

Speedup: 12.9×

Specification

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 53.0% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+161}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1100:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 11000000:\\ \;\;\;\;\frac{{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(4, \frac{\frac{b}{c}}{-4}, \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e+161)
   (- (/ c b) (/ b a))
   (if (<= b 8.6e-130)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (if (<= b 1100.0)
       (/ c (- b))
       (if (<= b 11000000.0)
         (/ (- (pow (pow (* a (* c -4.0)) 0.25) 2.0) b) (* a 2.0))
         (/ 1.0 (fma 4.0 (/ (/ b c) -4.0) (/ a b))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e+161) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8.6e-130) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else if (b <= 1100.0) {
		tmp = c / -b;
	} else if (b <= 11000000.0) {
		tmp = (pow(pow((a * (c * -4.0)), 0.25), 2.0) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / fma(4.0, ((b / c) / -4.0), (a / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e+161)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 8.6e-130)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	elseif (b <= 1100.0)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 11000000.0)
		tmp = Float64(Float64(((Float64(a * Float64(c * -4.0)) ^ 0.25) ^ 2.0) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / fma(4.0, Float64(Float64(b / c) / -4.0), Float64(a / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e+161], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-130], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1100.0], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 11000000.0], N[(N[(N[Power[N[Power[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(4.0 * N[(N[(b / c), $MachinePrecision] / -4.0), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.60000000000000001e161

   1. Initial program 23.9%

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

      1. *-commutative 23.9%

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

   3. Simplified 23.9%

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

   5. Taylor expanded in b around -inf 95.5%

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

      1. +-commutative 95.5%

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

      2. mul-1-neg 95.5%

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

      3. unsub-neg 95.5%

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

   7. Simplified 95.5%

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

   if -1.60000000000000001e161 < b < 8.60000000000000058e-130

   1. Initial program 86.5%

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

   if 8.60000000000000058e-130 < b < 1100

   1. Initial program 34.5%

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

      1. *-commutative 34.5%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in b around inf 60.3%

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

      1. associate-*r/ 60.3%

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

      2. neg-mul-1 60.3%

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

   7. Simplified 60.3%

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

   if 1100 < b < 1.1e7

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in b around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.7%

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

   7. Simplified 99.7%

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

      1. add-sqr-sqrt 100.0%

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

      2. pow2 100.0%

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

      3. pow1/2 100.0%

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

      4. sqrt-pow1 100.0%

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

      5. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 1.1e7 < b

   1. Initial program 13.2%

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

      1. *-commutative 13.2%

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

   3. Simplified 13.2%

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

      1. frac-2neg 13.2%

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

      2. div-inv 13.2%

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

   6. Applied egg-rr 13.3%

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

      1. fma-define 13.2%

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

      2. +-commutative 13.2%

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

      3. unpow2 13.2%

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

      4. rem-square-sqrt 7.7%

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

      5. hypot-define 21.3%

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

      6. *-commutative 21.3%

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

      7. *-commutative 21.3%

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

      8. *-commutative 21.3%

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

      9. associate-/r* 21.3%

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

      10. metadata-eval 21.3%

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

   8. Simplified 21.3%

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

      1. add-sqr-sqrt 3.5%

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

      2. pow2 3.5%

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

      3. pow1/2 3.5%

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

      4. sqrt-pow1 3.5%

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

      5. metadata-eval 3.5%

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

   10. Applied egg-rr 21.3%

       \[\leadsto \left(b - \mathsf{hypot}\left(b, \color{blue}{{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}}\right)\right) \cdot \frac{-0.5}{a} \]
   11. Step-by-step derivation

       1. associate-*r/ 21.3%

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

       2. clear-num 21.3%

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

       3. pow-pow 21.3%

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

       4. metadata-eval 21.3%

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

       5. pow1/2 21.3%

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

   12. Applied egg-rr 21.3%

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

   13. Taylor expanded in a around 0 0.0%

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

       1. fma-define 0.0%

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

       2. associate-/r* 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 95.4%

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

   15. Simplified 95.4%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(4, \frac{\frac{b}{c}}{-4}, \frac{a}{b}\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+161}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1100:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 11000000:\\ \;\;\;\;\frac{{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(4, \frac{\frac{b}{c}}{-4}, \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.02 \cdot 10^{-96}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-133}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1100:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 11000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(4, \frac{\frac{b}{c}}{-4}, \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))))
   (if (<= b -1.02e-96)
     (- (/ c b) (/ b a))
     (if (<= b 1.6e-133)
       t_0
       (if (<= b 1100.0)
         (/ c (- b))
         (if (<= b 11000000.0)
           t_0
           (/ 1.0 (fma 4.0 (/ (/ b c) -4.0) (/ a b)))))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -1.02e-96) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.6e-133) {
		tmp = t_0;
	} else if (b <= 1100.0) {
		tmp = c / -b;
	} else if (b <= 11000000.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 / fma(4.0, ((b / c) / -4.0), (a / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (b <= -1.02e-96)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.6e-133)
		tmp = t_0;
	elseif (b <= 1100.0)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 11000000.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 / fma(4.0, Float64(Float64(b / c) / -4.0), Float64(a / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.02e-96], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e-133], t$95$0, If[LessEqual[b, 1100.0], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 11000000.0], t$95$0, N[(1.0 / N[(4.0 * N[(N[(b / c), $MachinePrecision] / -4.0), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.02000000000000007e-96

   1. Initial program 63.6%

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

      1. *-commutative 63.6%

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

   3. Simplified 63.6%

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

   5. Taylor expanded in b around -inf 88.2%

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

      1. +-commutative 88.2%

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

      2. mul-1-neg 88.2%

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

      3. unsub-neg 88.2%

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

   7. Simplified 88.2%

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

   if -1.02000000000000007e-96 < b < 1.60000000000000006e-133 or 1100 < b < 1.1e7

   1. Initial program 82.7%

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

      1. *-commutative 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in b around 0 80.8%

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

      1. *-commutative 80.8%

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

      2. associate-*r* 80.8%

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

   7. Simplified 80.8%

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

      1. pow1/2 80.8%

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

      2. metadata-eval 80.8%

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

      3. pow-pow 80.6%

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

      4. +-commutative 80.6%

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

      5. unsub-neg 80.6%

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

      6. pow-pow 80.8%

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

      7. metadata-eval 80.8%

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

      8. pow1/2 80.8%

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

   9. Applied egg-rr 80.8%

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

   if 1.60000000000000006e-133 < b < 1100

   1. Initial program 34.5%

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

      1. *-commutative 34.5%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in b around inf 60.3%

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

      1. associate-*r/ 60.3%

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

      2. neg-mul-1 60.3%

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

   7. Simplified 60.3%

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

   if 1.1e7 < b

   1. Initial program 13.2%

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

      1. *-commutative 13.2%

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

   3. Simplified 13.2%

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

      1. frac-2neg 13.2%

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

      2. div-inv 13.2%

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

   6. Applied egg-rr 13.3%

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

      1. fma-define 13.2%

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

      2. +-commutative 13.2%

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

      3. unpow2 13.2%

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

      4. rem-square-sqrt 7.7%

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

      5. hypot-define 21.3%

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

      6. *-commutative 21.3%

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

      7. *-commutative 21.3%

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

      8. *-commutative 21.3%

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

      9. associate-/r* 21.3%

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

      10. metadata-eval 21.3%

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

   8. Simplified 21.3%

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

      1. add-sqr-sqrt 3.5%

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

      2. pow2 3.5%

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

      3. pow1/2 3.5%

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

      4. sqrt-pow1 3.5%

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

      5. metadata-eval 3.5%

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

   10. Applied egg-rr 21.3%

       \[\leadsto \left(b - \mathsf{hypot}\left(b, \color{blue}{{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}}\right)\right) \cdot \frac{-0.5}{a} \]
   11. Step-by-step derivation

       1. associate-*r/ 21.3%

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

       2. clear-num 21.3%

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

       3. pow-pow 21.3%

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

       4. metadata-eval 21.3%

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

       5. pow1/2 21.3%

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

   12. Applied egg-rr 21.3%

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

   13. Taylor expanded in a around 0 0.0%

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

       1. fma-define 0.0%

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

       2. associate-/r* 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 95.4%

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

   15. Simplified 95.4%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(4, \frac{\frac{b}{c}}{-4}, \frac{a}{b}\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-96}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1100:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 11000000:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(4, \frac{\frac{b}{c}}{-4}, \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+161}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1100:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 21000000:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(4, \frac{\frac{b}{c}}{-4}, \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e+161)
   (- (/ c b) (/ b a))
   (if (<= b 5.2e-130)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (if (<= b 1100.0)
       (/ c (- b))
       (if (<= b 21000000.0)
         (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
         (/ 1.0 (fma 4.0 (/ (/ b c) -4.0) (/ a b))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e+161) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.2e-130) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else if (b <= 1100.0) {
		tmp = c / -b;
	} else if (b <= 21000000.0) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / fma(4.0, ((b / c) / -4.0), (a / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e+161)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 5.2e-130)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	elseif (b <= 1100.0)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 21000000.0)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / fma(4.0, Float64(Float64(b / c) / -4.0), Float64(a / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e+161], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-130], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1100.0], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 21000000.0], 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[(4.0 * N[(N[(b / c), $MachinePrecision] / -4.0), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.60000000000000001e161

   1. Initial program 23.9%

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

      1. *-commutative 23.9%

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

   3. Simplified 23.9%

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

   5. Taylor expanded in b around -inf 95.5%

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

      1. +-commutative 95.5%

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

      2. mul-1-neg 95.5%

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

      3. unsub-neg 95.5%

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

   7. Simplified 95.5%

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

   if -1.60000000000000001e161 < b < 5.2000000000000001e-130

   1. Initial program 86.5%

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

   if 5.2000000000000001e-130 < b < 1100

   1. Initial program 34.5%

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

      1. *-commutative 34.5%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in b around inf 60.3%

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

      1. associate-*r/ 60.3%

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

      2. neg-mul-1 60.3%

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

   7. Simplified 60.3%

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

   if 1100 < b < 2.1e7

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in b around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.7%

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

   7. Simplified 99.7%

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

      1. pow1/2 99.7%

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

      2. metadata-eval 99.7%

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

      3. pow-pow 100.0%

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

      4. +-commutative 100.0%

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

      5. unsub-neg 100.0%

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

      6. pow-pow 99.7%

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

      7. metadata-eval 99.7%

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

      8. pow1/2 99.7%

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

   9. Applied egg-rr 99.7%

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

   if 2.1e7 < b

   1. Initial program 13.2%

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

      1. *-commutative 13.2%

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

   3. Simplified 13.2%

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

      1. frac-2neg 13.2%

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

      2. div-inv 13.2%

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

   6. Applied egg-rr 13.3%

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

      1. fma-define 13.2%

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

      2. +-commutative 13.2%

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

      3. unpow2 13.2%

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

      4. rem-square-sqrt 7.7%

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

      5. hypot-define 21.3%

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

      6. *-commutative 21.3%

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

      7. *-commutative 21.3%

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

      8. *-commutative 21.3%

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

      9. associate-/r* 21.3%

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

      10. metadata-eval 21.3%

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

   8. Simplified 21.3%

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

      1. add-sqr-sqrt 3.5%

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

      2. pow2 3.5%

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

      3. pow1/2 3.5%

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

      4. sqrt-pow1 3.5%

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

      5. metadata-eval 3.5%

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

   10. Applied egg-rr 21.3%

       \[\leadsto \left(b - \mathsf{hypot}\left(b, \color{blue}{{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}}\right)\right) \cdot \frac{-0.5}{a} \]
   11. Step-by-step derivation

       1. associate-*r/ 21.3%

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

       2. clear-num 21.3%

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

       3. pow-pow 21.3%

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

       4. metadata-eval 21.3%

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

       5. pow1/2 21.3%

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

   12. Applied egg-rr 21.3%

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

   13. Taylor expanded in a around 0 0.0%

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

       1. fma-define 0.0%

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

       2. associate-/r* 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 95.4%

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

   15. Simplified 95.4%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(4, \frac{\frac{b}{c}}{-4}, \frac{a}{b}\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+161}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1100:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 21000000:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(4, \frac{\frac{b}{c}}{-4}, \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(4, \frac{\frac{b}{c}}{-4}, \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (- (/ c b) (/ b a))
   (/ 1.0 (fma 4.0 (/ (/ b c) -4.0) (/ a b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = 1.0 / fma(4.0, ((b / c) / -4.0), (a / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(1.0 / fma(4.0, Float64(Float64(b / c) / -4.0), Float64(a / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(4.0 * N[(N[(b / c), $MachinePrecision] / -4.0), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 67.7%

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

      1. *-commutative 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in b around -inf 69.9%

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

      1. +-commutative 69.9%

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

      2. mul-1-neg 69.9%

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

      3. unsub-neg 69.9%

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

   7. Simplified 69.9%

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

   if -4.999999999999985e-310 < b

   1. Initial program 32.8%

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

      1. *-commutative 32.8%

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

   3. Simplified 32.8%

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

      1. frac-2neg 32.8%

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

      2. div-inv 32.7%

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

   6. Applied egg-rr 32.7%

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

      1. fma-define 32.7%

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

      2. +-commutative 32.7%

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

      3. unpow2 32.7%

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

      4. rem-square-sqrt 29.2%

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

      5. hypot-define 37.1%

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

      6. *-commutative 37.1%

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

      7. *-commutative 37.1%

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

      8. *-commutative 37.1%

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

      9. associate-/r* 37.1%

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

      10. metadata-eval 37.1%

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

   8. Simplified 37.1%

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

      1. add-sqr-sqrt 25.9%

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

      2. pow2 25.9%

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

      3. pow1/2 25.9%

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

      4. sqrt-pow1 25.9%

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

      5. metadata-eval 25.9%

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

   10. Applied egg-rr 37.1%

       \[\leadsto \left(b - \mathsf{hypot}\left(b, \color{blue}{{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}}\right)\right) \cdot \frac{-0.5}{a} \]
   11. Step-by-step derivation

       1. associate-*r/ 37.1%

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

       2. clear-num 37.1%

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

       3. pow-pow 37.1%

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

       4. metadata-eval 37.1%

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

       5. pow1/2 37.1%

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

   12. Applied egg-rr 37.1%

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

   13. Taylor expanded in a around 0 0.0%

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

       1. fma-define 0.0%

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

       2. associate-/r* 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 69.9%

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

   15. Simplified 69.9%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(4, \frac{\frac{b}{c}}{-4}, \frac{a}{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(4, \frac{\frac{b}{c}}{-4}, \frac{a}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 67.7%

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

      1. *-commutative 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in b around -inf 69.9%

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

      1. +-commutative 69.9%

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

      2. mul-1-neg 69.9%

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

      3. unsub-neg 69.9%

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

   7. Simplified 69.9%

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

   if -4.999999999999985e-310 < b

   1. Initial program 32.8%

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

      1. *-commutative 32.8%

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

   3. Simplified 32.8%

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

   5. Taylor expanded in b around inf 69.4%

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

      1. associate-*r/ 69.4%

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

      2. neg-mul-1 69.4%

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

   7. Simplified 69.4%

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

4. Final simplification 69.7%

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

Alternative 6: 42.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 5.4e-5) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.4e-5) {
		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 <= 5.4d-5) 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 <= 5.4e-5) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.4e-5:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.4e-5)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.4e-5)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.4e-5], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.3999999999999998e-5

   1. Initial program 65.2%

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

      1. *-commutative 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in b around -inf 51.8%

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

      1. associate-*r/ 51.8%

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

      2. mul-1-neg 51.8%

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

   7. Simplified 51.8%

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

   if 5.3999999999999998e-5 < b

   1. Initial program 18.4%

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

      1. *-commutative 18.4%

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

   3. Simplified 18.4%

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

   5. Taylor expanded in b around -inf 2.5%

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

   6. Taylor expanded in b around 0 25.5%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-296}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.08e-296) (/ (- b) a) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.08e-296) {
		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 <= 1.08d-296) 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 <= 1.08e-296) {
		tmp = -b / a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.08e-296:
		tmp = -b / a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.08e-296)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.08e-296)
		tmp = -b / a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.08e-296], N[((-b) / a), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.08e-296

   1. Initial program 67.2%

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

      1. *-commutative 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in b around -inf 68.8%

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

      1. associate-*r/ 68.8%

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

      2. mul-1-neg 68.8%

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

   7. Simplified 68.8%

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

   if 1.08e-296 < b

   1. Initial program 33.0%

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

      1. *-commutative 33.0%

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

   3. Simplified 33.0%

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

   5. Taylor expanded in b around inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. neg-mul-1 69.9%

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

   7. Simplified 69.9%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-296}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 10.9% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

   1. *-commutative 50.2%

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

3. Simplified 50.2%

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

5. Taylor expanded in b around -inf 34.4%

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

6. Taylor expanded in b around 0 10.3%

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

7. Final simplification 10.3%

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

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024040 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (if (< b 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))