The quadratic formula (r1)

Percentage Accurate: 51.7% → 84.2%

Time: 16.7s

Alternatives: 8

Speedup: 12.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.7% 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: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\frac{b + \mathsf{fma}\left(-2, a \cdot \frac{c}{b}, b\right)}{-2 \cdot a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e+154)
   (/ (+ b (fma -2.0 (* a (/ c b)) b)) (* -2.0 a))
   (if (<= b 6.2e-6)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+154) {
		tmp = (b + fma(-2.0, (a * (c / b)), b)) / (-2.0 * a);
	} else if (b <= 6.2e-6) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e+154)
		tmp = Float64(Float64(b + fma(-2.0, Float64(a * Float64(c / b)), b)) / Float64(-2.0 * a));
	elseif (b <= 6.2e-6)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e+154], N[(N[(b + N[(-2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.2e-6], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.00000000000000015e154

   1. Initial program 29.7%

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

      1. +-commutative 29.7%

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

      2. sqr-neg 29.7%

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

      3. unsub-neg 29.7%

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

      4. sqr-neg 29.7%

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

      5. sub-neg 29.7%

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

      6. +-commutative 29.7%

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

      7. *-commutative 29.7%

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

      8. associate-*r* 29.7%

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

      9. distribute-rgt-neg-in 29.7%

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

      10. fma-define 30.0%

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

      11. *-commutative 30.0%

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

      12. distribute-rgt-neg-in 30.0%

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

      13. metadata-eval 30.0%

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

   3. Simplified 30.0%

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

      1. frac-2neg 30.0%

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

      2. div-inv 30.0%

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

      3. sub-neg 30.0%

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

      4. distribute-neg-in 30.0%

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

      5. pow2 30.0%

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

      6. add-sqr-sqrt 30.0%

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

      7. sqrt-unprod 30.0%

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

      8. sqr-neg 30.0%

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

      9. sqrt-prod 0.0%

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

      10. add-sqr-sqrt 30.0%

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

      11. add-sqr-sqrt 30.0%

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

      12. sqrt-unprod 0.0%

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

      13. sqr-neg 0.0%

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

      14. sqrt-prod 0.0%

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

      15. add-sqr-sqrt 30.0%

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

      16. distribute-rgt-neg-in 30.0%

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

      17. metadata-eval 30.0%

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

   6. Applied egg-rr 30.0%

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

   7. Taylor expanded in a around 0 2.1%

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

      1. un-div-inv 2.1%

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

      2. +-commutative 2.1%

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

      3. add-sqr-sqrt 9.4%

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

      4. sqrt-unprod 0.3%

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

      5. sqr-neg 0.3%

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

      6. sqrt-unprod 0.1%

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

      7. add-sqr-sqrt 85.9%

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

      8. +-commutative 85.9%

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

      9. fma-define 85.9%

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

      10. associate-/l* 95.6%

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

   9. Applied egg-rr 95.6%

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

   if -4.00000000000000015e154 < b < 6.1999999999999999e-6

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

      2. +-commutative 82.1%

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

      3. sqr-neg 82.1%

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

      4. unsub-neg 82.1%

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

      5. sqr-neg 82.1%

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

      6. fma-neg 82.2%

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

      7. distribute-lft-neg-in 82.2%

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

      8. *-commutative 82.2%

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

      9. *-commutative 82.2%

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

      10. distribute-rgt-neg-in 82.2%

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

      11. metadata-eval 82.2%

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

   3. Simplified 82.2%

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

   if 6.1999999999999999e-6 < b

   1. Initial program 14.7%

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

      1. *-commutative 14.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in b around inf 87.9%

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

      1. associate-*r/ 87.9%

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

      2. mul-1-neg 87.9%

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

   7. Simplified 87.9%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\frac{b + \mathsf{fma}\left(-2, a \cdot \frac{c}{b}, b\right)}{-2 \cdot a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{-a}\\ t_1 := \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -2.55 \cdot 10^{-90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ b (- a))) (t_1 (* (- b (sqrt (* c (* a -4.0)))) (/ -0.5 a))))
   (if (<= b -1.6e-29)
     t_0
     (if (<= b -2.55e-90)
       t_1
       (if (<= b -1.05e-129) t_0 (if (<= b 2.4e-8) t_1 (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = b / -a;
	double t_1 = (b - sqrt((c * (a * -4.0)))) * (-0.5 / a);
	double tmp;
	if (b <= -1.6e-29) {
		tmp = t_0;
	} else if (b <= -2.55e-90) {
		tmp = t_1;
	} else if (b <= -1.05e-129) {
		tmp = t_0;
	} else if (b <= 2.4e-8) {
		tmp = t_1;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b / -a
    t_1 = (b - sqrt((c * (a * (-4.0d0))))) * ((-0.5d0) / a)
    if (b <= (-1.6d-29)) then
        tmp = t_0
    else if (b <= (-2.55d-90)) then
        tmp = t_1
    else if (b <= (-1.05d-129)) then
        tmp = t_0
    else if (b <= 2.4d-8) then
        tmp = t_1
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b / -a;
	double t_1 = (b - Math.sqrt((c * (a * -4.0)))) * (-0.5 / a);
	double tmp;
	if (b <= -1.6e-29) {
		tmp = t_0;
	} else if (b <= -2.55e-90) {
		tmp = t_1;
	} else if (b <= -1.05e-129) {
		tmp = t_0;
	} else if (b <= 2.4e-8) {
		tmp = t_1;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b / -a
	t_1 = (b - math.sqrt((c * (a * -4.0)))) * (-0.5 / a)
	tmp = 0
	if b <= -1.6e-29:
		tmp = t_0
	elif b <= -2.55e-90:
		tmp = t_1
	elif b <= -1.05e-129:
		tmp = t_0
	elif b <= 2.4e-8:
		tmp = t_1
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b / Float64(-a))
	t_1 = Float64(Float64(b - sqrt(Float64(c * Float64(a * -4.0)))) * Float64(-0.5 / a))
	tmp = 0.0
	if (b <= -1.6e-29)
		tmp = t_0;
	elseif (b <= -2.55e-90)
		tmp = t_1;
	elseif (b <= -1.05e-129)
		tmp = t_0;
	elseif (b <= 2.4e-8)
		tmp = t_1;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b / -a;
	t_1 = (b - sqrt((c * (a * -4.0)))) * (-0.5 / a);
	tmp = 0.0;
	if (b <= -1.6e-29)
		tmp = t_0;
	elseif (b <= -2.55e-90)
		tmp = t_1;
	elseif (b <= -1.05e-129)
		tmp = t_0;
	elseif (b <= 2.4e-8)
		tmp = t_1;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b / (-a)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.6e-29], t$95$0, If[LessEqual[b, -2.55e-90], t$95$1, If[LessEqual[b, -1.05e-129], t$95$0, If[LessEqual[b, 2.4e-8], t$95$1, N[((-c) / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.6e-29 or -2.5499999999999998e-90 < b < -1.05e-129

   1. Initial program 63.9%

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

      1. *-commutative 63.9%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in b around -inf 87.6%

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

      1. associate-*r/ 87.6%

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

      2. mul-1-neg 87.6%

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

   7. Simplified 87.6%

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

   if -1.6e-29 < b < -2.5499999999999998e-90 or -1.05e-129 < b < 2.39999999999999998e-8

   1. Initial program 77.1%

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

      1. *-commutative 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in b around 0 65.8%

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

      1. *-commutative 65.8%

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

      2. associate-*r* 65.8%

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

   7. Simplified 65.8%

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

      1. *-commutative 65.8%

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

      2. sqrt-prod 35.9%

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

   9. Applied egg-rr 35.9%

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

       1. frac-2neg 35.9%

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

       2. div-inv 35.9%

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

       3. distribute-neg-in 35.9%

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

       4. add-sqr-sqrt 20.1%

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

       5. sqrt-unprod 35.8%

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

       6. sqr-neg 35.8%

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

       7. sqrt-prod 15.8%

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

       8. add-sqr-sqrt 35.2%

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

       9. sub-neg 35.2%

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

       10. add-sqr-sqrt 19.5%

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

       11. *-commutative 19.5%

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

       12. sqrt-unprod 35.3%

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

       13. sqr-neg 35.3%

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

       14. sqrt-prod 15.7%

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

       15. add-sqr-sqrt 35.9%

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

       16. sqrt-prod 65.7%

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

       17. distribute-rgt-neg-in 65.7%

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

       18. metadata-eval 65.7%

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

   11. Applied egg-rr 65.7%

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

       1. *-commutative 65.7%

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

       2. associate-*l/ 65.7%

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

       3. metadata-eval 65.7%

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

       4. *-commutative 65.7%

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

       5. associate-*r* 65.7%

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

   13. Simplified 65.7%

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

   if 2.39999999999999998e-8 < b

   1. Initial program 14.7%

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

      1. *-commutative 14.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in b around inf 87.9%

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

      1. associate-*r/ 87.9%

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

      2. mul-1-neg 87.9%

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

   7. Simplified 87.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq -2.55 \cdot 10^{-90}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{-a}\\ t_1 := \frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{-2 \cdot a}\\ \mathbf{if}\;b \leq -3.85 \cdot 10^{-31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -2.7 \cdot 10^{-90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ b (- a))) (t_1 (/ (- b (sqrt (* c (* a -4.0)))) (* -2.0 a))))
   (if (<= b -3.85e-31)
     t_0
     (if (<= b -2.7e-90)
       t_1
       (if (<= b -2e-129) t_0 (if (<= b 4.8e-8) t_1 (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = b / -a;
	double t_1 = (b - sqrt((c * (a * -4.0)))) / (-2.0 * a);
	double tmp;
	if (b <= -3.85e-31) {
		tmp = t_0;
	} else if (b <= -2.7e-90) {
		tmp = t_1;
	} else if (b <= -2e-129) {
		tmp = t_0;
	} else if (b <= 4.8e-8) {
		tmp = t_1;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b / -a
    t_1 = (b - sqrt((c * (a * (-4.0d0))))) / ((-2.0d0) * a)
    if (b <= (-3.85d-31)) then
        tmp = t_0
    else if (b <= (-2.7d-90)) then
        tmp = t_1
    else if (b <= (-2d-129)) then
        tmp = t_0
    else if (b <= 4.8d-8) then
        tmp = t_1
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b / -a;
	double t_1 = (b - Math.sqrt((c * (a * -4.0)))) / (-2.0 * a);
	double tmp;
	if (b <= -3.85e-31) {
		tmp = t_0;
	} else if (b <= -2.7e-90) {
		tmp = t_1;
	} else if (b <= -2e-129) {
		tmp = t_0;
	} else if (b <= 4.8e-8) {
		tmp = t_1;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b / -a
	t_1 = (b - math.sqrt((c * (a * -4.0)))) / (-2.0 * a)
	tmp = 0
	if b <= -3.85e-31:
		tmp = t_0
	elif b <= -2.7e-90:
		tmp = t_1
	elif b <= -2e-129:
		tmp = t_0
	elif b <= 4.8e-8:
		tmp = t_1
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b / Float64(-a))
	t_1 = Float64(Float64(b - sqrt(Float64(c * Float64(a * -4.0)))) / Float64(-2.0 * a))
	tmp = 0.0
	if (b <= -3.85e-31)
		tmp = t_0;
	elseif (b <= -2.7e-90)
		tmp = t_1;
	elseif (b <= -2e-129)
		tmp = t_0;
	elseif (b <= 4.8e-8)
		tmp = t_1;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b / -a;
	t_1 = (b - sqrt((c * (a * -4.0)))) / (-2.0 * a);
	tmp = 0.0;
	if (b <= -3.85e-31)
		tmp = t_0;
	elseif (b <= -2.7e-90)
		tmp = t_1;
	elseif (b <= -2e-129)
		tmp = t_0;
	elseif (b <= 4.8e-8)
		tmp = t_1;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b / (-a)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.85e-31], t$95$0, If[LessEqual[b, -2.7e-90], t$95$1, If[LessEqual[b, -2e-129], t$95$0, If[LessEqual[b, 4.8e-8], t$95$1, N[((-c) / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.85000000000000006e-31 or -2.69999999999999996e-90 < b < -1.9999999999999999e-129

   1. Initial program 63.9%

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

      1. *-commutative 63.9%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in b around -inf 87.6%

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

      1. associate-*r/ 87.6%

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

      2. mul-1-neg 87.6%

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

   7. Simplified 87.6%

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

   if -3.85000000000000006e-31 < b < -2.69999999999999996e-90 or -1.9999999999999999e-129 < b < 4.79999999999999997e-8

   1. Initial program 77.1%

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

      1. *-commutative 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in b around 0 65.8%

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

      1. *-commutative 65.8%

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

      2. associate-*r* 65.8%

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

   7. Simplified 65.8%

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

      1. *-un-lft-identity 65.8%

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

      2. times-frac 65.7%

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

      3. add-sqr-sqrt 34.9%

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

      4. sqrt-unprod 65.4%

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

      5. sqr-neg 65.4%

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

      6. sqrt-unprod 30.9%

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

      7. add-sqr-sqrt 63.4%

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

   9. Applied egg-rr 63.4%

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

       1. frac-2neg 63.4%

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

       2. metadata-eval 63.4%

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

       3. frac-2neg 63.4%

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

       4. metadata-eval 63.4%

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

       5. frac-times 63.5%

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

   11. Applied egg-rr 65.8%

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

       1. neg-mul-1 65.8%

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

       2. distribute-frac-neg 65.8%

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

       3. distribute-neg-frac2 65.8%

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

       4. *-commutative 65.8%

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

       5. associate-*r* 65.8%

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

       6. distribute-rgt-neg-in 65.8%

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

       7. metadata-eval 65.8%

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

   13. Simplified 65.8%

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

   if 4.79999999999999997e-8 < b

   1. Initial program 14.7%

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

      1. *-commutative 14.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in b around inf 87.9%

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

      1. associate-*r/ 87.9%

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

      2. mul-1-neg 87.9%

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

   7. Simplified 87.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.85 \cdot 10^{-31}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq -2.7 \cdot 10^{-90}:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{-2 \cdot a}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{-2 \cdot a}\\ \mathbf{if}\;b \leq -5.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{b + \mathsf{fma}\left(-2, a \cdot \frac{c}{b}, b\right)}{-2 \cdot a}\\ \mathbf{elif}\;b \leq -2.3 \cdot 10^{-90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- b (sqrt (* c (* a -4.0)))) (* -2.0 a))))
   (if (<= b -5.7e-30)
     (/ (+ b (fma -2.0 (* a (/ c b)) b)) (* -2.0 a))
     (if (<= b -2.3e-90)
       t_0
       (if (<= b -2e-129) (/ b (- a)) (if (<= b 2.4e-8) t_0 (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (b - sqrt((c * (a * -4.0)))) / (-2.0 * a);
	double tmp;
	if (b <= -5.7e-30) {
		tmp = (b + fma(-2.0, (a * (c / b)), b)) / (-2.0 * a);
	} else if (b <= -2.3e-90) {
		tmp = t_0;
	} else if (b <= -2e-129) {
		tmp = b / -a;
	} else if (b <= 2.4e-8) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b - sqrt(Float64(c * Float64(a * -4.0)))) / Float64(-2.0 * a))
	tmp = 0.0
	if (b <= -5.7e-30)
		tmp = Float64(Float64(b + fma(-2.0, Float64(a * Float64(c / b)), b)) / Float64(-2.0 * a));
	elseif (b <= -2.3e-90)
		tmp = t_0;
	elseif (b <= -2e-129)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 2.4e-8)
		tmp = t_0;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5.7e-30], N[(N[(b + N[(-2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -2.3e-90], t$95$0, If[LessEqual[b, -2e-129], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 2.4e-8], t$95$0, N[((-c) / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -5.69999999999999977e-30

   1. Initial program 61.5%

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

      1. +-commutative 61.5%

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

      2. sqr-neg 61.5%

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

      3. unsub-neg 61.5%

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

      4. sqr-neg 61.5%

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

      5. sub-neg 61.5%

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

      6. +-commutative 61.5%

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

      7. *-commutative 61.5%

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

      8. associate-*r* 61.5%

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

      9. distribute-rgt-neg-in 61.5%

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

      10. fma-define 61.7%

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

      11. *-commutative 61.7%

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

      12. distribute-rgt-neg-in 61.7%

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

      13. metadata-eval 61.7%

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

   3. Simplified 61.7%

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

      1. frac-2neg 61.7%

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

      2. div-inv 61.6%

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

      3. sub-neg 61.6%

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

      4. distribute-neg-in 61.6%

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

      5. pow2 61.6%

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

      6. add-sqr-sqrt 61.5%

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

      7. sqrt-unprod 61.6%

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

      8. sqr-neg 61.6%

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

      9. sqrt-prod 0.0%

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

      10. add-sqr-sqrt 19.0%

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

      11. add-sqr-sqrt 21.4%

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

      12. sqrt-unprod 5.3%

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

      13. sqr-neg 5.3%

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

      14. sqrt-prod 0.0%

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

      15. add-sqr-sqrt 61.6%

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

      16. distribute-rgt-neg-in 61.6%

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

      17. metadata-eval 61.6%

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

   6. Applied egg-rr 61.6%

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

   7. Taylor expanded in a around 0 2.3%

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

      1. un-div-inv 2.3%

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

      2. +-commutative 2.3%

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

      3. add-sqr-sqrt 8.0%

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

      4. sqrt-unprod 1.4%

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

      5. sqr-neg 1.4%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 82.4%

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

      8. +-commutative 82.4%

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

      9. fma-define 82.4%

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

      10. associate-/l* 86.9%

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

   9. Applied egg-rr 86.9%

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

   if -5.69999999999999977e-30 < b < -2.2999999999999998e-90 or -1.9999999999999999e-129 < b < 2.39999999999999998e-8

   1. Initial program 77.1%

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

      1. *-commutative 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in b around 0 65.8%

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

      1. *-commutative 65.8%

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

      2. associate-*r* 65.8%

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

   7. Simplified 65.8%

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

      1. *-un-lft-identity 65.8%

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

      2. times-frac 65.7%

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

      3. add-sqr-sqrt 34.9%

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

      4. sqrt-unprod 65.4%

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

      5. sqr-neg 65.4%

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

      6. sqrt-unprod 30.9%

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

      7. add-sqr-sqrt 63.4%

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

   9. Applied egg-rr 63.4%

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

       1. frac-2neg 63.4%

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

       2. metadata-eval 63.4%

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

       3. frac-2neg 63.4%

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

       4. metadata-eval 63.4%

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

       5. frac-times 63.5%

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

   11. Applied egg-rr 65.8%

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

       1. neg-mul-1 65.8%

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

       2. distribute-frac-neg 65.8%

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

       3. distribute-neg-frac2 65.8%

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

       4. *-commutative 65.8%

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

       5. associate-*r* 65.8%

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

       6. distribute-rgt-neg-in 65.8%

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

       7. metadata-eval 65.8%

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

   13. Simplified 65.8%

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

   if -2.2999999999999998e-90 < b < -1.9999999999999999e-129

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if 2.39999999999999998e-8 < b

   1. Initial program 14.7%

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

      1. *-commutative 14.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in b around inf 87.9%

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

      1. associate-*r/ 87.9%

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

      2. mul-1-neg 87.9%

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

   7. Simplified 87.9%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{b + \mathsf{fma}\left(-2, a \cdot \frac{c}{b}, b\right)}{-2 \cdot a}\\ \mathbf{elif}\;b \leq -2.3 \cdot 10^{-90}:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{-2 \cdot a}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+153}:\\ \;\;\;\;\frac{b + \mathsf{fma}\left(-2, a \cdot \frac{c}{b}, b\right)}{-2 \cdot a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e+153)
   (/ (+ b (fma -2.0 (* a (/ c b)) b)) (* -2.0 a))
   (if (<= b 2.4e-8)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e+153) {
		tmp = (b + fma(-2.0, (a * (c / b)), b)) / (-2.0 * a);
	} else if (b <= 2.4e-8) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e+153)
		tmp = Float64(Float64(b + fma(-2.0, Float64(a * Float64(c / b)), b)) / Float64(-2.0 * a));
	elseif (b <= 2.4e-8)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e+153], N[(N[(b + N[(-2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-8], 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[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.00000000000000019e153

   1. Initial program 29.7%

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

      1. +-commutative 29.7%

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

      2. sqr-neg 29.7%

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

      3. unsub-neg 29.7%

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

      4. sqr-neg 29.7%

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

      5. sub-neg 29.7%

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

      6. +-commutative 29.7%

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

      7. *-commutative 29.7%

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

      8. associate-*r* 29.7%

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

      9. distribute-rgt-neg-in 29.7%

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

      10. fma-define 30.0%

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

      11. *-commutative 30.0%

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

      12. distribute-rgt-neg-in 30.0%

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

      13. metadata-eval 30.0%

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

   3. Simplified 30.0%

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

      1. frac-2neg 30.0%

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

      2. div-inv 30.0%

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

      3. sub-neg 30.0%

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

      4. distribute-neg-in 30.0%

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

      5. pow2 30.0%

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

      6. add-sqr-sqrt 30.0%

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

      7. sqrt-unprod 30.0%

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

      8. sqr-neg 30.0%

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

      9. sqrt-prod 0.0%

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

      10. add-sqr-sqrt 30.0%

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

      11. add-sqr-sqrt 30.0%

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

      12. sqrt-unprod 0.0%

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

      13. sqr-neg 0.0%

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

      14. sqrt-prod 0.0%

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

      15. add-sqr-sqrt 30.0%

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

      16. distribute-rgt-neg-in 30.0%

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

      17. metadata-eval 30.0%

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

   6. Applied egg-rr 30.0%

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

   7. Taylor expanded in a around 0 2.1%

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

      1. un-div-inv 2.1%

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

      2. +-commutative 2.1%

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

      3. add-sqr-sqrt 9.4%

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

      4. sqrt-unprod 0.3%

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

      5. sqr-neg 0.3%

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

      6. sqrt-unprod 0.1%

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

      7. add-sqr-sqrt 85.9%

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

      8. +-commutative 85.9%

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

      9. fma-define 85.9%

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

      10. associate-/l* 95.6%

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

   9. Applied egg-rr 95.6%

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

   if -3.00000000000000019e153 < b < 2.39999999999999998e-8

   1. Initial program 82.1%

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

   if 2.39999999999999998e-8 < b

   1. Initial program 14.7%

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

      1. *-commutative 14.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in b around inf 87.9%

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

      1. associate-*r/ 87.9%

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

      2. mul-1-neg 87.9%

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

   7. Simplified 87.9%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+153}:\\ \;\;\;\;\frac{b + \mathsf{fma}\left(-2, a \cdot \frac{c}{b}, b\right)}{-2 \cdot a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-254}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.4e-254) (/ b (- a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e-254) {
		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 <= 4.4d-254) 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 <= 4.4e-254) {
		tmp = b / -a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.4e-254:
		tmp = b / -a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.4e-254)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.4e-254)
		tmp = b / -a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.4e-254], N[(b / (-a)), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.4000000000000002e-254

   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 67.0%

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

      1. associate-*r/ 67.0%

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

      2. mul-1-neg 67.0%

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

   7. Simplified 67.0%

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

   if 4.4000000000000002e-254 < b

   1. Initial program 31.0%

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

      1. *-commutative 31.0%

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

   3. Simplified 31.0%

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

   5. Taylor expanded in b around inf 68.7%

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

      1. associate-*r/ 68.7%

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

      2. mul-1-neg 68.7%

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

   7. Simplified 68.7%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-254}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

   1. *-commutative 53.5%

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

3. Simplified 53.5%

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

5. Taylor expanded in b around -inf 38.8%

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

   1. associate-*r/ 38.8%

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

   2. mul-1-neg 38.8%

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

7. Simplified 38.8%

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

8. Final simplification 38.8%

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

Alternative 8: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

   1. *-commutative 53.5%

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

3. Simplified 53.5%

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

   1. clear-num 53.4%

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

   2. inv-pow 53.4%

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

6. Applied egg-rr 24.7%

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

   1. unpow-1 24.7%

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

8. Simplified 24.7%

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

9. Taylor expanded in a around 0 2.2%

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

10. Final simplification 2.2%

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

Developer target: 70.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b < 0.0d0) then
        tmp = (-b + t_0) / (2.0d0 * a)
    else
        tmp = c / (a * ((-b - t_0) / (2.0d0 * a)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024072 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :alt
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

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