The quadratic formula (r1)

Percentage Accurate: 53.3% → 90.4%

Time: 11.5s

Alternatives: 9

Speedup: 19.1×

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: 53.3% 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: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+158}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-281}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{-0.25}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e+158)
   (/ (- b) a)
   (if (<= b 4.3e-281)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (if (<= b 1.8e+74)
       (/ (* -0.5 (/ c -0.25)) (- (- b) (sqrt (fma b b (* c (* a -4.0))))))
       (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+158) {
		tmp = -b / a;
	} else if (b <= 4.3e-281) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else if (b <= 1.8e+74) {
		tmp = (-0.5 * (c / -0.25)) / (-b - sqrt(fma(b, b, (c * (a * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e+158)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 4.3e-281)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	elseif (b <= 1.8e+74)
		tmp = Float64(Float64(-0.5 * Float64(c / -0.25)) / Float64(Float64(-b) - sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e+158], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 4.3e-281], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e+74], N[(N[(-0.5 * N[(c / -0.25), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.80000000000000001e158

   1. Initial program 33.1%

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

   2. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

   if -2.80000000000000001e158 < b < 4.30000000000000023e-281

   1. Initial program 90.1%

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

   if 4.30000000000000023e-281 < b < 1.79999999999999994e74

   1. Initial program 61.0%

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

      1. prod-diff 60.7%

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

      2. *-commutative 60.7%

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

      3. fma-def 60.7%

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

      4. associate-+l+ 60.7%

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

      5. *-commutative 60.7%

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

      6. distribute-rgt-neg-in 60.7%

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

      7. fma-def 60.6%

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

      8. *-commutative 60.6%

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

      9. distribute-rgt-neg-in 60.6%

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

      10. metadata-eval 60.6%

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

      11. *-commutative 60.6%

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

      12. fma-udef 60.7%

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

      13. distribute-lft-neg-in 60.7%

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

      14. distribute-rgt-neg-in 60.7%

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

      15. fma-def 60.6%

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

   3. Applied egg-rr 60.6%

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

      1. flip-+ 60.6%

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

      2. add-sqr-sqrt 60.6%

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

      3. fma-def 60.5%

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

      4. *-commutative 60.5%

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

      5. fma-def 60.5%

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

   5. Applied egg-rr 60.5%

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

   7. Simplified 78.1%

      \[\leadsto \frac{\color{blue}{\frac{0 - \mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}}}}{2 \cdot a} \]
   8. Step-by-step derivation

      1. div-inv 77.9%

         \[\leadsto \color{blue}{\frac{0 - \mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \cdot \frac{1}{2 \cdot a}} \]

      2. sub0-neg 77.9%

         \[\leadsto \frac{\color{blue}{-\mathsf{fma}\left(c, -4 \cdot a, 0\right)}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \cdot \frac{1}{2 \cdot a} \]

      3. *-commutative 77.9%

         \[\leadsto \frac{-\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \cdot \frac{1}{\color{blue}{a \cdot 2}} \]

   9. Applied egg-rr 77.9%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \cdot \frac{1}{a \cdot 2}} \]
   10. Step-by-step derivation

       1. associate-*l/ 78.1%

          \[\leadsto \color{blue}{\frac{\left(-\mathsf{fma}\left(c, -4 \cdot a, 0\right)\right) \cdot \frac{1}{a \cdot 2}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}}} \]

       2. associate-*r/ 78.2%

          \[\leadsto \frac{\color{blue}{\frac{\left(-\mathsf{fma}\left(c, -4 \cdot a, 0\right)\right) \cdot 1}{a \cdot 2}}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       3. *-rgt-identity 78.2%

          \[\leadsto \frac{\frac{\color{blue}{-\mathsf{fma}\left(c, -4 \cdot a, 0\right)}}{a \cdot 2}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       4. neg-mul-1 78.2%

          \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \mathsf{fma}\left(c, -4 \cdot a, 0\right)}}{a \cdot 2}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       5. *-commutative 78.2%

          \[\leadsto \frac{\frac{-1 \cdot \mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\color{blue}{2 \cdot a}}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       6. times-frac 78.2%

          \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \frac{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{a}}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       7. metadata-eval 78.2%

          \[\leadsto \frac{\color{blue}{-0.5} \cdot \frac{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       8. fma-udef 78.2%

          \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{c \cdot \left(-4 \cdot a\right) + 0}}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       9. +-rgt-identity 78.2%

          \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{c \cdot \left(-4 \cdot a\right)}}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       10. *-commutative 78.2%

           \[\leadsto \frac{-0.5 \cdot \frac{c \cdot \color{blue}{\left(a \cdot -4\right)}}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       11. fma-udef 78.2%

           \[\leadsto \frac{-0.5 \cdot \frac{c \cdot \left(a \cdot -4\right)}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right) + 0}\right)}} \]

       12. +-rgt-identity 78.2%

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

       13. *-commutative 78.2%

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

   11. Simplified 78.2%

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

       1. expm1-log1p-u 55.8%

          \[\leadsto \frac{-0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c \cdot \left(a \cdot -4\right)}{a}\right)\right)}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} \]

       2. expm1-udef 21.4%

          \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c \cdot \left(a \cdot -4\right)}{a}\right)} - 1\right)}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} \]

       3. associate-/l* 21.5%

          \[\leadsto \frac{-0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{c}{\frac{a}{a \cdot -4}}}\right)} - 1\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} \]

   13. Applied egg-rr 21.5%

       \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c}{\frac{a}{a \cdot -4}}\right)} - 1\right)}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} \]
   14. Step-by-step derivation

       1. expm1-def 65.2%

          \[\leadsto \frac{-0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{\frac{a}{a \cdot -4}}\right)\right)}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} \]

       2. expm1-log1p 87.7%

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

       3. associate-/r* 87.7%

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

       4. *-inverses 87.7%

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

       5. metadata-eval 87.7%

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

   15. Simplified 87.7%

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

   if 1.79999999999999994e74 < b

   1. Initial program 3.8%

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

   2. Taylor expanded in b around inf 95.8%

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

      1. associate-*r/ 95.8%

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

      2. neg-mul-1 95.8%

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

   4. Simplified 95.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+158}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-281}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{-0.25}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 88.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+158}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 54:\\ \;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e+158)
   (/ (- b) a)
   (if (<= b 4.2e-281)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (if (<= b 54.0)
       (/ (/ c 0.5) (- (- b) (hypot b (sqrt (* a (* c -4.0))))))
       (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+158) {
		tmp = -b / a;
	} else if (b <= 4.2e-281) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else if (b <= 54.0) {
		tmp = (c / 0.5) / (-b - hypot(b, sqrt((a * (c * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+158) {
		tmp = -b / a;
	} else if (b <= 4.2e-281) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else if (b <= 54.0) {
		tmp = (c / 0.5) / (-b - Math.hypot(b, Math.sqrt((a * (c * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e+158:
		tmp = -b / a
	elif b <= 4.2e-281:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	elif b <= 54.0:
		tmp = (c / 0.5) / (-b - math.hypot(b, math.sqrt((a * (c * -4.0)))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e+158)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 4.2e-281)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	elseif (b <= 54.0)
		tmp = Float64(Float64(c / 0.5) / Float64(Float64(-b) - hypot(b, sqrt(Float64(a * Float64(c * -4.0))))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.8e+158)
		tmp = -b / a;
	elseif (b <= 4.2e-281)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	elseif (b <= 54.0)
		tmp = (c / 0.5) / (-b - hypot(b, sqrt((a * (c * -4.0)))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e+158], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 4.2e-281], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 54.0], N[(N[(c / 0.5), $MachinePrecision] / N[((-b) - N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.80000000000000001e158

   1. Initial program 33.1%

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

   2. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

   if -2.80000000000000001e158 < b < 4.1999999999999998e-281

   1. Initial program 90.1%

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

   if 4.1999999999999998e-281 < b < 54

   1. Initial program 71.2%

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

      1. prod-diff 70.9%

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

      2. *-commutative 70.9%

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

      3. fma-def 70.9%

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

      4. associate-+l+ 70.9%

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

      5. *-commutative 70.9%

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

      6. distribute-rgt-neg-in 70.9%

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

      7. fma-def 70.8%

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

      8. *-commutative 70.8%

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

      9. distribute-rgt-neg-in 70.8%

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

      10. metadata-eval 70.8%

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

      11. *-commutative 70.8%

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

      12. fma-udef 70.9%

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

      13. distribute-lft-neg-in 70.9%

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

      14. distribute-rgt-neg-in 70.9%

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

      15. fma-def 70.8%

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

   3. Applied egg-rr 70.8%

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

      1. flip-+ 70.8%

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

      2. add-sqr-sqrt 70.8%

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

      3. fma-def 70.7%

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

      4. *-commutative 70.7%

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

      5. fma-def 70.7%

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

   5. Applied egg-rr 70.7%

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

   7. Simplified 78.4%

      \[\leadsto \frac{\color{blue}{\frac{0 - \mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}}}}{2 \cdot a} \]
   8. Step-by-step derivation

      1. div-inv 78.3%

         \[\leadsto \color{blue}{\frac{0 - \mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \cdot \frac{1}{2 \cdot a}} \]

      2. sub0-neg 78.3%

         \[\leadsto \frac{\color{blue}{-\mathsf{fma}\left(c, -4 \cdot a, 0\right)}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \cdot \frac{1}{2 \cdot a} \]

      3. *-commutative 78.3%

         \[\leadsto \frac{-\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \cdot \frac{1}{\color{blue}{a \cdot 2}} \]

   9. Applied egg-rr 78.3%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \cdot \frac{1}{a \cdot 2}} \]
   10. Step-by-step derivation

       1. associate-*l/ 78.4%

          \[\leadsto \color{blue}{\frac{\left(-\mathsf{fma}\left(c, -4 \cdot a, 0\right)\right) \cdot \frac{1}{a \cdot 2}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}}} \]

       2. associate-*r/ 78.5%

          \[\leadsto \frac{\color{blue}{\frac{\left(-\mathsf{fma}\left(c, -4 \cdot a, 0\right)\right) \cdot 1}{a \cdot 2}}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       3. *-rgt-identity 78.5%

          \[\leadsto \frac{\frac{\color{blue}{-\mathsf{fma}\left(c, -4 \cdot a, 0\right)}}{a \cdot 2}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       4. neg-mul-1 78.5%

          \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \mathsf{fma}\left(c, -4 \cdot a, 0\right)}}{a \cdot 2}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       5. *-commutative 78.5%

          \[\leadsto \frac{\frac{-1 \cdot \mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\color{blue}{2 \cdot a}}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       6. times-frac 78.5%

          \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \frac{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{a}}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       7. metadata-eval 78.5%

          \[\leadsto \frac{\color{blue}{-0.5} \cdot \frac{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       8. fma-udef 78.5%

          \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{c \cdot \left(-4 \cdot a\right) + 0}}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       9. +-rgt-identity 78.5%

          \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{c \cdot \left(-4 \cdot a\right)}}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       10. *-commutative 78.5%

           \[\leadsto \frac{-0.5 \cdot \frac{c \cdot \color{blue}{\left(a \cdot -4\right)}}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -4 \cdot a, 0\right)\right)}} \]

       11. fma-udef 78.5%

           \[\leadsto \frac{-0.5 \cdot \frac{c \cdot \left(a \cdot -4\right)}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right) + 0}\right)}} \]

       12. +-rgt-identity 78.5%

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

       13. *-commutative 78.5%

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

   11. Simplified 78.5%

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

       1. expm1-log1p-u 62.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.5 \cdot \frac{c \cdot \left(a \cdot -4\right)}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right)\right)} \]

       2. expm1-udef 14.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5 \cdot \frac{c \cdot \left(a \cdot -4\right)}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right)} - 1} \]

   13. Applied egg-rr 15.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5}{\frac{\left(-b\right) - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}{\frac{c}{\frac{a}{a \cdot -4}}}}\right)} - 1} \]
   14. Step-by-step derivation

       1. expm1-def 71.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.5}{\frac{\left(-b\right) - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}{\frac{c}{\frac{a}{a \cdot -4}}}}\right)\right)} \]

       2. expm1-log1p 87.1%

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

       3. associate-/r/ 87.0%

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

       4. associate-*l/ 87.1%

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

       5. *-commutative 87.1%

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

       6. associate-*l/ 87.1%

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

       7. associate-/l* 87.1%

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

       8. associate-/r* 87.1%

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

       9. *-inverses 87.1%

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

       10. metadata-eval 87.1%

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

       11. metadata-eval 87.1%

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

       12. associate-*r* 87.1%

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

       13. *-commutative 87.1%

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

       14. associate-*l* 87.1%

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

   15. Simplified 87.1%

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

   if 54 < b

   1. Initial program 8.5%

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

   2. Taylor expanded in b around inf 94.3%

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

      1. associate-*r/ 94.3%

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

      2. neg-mul-1 94.3%

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

   4. Simplified 94.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+158}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 54:\\ \;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+158}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 220:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e+158)
   (/ (- b) a)
   (if (<= b 220.0)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+158) {
		tmp = -b / a;
	} else if (b <= 220.0) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.8d+158)) then
        tmp = -b / a
    else if (b <= 220.0d0) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+158) {
		tmp = -b / a;
	} else if (b <= 220.0) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e+158:
		tmp = -b / a
	elif b <= 220.0:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e+158)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 220.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.8e+158)
		tmp = -b / a;
	elseif (b <= 220.0)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e+158], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 220.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.80000000000000001e158

   1. Initial program 33.1%

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

   2. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

   if -2.80000000000000001e158 < b < 220

   1. Initial program 84.1%

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

   if 220 < b

   1. Initial program 8.5%

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

   2. Taylor expanded in b around inf 94.3%

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

      1. associate-*r/ 94.3%

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

      2. neg-mul-1 94.3%

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

   4. Simplified 94.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+158}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 220:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 11.2:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.4e-12)
   (- (/ c b) (/ b a))
   (if (<= b 11.2) (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.4e-12) {
		tmp = (c / b) - (b / a);
	} else if (b <= 11.2) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.4d-12)) then
        tmp = (c / b) - (b / a)
    else if (b <= 11.2d0) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.4e-12) {
		tmp = (c / b) - (b / a);
	} else if (b <= 11.2) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.4e-12:
		tmp = (c / b) - (b / a)
	elif b <= 11.2:
		tmp = (math.sqrt((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 <= -5.4e-12)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 11.2)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.4e-12)
		tmp = (c / b) - (b / a);
	elseif (b <= 11.2)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.4e-12], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 11.2], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $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 < -5.39999999999999961e-12

   1. Initial program 69.8%

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

   2. Taylor expanded in b around -inf 94.9%

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

      1. +-commutative 94.9%

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

      2. mul-1-neg 94.9%

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

      3. unsub-neg 94.9%

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

   4. Simplified 94.9%

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

   if -5.39999999999999961e-12 < b < 11.199999999999999

   1. Initial program 79.2%

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

      1. prod-diff 78.9%

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

      2. *-commutative 78.9%

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

      3. fma-def 78.9%

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

      4. associate-+l+ 78.9%

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

      5. *-commutative 78.9%

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

      6. distribute-rgt-neg-in 78.9%

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

      7. fma-def 78.9%

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

      8. *-commutative 78.9%

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

      9. distribute-rgt-neg-in 78.9%

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

      10. metadata-eval 78.9%

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

      11. *-commutative 78.9%

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

      12. fma-udef 78.9%

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

      13. distribute-lft-neg-in 78.9%

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

      14. distribute-rgt-neg-in 78.9%

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

      15. fma-def 78.9%

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

   3. Applied egg-rr 78.9%

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

   4. Taylor expanded in b around 0 69.7%

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

      1. neg-mul-1 69.7%

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

      2. unsub-neg 69.7%

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

      3. distribute-rgt-out 70.0%

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

      4. *-commutative 70.0%

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

      5. metadata-eval 70.0%

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

      6. associate-*l* 70.0%

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

      7. *-commutative 70.0%

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

   6. Simplified 70.0%

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

   if 11.199999999999999 < b

   1. Initial program 8.5%

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

   2. Taylor expanded in b around inf 94.3%

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

      1. associate-*r/ 94.3%

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

      2. neg-mul-1 94.3%

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

   4. Simplified 94.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 11.2:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 36:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e-91)
   (- (/ c b) (/ b a))
   (if (<= b 36.0) (* 0.5 (/ (sqrt (* c (* a -4.0))) a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-91) {
		tmp = (c / b) - (b / a);
	} else if (b <= 36.0) {
		tmp = 0.5 * (sqrt((c * (a * -4.0))) / 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 <= (-8.2d-91)) then
        tmp = (c / b) - (b / a)
    else if (b <= 36.0d0) then
        tmp = 0.5d0 * (sqrt((c * (a * (-4.0d0)))) / 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 <= -8.2e-91) {
		tmp = (c / b) - (b / a);
	} else if (b <= 36.0) {
		tmp = 0.5 * (Math.sqrt((c * (a * -4.0))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.2e-91:
		tmp = (c / b) - (b / a)
	elif b <= 36.0:
		tmp = 0.5 * (math.sqrt((c * (a * -4.0))) / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e-91)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 36.0)
		tmp = Float64(0.5 * Float64(sqrt(Float64(c * Float64(a * -4.0))) / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.2e-91)
		tmp = (c / b) - (b / a);
	elseif (b <= 36.0)
		tmp = 0.5 * (sqrt((c * (a * -4.0))) / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e-91], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 36.0], N[(0.5 * N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.20000000000000048e-91

   1. Initial program 70.9%

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

   2. Taylor expanded in b around -inf 86.9%

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

      1. +-commutative 86.9%

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

      2. mul-1-neg 86.9%

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

      3. unsub-neg 86.9%

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

   4. Simplified 86.9%

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

   if -8.20000000000000048e-91 < b < 36

   1. Initial program 79.5%

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

      1. prod-diff 79.2%

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

      2. *-commutative 79.2%

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

      3. fma-def 79.2%

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

      4. associate-+l+ 79.2%

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

      5. *-commutative 79.2%

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

      6. distribute-rgt-neg-in 79.2%

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

      7. fma-def 79.2%

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

      8. *-commutative 79.2%

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

      9. distribute-rgt-neg-in 79.2%

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

      10. metadata-eval 79.2%

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

      11. *-commutative 79.2%

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

      12. fma-udef 79.2%

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

      13. distribute-lft-neg-in 79.2%

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

      14. distribute-rgt-neg-in 79.2%

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

      15. fma-def 79.2%

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

   3. Applied egg-rr 79.2%

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

   4. Taylor expanded in b around 0 73.2%

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

      1. associate-*l/ 73.3%

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

      2. *-lft-identity 73.3%

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

      3. distribute-rgt-out 73.5%

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

      4. *-commutative 73.5%

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

      5. metadata-eval 73.5%

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

      6. associate-*l* 73.5%

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

      7. *-commutative 73.5%

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

   6. Simplified 73.5%

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

   if 36 < b

   1. Initial program 8.5%

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

   2. Taylor expanded in b around inf 94.3%

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

      1. associate-*r/ 94.3%

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

      2. neg-mul-1 94.3%

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

   4. Simplified 94.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 36:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 67.2% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 75.4%

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

   2. Taylor expanded in b around -inf 67.2%

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

      1. +-commutative 67.2%

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

      2. mul-1-neg 67.2%

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

      3. unsub-neg 67.2%

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

   4. Simplified 67.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 34.6%

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

   2. Taylor expanded in b around inf 65.7%

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

      1. associate-*r/ 65.7%

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

      2. neg-mul-1 65.7%

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

   4. Simplified 65.7%

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

4. Final simplification 66.4%

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

Alternative 7: 67.0% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.85 \cdot 10^{-301}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.85e-301) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.85e-301) {
		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 <= 2.85d-301) 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 <= 2.85e-301) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.85e-301:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.85e-301)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.85e-301)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.85e-301], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.85000000000000018e-301

   1. Initial program 75.6%

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

   2. Taylor expanded in b around -inf 66.3%

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

      1. associate-*r/ 66.3%

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

      2. mul-1-neg 66.3%

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

   4. Simplified 66.3%

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

   if 2.85000000000000018e-301 < b

   1. Initial program 34.1%

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

   2. Taylor expanded in b around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. neg-mul-1 66.2%

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

   4. Simplified 66.2%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.85 \cdot 10^{-301}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8: 35.7% 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(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 53.9%

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

2. Taylor expanded in b around -inf 33.0%

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

   1. associate-*r/ 33.0%

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

   2. mul-1-neg 33.0%

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

4. Simplified 33.0%

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

5. Final simplification 33.0%

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

Alternative 9: 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 53.9%

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

   1. clear-num 53.8%

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

   2. associate-/r/ 53.8%

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

   3. associate-/r* 53.8%

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

   4. metadata-eval 53.8%

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

   5. add-sqr-sqrt 35.5%

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

   6. sqrt-unprod 51.6%

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

   7. sqr-neg 51.6%

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

   8. sqrt-prod 16.1%

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

   9. add-sqr-sqrt 34.1%

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

   10. sub-neg 34.1%

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

   11. add-sqr-sqrt 31.2%

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

   12. hypot-def 29.2%

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

   13. *-commutative 29.2%

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

   14. distribute-rgt-neg-in 29.2%

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

   15. *-commutative 29.2%

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

   16. distribute-rgt-neg-in 29.2%

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

   17. metadata-eval 29.2%

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

3. Applied egg-rr 29.2%

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

4. Taylor expanded in a around 0 2.6%

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

5. Final simplification 2.6%

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

Developer target: 70.4% accurate, 1.0× speedup

Math

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

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 2023293 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (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)))