The quadratic formula (r1)

Percentage Accurate: 52.9% → 86.2%

Time: 15.1s

Alternatives: 8

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: 52.9% 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: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e+139)
   (/ (- b) a)
   (if (<= b 1.2e-50)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ 1.0 (- (/ a b) (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+139) {
		tmp = -b / a;
	} else if (b <= 1.2e-50) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	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+139)) then
        tmp = -b / a
    else if (b <= 1.2d-50) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (a * 2.0d0)
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+139) {
		tmp = -b / a;
	} else if (b <= 1.2e-50) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e+139:
		tmp = -b / a
	elif b <= 1.2e-50:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e+139)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.2e-50)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.8e+139)
		tmp = -b / a;
	elseif (b <= 1.2e-50)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = 1.0 / ((a / b) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e+139], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.2e-50], 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[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.7999999999999998e139

   1. Initial program 41.9%

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

      1. *-commutative 41.9%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in b around -inf 94.9%

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

      1. associate-*r/ 94.9%

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

      2. mul-1-neg 94.9%

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

   7. Simplified 94.9%

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

   if -2.7999999999999998e139 < b < 1.20000000000000001e-50

   1. Initial program 81.0%

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

   if 1.20000000000000001e-50 < b

   1. Initial program 16.1%

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

      1. *-commutative 16.1%

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

   3. Simplified 16.1%

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

      1. prod-diff 15.8%

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

      2. *-commutative 15.8%

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

      3. fma-def 15.8%

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

      4. associate-+l+ 15.8%

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

      5. pow2 15.8%

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

      6. distribute-lft-neg-in 15.8%

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

      7. *-commutative 15.8%

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

      8. distribute-rgt-neg-in 15.8%

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

      9. metadata-eval 15.8%

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

      10. associate-*r* 15.8%

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

      11. *-commutative 15.8%

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

      12. *-commutative 15.8%

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

      13. fma-udef 15.8%

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

   6. Applied egg-rr 15.8%

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

      1. +-commutative 15.8%

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

      2. associate-+r+ 15.8%

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

      3. associate-+l+ 15.8%

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

      4. associate-*r* 15.8%

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

      5. associate-*r* 15.8%

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

      6. distribute-lft-out 15.8%

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

      7. metadata-eval 15.8%

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

      8. +-commutative 15.8%

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

      9. unpow2 15.8%

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

      10. fma-udef 15.8%

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

      11. associate-*l* 15.8%

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

      12. *-commutative 15.8%

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

   8. Simplified 15.8%

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

      1. associate-*l* 15.8%

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

      2. fma-def 15.8%

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

   10. Applied egg-rr 15.8%

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

       1. clear-num 15.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}}} \]

       2. inv-pow 15.8%

          \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}\right)}^{-1}} \]

       3. neg-mul-1 15.8%

          \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}\right)}^{-1} \]

       4. fma-def 15.8%

          \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}\right)}^{-1} \]

   12. Applied egg-rr 15.8%

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

       1. unpow-1 15.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       2. *-lft-identity 15.8%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{1 \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       3. times-frac 15.8%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{1} \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       4. /-rgt-identity 15.8%

          \[\leadsto \frac{1}{\color{blue}{a} \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}} \]

       5. fma-udef 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{-1 \cdot b + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}}} \]

       6. *-commutative 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{b \cdot -1} + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}} \]

       7. fma-def 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{\mathsf{fma}\left(b, -1, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       8. fma-def 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{a \cdot \left(c \cdot -8\right) + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}\right)}} \]

       9. associate-*r* 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -8} + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       10. *-commutative 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{-8 \cdot \left(a \cdot c\right)} + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       11. fma-udef 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot 4\right)\right)}}\right)}} \]

       12. unpow2 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left(\color{blue}{{b}^{2}} + c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       13. associate-*r* 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}\right)}} \]

       14. *-commutative 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{\left(a \cdot c\right)} \cdot 4\right)}\right)}} \]

       15. *-commutative 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}\right)}} \]

       16. +-commutative 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \color{blue}{\left(4 \cdot \left(a \cdot c\right) + {b}^{2}\right)}}\right)}} \]

   14. Simplified 16.1%

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

   15. Taylor expanded in b around inf 83.0%

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

       1. +-commutative 83.0%

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

       2. mul-1-neg 83.0%

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

       3. unsub-neg 83.0%

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

   17. Simplified 83.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b} - \frac{b}{a}\\ t_1 := \frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.18 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (/ c b) (/ b a)))
        (t_1 (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))))
   (if (<= b -1.18e+80)
     t_0
     (if (<= b -7.2e+14)
       t_1
       (if (<= b -2.1e-8)
         t_0
         (if (<= b 2.8e-53) t_1 (/ 1.0 (- (/ a b) (/ b c)))))))))

C

double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -1.18e+80) {
		tmp = t_0;
	} else if (b <= -7.2e+14) {
		tmp = t_1;
	} else if (b <= -2.1e-8) {
		tmp = t_0;
	} else if (b <= 2.8e-53) {
		tmp = t_1;
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	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 = (c / b) - (b / a)
    t_1 = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    if (b <= (-1.18d+80)) then
        tmp = t_0
    else if (b <= (-7.2d+14)) then
        tmp = t_1
    else if (b <= (-2.1d-8)) then
        tmp = t_0
    else if (b <= 2.8d-53) then
        tmp = t_1
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -1.18e+80) {
		tmp = t_0;
	} else if (b <= -7.2e+14) {
		tmp = t_1;
	} else if (b <= -2.1e-8) {
		tmp = t_0;
	} else if (b <= 2.8e-53) {
		tmp = t_1;
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (c / b) - (b / a)
	t_1 = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	tmp = 0
	if b <= -1.18e+80:
		tmp = t_0
	elif b <= -7.2e+14:
		tmp = t_1
	elif b <= -2.1e-8:
		tmp = t_0
	elif b <= 2.8e-53:
		tmp = t_1
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / b) - Float64(b / a))
	t_1 = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (b <= -1.18e+80)
		tmp = t_0;
	elseif (b <= -7.2e+14)
		tmp = t_1;
	elseif (b <= -2.1e-8)
		tmp = t_0;
	elseif (b <= 2.8e-53)
		tmp = t_1;
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (c / b) - (b / a);
	t_1 = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	tmp = 0.0;
	if (b <= -1.18e+80)
		tmp = t_0;
	elseif (b <= -7.2e+14)
		tmp = t_1;
	elseif (b <= -2.1e-8)
		tmp = t_0;
	elseif (b <= 2.8e-53)
		tmp = t_1;
	else
		tmp = 1.0 / ((a / b) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.18e+80], t$95$0, If[LessEqual[b, -7.2e+14], t$95$1, If[LessEqual[b, -2.1e-8], t$95$0, If[LessEqual[b, 2.8e-53], t$95$1, N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.18e80 or -7.2e14 < b < -2.09999999999999994e-8

   1. Initial program 60.9%

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

      1. *-commutative 60.9%

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

   3. Simplified 60.9%

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

   5. Taylor expanded in b around -inf 94.8%

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

      1. +-commutative 94.8%

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

      2. mul-1-neg 94.8%

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

      3. unsub-neg 94.8%

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

   7. Simplified 94.8%

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

   if -1.18e80 < b < -7.2e14 or -2.09999999999999994e-8 < b < 2.79999999999999985e-53

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

   3. Simplified 78.2%

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

      1. prod-diff 78.0%

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

      2. *-commutative 78.0%

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

      3. fma-def 78.0%

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

      4. associate-+l+ 78.0%

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

      5. pow2 78.0%

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

      6. distribute-lft-neg-in 78.0%

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

      7. *-commutative 78.0%

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

      8. distribute-rgt-neg-in 78.0%

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

      9. metadata-eval 78.0%

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

      10. associate-*r* 78.0%

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

      11. *-commutative 78.0%

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

      12. *-commutative 78.0%

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

      13. fma-udef 78.0%

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

   6. Applied egg-rr 78.0%

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

      1. +-commutative 78.0%

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

      2. associate-+r+ 78.0%

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

      3. associate-+l+ 78.0%

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

      4. associate-*r* 78.0%

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

      5. associate-*r* 78.0%

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

      6. distribute-lft-out 78.0%

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

      7. metadata-eval 78.0%

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

      8. +-commutative 78.0%

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

      9. unpow2 78.0%

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

      10. fma-udef 78.0%

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

      11. associate-*l* 78.0%

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

      12. *-commutative 78.0%

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

   8. Simplified 78.0%

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

   9. Taylor expanded in b around 0 70.0%

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

       1. mul-1-neg 70.0%

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

       2. unsub-neg 70.0%

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

       3. associate-*r* 70.0%

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

       4. associate-*r* 70.0%

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

       5. distribute-rgt-in 70.2%

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

       6. distribute-rgt-out 70.2%

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

       7. metadata-eval 70.2%

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

   11. Simplified 70.2%

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

   if 2.79999999999999985e-53 < b

   1. Initial program 16.1%

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

      1. *-commutative 16.1%

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

   3. Simplified 16.1%

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

      1. prod-diff 15.8%

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

      2. *-commutative 15.8%

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

      3. fma-def 15.8%

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

      4. associate-+l+ 15.8%

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

      5. pow2 15.8%

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

      6. distribute-lft-neg-in 15.8%

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

      7. *-commutative 15.8%

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

      8. distribute-rgt-neg-in 15.8%

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

      9. metadata-eval 15.8%

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

      10. associate-*r* 15.8%

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

      11. *-commutative 15.8%

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

      12. *-commutative 15.8%

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

      13. fma-udef 15.8%

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

   6. Applied egg-rr 15.8%

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

      1. +-commutative 15.8%

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

      2. associate-+r+ 15.8%

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

      3. associate-+l+ 15.8%

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

      4. associate-*r* 15.8%

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

      5. associate-*r* 15.8%

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

      6. distribute-lft-out 15.8%

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

      7. metadata-eval 15.8%

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

      8. +-commutative 15.8%

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

      9. unpow2 15.8%

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

      10. fma-udef 15.8%

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

      11. associate-*l* 15.8%

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

      12. *-commutative 15.8%

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

   8. Simplified 15.8%

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

      1. associate-*l* 15.8%

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

      2. fma-def 15.8%

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

   10. Applied egg-rr 15.8%

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

       1. clear-num 15.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}}} \]

       2. inv-pow 15.8%

          \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}\right)}^{-1}} \]

       3. neg-mul-1 15.8%

          \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}\right)}^{-1} \]

       4. fma-def 15.8%

          \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}\right)}^{-1} \]

   12. Applied egg-rr 15.8%

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

       1. unpow-1 15.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       2. *-lft-identity 15.8%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{1 \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       3. times-frac 15.8%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{1} \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       4. /-rgt-identity 15.8%

          \[\leadsto \frac{1}{\color{blue}{a} \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}} \]

       5. fma-udef 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{-1 \cdot b + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}}} \]

       6. *-commutative 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{b \cdot -1} + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}} \]

       7. fma-def 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{\mathsf{fma}\left(b, -1, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       8. fma-def 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{a \cdot \left(c \cdot -8\right) + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}\right)}} \]

       9. associate-*r* 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -8} + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       10. *-commutative 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{-8 \cdot \left(a \cdot c\right)} + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       11. fma-udef 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot 4\right)\right)}}\right)}} \]

       12. unpow2 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left(\color{blue}{{b}^{2}} + c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       13. associate-*r* 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}\right)}} \]

       14. *-commutative 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{\left(a \cdot c\right)} \cdot 4\right)}\right)}} \]

       15. *-commutative 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}\right)}} \]

       16. +-commutative 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \color{blue}{\left(4 \cdot \left(a \cdot c\right) + {b}^{2}\right)}}\right)}} \]

   14. Simplified 16.1%

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

   15. Taylor expanded in b around inf 83.0%

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

       1. +-commutative 83.0%

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

       2. mul-1-neg 83.0%

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

       3. unsub-neg 83.0%

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

   17. Simplified 83.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+80}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b} - \frac{b}{a}\\ t_1 := 0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{if}\;b \leq -8.6 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.3 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (/ c b) (/ b a))) (t_1 (* 0.5 (/ (sqrt (* c (* a -4.0))) a))))
   (if (<= b -8.6e+38)
     t_0
     (if (<= b -7.2e+14)
       t_1
       (if (<= b -1.3e-8)
         t_0
         (if (<= b 2.55e-51) t_1 (/ 1.0 (- (/ a b) (/ b c)))))))))

C

double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = 0.5 * (sqrt((c * (a * -4.0))) / a);
	double tmp;
	if (b <= -8.6e+38) {
		tmp = t_0;
	} else if (b <= -7.2e+14) {
		tmp = t_1;
	} else if (b <= -1.3e-8) {
		tmp = t_0;
	} else if (b <= 2.55e-51) {
		tmp = t_1;
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	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 = (c / b) - (b / a)
    t_1 = 0.5d0 * (sqrt((c * (a * (-4.0d0)))) / a)
    if (b <= (-8.6d+38)) then
        tmp = t_0
    else if (b <= (-7.2d+14)) then
        tmp = t_1
    else if (b <= (-1.3d-8)) then
        tmp = t_0
    else if (b <= 2.55d-51) then
        tmp = t_1
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = 0.5 * (Math.sqrt((c * (a * -4.0))) / a);
	double tmp;
	if (b <= -8.6e+38) {
		tmp = t_0;
	} else if (b <= -7.2e+14) {
		tmp = t_1;
	} else if (b <= -1.3e-8) {
		tmp = t_0;
	} else if (b <= 2.55e-51) {
		tmp = t_1;
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (c / b) - (b / a)
	t_1 = 0.5 * (math.sqrt((c * (a * -4.0))) / a)
	tmp = 0
	if b <= -8.6e+38:
		tmp = t_0
	elif b <= -7.2e+14:
		tmp = t_1
	elif b <= -1.3e-8:
		tmp = t_0
	elif b <= 2.55e-51:
		tmp = t_1
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / b) - Float64(b / a))
	t_1 = Float64(0.5 * Float64(sqrt(Float64(c * Float64(a * -4.0))) / a))
	tmp = 0.0
	if (b <= -8.6e+38)
		tmp = t_0;
	elseif (b <= -7.2e+14)
		tmp = t_1;
	elseif (b <= -1.3e-8)
		tmp = t_0;
	elseif (b <= 2.55e-51)
		tmp = t_1;
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (c / b) - (b / a);
	t_1 = 0.5 * (sqrt((c * (a * -4.0))) / a);
	tmp = 0.0;
	if (b <= -8.6e+38)
		tmp = t_0;
	elseif (b <= -7.2e+14)
		tmp = t_1;
	elseif (b <= -1.3e-8)
		tmp = t_0;
	elseif (b <= 2.55e-51)
		tmp = t_1;
	else
		tmp = 1.0 / ((a / b) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8.6e+38], t$95$0, If[LessEqual[b, -7.2e+14], t$95$1, If[LessEqual[b, -1.3e-8], t$95$0, If[LessEqual[b, 2.55e-51], t$95$1, N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.5999999999999994e38 or -7.2e14 < b < -1.3000000000000001e-8

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in b around -inf 92.2%

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

      1. +-commutative 92.2%

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

      2. mul-1-neg 92.2%

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

      3. unsub-neg 92.2%

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

   7. Simplified 92.2%

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

   if -8.5999999999999994e38 < b < -7.2e14 or -1.3000000000000001e-8 < b < 2.5499999999999999e-51

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

   3. Simplified 77.2%

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

      1. prod-diff 76.9%

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

      2. *-commutative 76.9%

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

      3. fma-def 76.9%

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

      4. associate-+l+ 76.9%

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

      5. pow2 76.9%

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

      6. distribute-lft-neg-in 76.9%

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

      7. *-commutative 76.9%

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

      8. distribute-rgt-neg-in 76.9%

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

      9. metadata-eval 76.9%

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

      10. associate-*r* 76.9%

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

      11. *-commutative 76.9%

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

      12. *-commutative 76.9%

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

      13. fma-udef 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. +-commutative 76.9%

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

      2. associate-+r+ 76.9%

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

      3. associate-+l+ 76.9%

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

      4. associate-*r* 76.9%

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

      5. associate-*r* 76.9%

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

      6. distribute-lft-out 76.9%

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

      7. metadata-eval 76.9%

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

      8. +-commutative 76.9%

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

      9. unpow2 76.9%

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

      10. fma-udef 76.9%

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

      11. associate-*l* 76.9%

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

      12. *-commutative 76.9%

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

   8. Simplified 76.9%

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

   9. Taylor expanded in b around 0 68.1%

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

       1. associate-*l/ 68.3%

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

       2. *-lft-identity 68.3%

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

       3. associate-*r* 68.3%

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

       4. associate-*r* 68.3%

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

       5. distribute-rgt-in 68.6%

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

       6. distribute-rgt-out 68.6%

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

       7. metadata-eval 68.6%

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

   11. Simplified 68.6%

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

   if 2.5499999999999999e-51 < b

   1. Initial program 16.1%

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

      1. *-commutative 16.1%

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

   3. Simplified 16.1%

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

      1. prod-diff 15.8%

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

      2. *-commutative 15.8%

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

      3. fma-def 15.8%

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

      4. associate-+l+ 15.8%

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

      5. pow2 15.8%

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

      6. distribute-lft-neg-in 15.8%

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

      7. *-commutative 15.8%

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

      8. distribute-rgt-neg-in 15.8%

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

      9. metadata-eval 15.8%

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

      10. associate-*r* 15.8%

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

      11. *-commutative 15.8%

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

      12. *-commutative 15.8%

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

      13. fma-udef 15.8%

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

   6. Applied egg-rr 15.8%

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

      1. +-commutative 15.8%

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

      2. associate-+r+ 15.8%

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

      3. associate-+l+ 15.8%

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

      4. associate-*r* 15.8%

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

      5. associate-*r* 15.8%

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

      6. distribute-lft-out 15.8%

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

      7. metadata-eval 15.8%

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

      8. +-commutative 15.8%

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

      9. unpow2 15.8%

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

      10. fma-udef 15.8%

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

      11. associate-*l* 15.8%

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

      12. *-commutative 15.8%

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

   8. Simplified 15.8%

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

      1. associate-*l* 15.8%

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

      2. fma-def 15.8%

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

   10. Applied egg-rr 15.8%

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

       1. clear-num 15.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}}} \]

       2. inv-pow 15.8%

          \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}\right)}^{-1}} \]

       3. neg-mul-1 15.8%

          \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}\right)}^{-1} \]

       4. fma-def 15.8%

          \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}\right)}^{-1} \]

   12. Applied egg-rr 15.8%

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

       1. unpow-1 15.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       2. *-lft-identity 15.8%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{1 \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       3. times-frac 15.8%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{1} \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       4. /-rgt-identity 15.8%

          \[\leadsto \frac{1}{\color{blue}{a} \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}} \]

       5. fma-udef 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{-1 \cdot b + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}}} \]

       6. *-commutative 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{b \cdot -1} + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}} \]

       7. fma-def 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{\mathsf{fma}\left(b, -1, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       8. fma-def 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{a \cdot \left(c \cdot -8\right) + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}\right)}} \]

       9. associate-*r* 15.8%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -8} + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       10. *-commutative 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{-8 \cdot \left(a \cdot c\right)} + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       11. fma-udef 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot 4\right)\right)}}\right)}} \]

       12. unpow2 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left(\color{blue}{{b}^{2}} + c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       13. associate-*r* 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}\right)}} \]

       14. *-commutative 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{\left(a \cdot c\right)} \cdot 4\right)}\right)}} \]

       15. *-commutative 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}\right)}} \]

       16. +-commutative 15.8%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \color{blue}{\left(4 \cdot \left(a \cdot c\right) + {b}^{2}\right)}}\right)}} \]

   14. Simplified 16.1%

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

   15. Taylor expanded in b around inf 83.0%

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

       1. +-commutative 83.0%

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

       2. mul-1-neg 83.0%

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

       3. unsub-neg 83.0%

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

   17. Simplified 83.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{+38}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{elif}\;b \leq -1.3 \cdot 10^{-8}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-51}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 10.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	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 = 1.0d0 / ((a / b) - (b / c))
    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 = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	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 = 1.0 / ((a / b) - (b / c));
	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[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.5%

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

      1. *-commutative 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in b around -inf 62.3%

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

      1. +-commutative 62.3%

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

      2. mul-1-neg 62.3%

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

      3. unsub-neg 62.3%

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

   7. Simplified 62.3%

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

   if -4.999999999999985e-310 < b

   1. Initial program 33.6%

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

      1. *-commutative 33.6%

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

   3. Simplified 33.6%

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

      1. prod-diff 33.2%

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

      2. *-commutative 33.2%

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

      3. fma-def 33.2%

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

      4. associate-+l+ 33.2%

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

      5. pow2 33.2%

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

      6. distribute-lft-neg-in 33.2%

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

      7. *-commutative 33.2%

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

      8. distribute-rgt-neg-in 33.2%

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

      9. metadata-eval 33.2%

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

      10. associate-*r* 33.2%

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

      11. *-commutative 33.2%

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

      12. *-commutative 33.2%

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

      13. fma-udef 33.2%

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

   6. Applied egg-rr 33.2%

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

      1. +-commutative 33.2%

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

      2. associate-+r+ 33.2%

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

      3. associate-+l+ 33.2%

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

      4. associate-*r* 33.2%

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

      5. associate-*r* 33.2%

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

      6. distribute-lft-out 33.2%

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

      7. metadata-eval 33.2%

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

      8. +-commutative 33.2%

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

      9. unpow2 33.2%

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

      10. fma-udef 33.2%

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

      11. associate-*l* 33.2%

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

      12. *-commutative 33.2%

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

   8. Simplified 33.2%

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

      1. associate-*l* 33.2%

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

      2. fma-def 33.2%

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

   10. Applied egg-rr 33.2%

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

       1. clear-num 33.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}}} \]

       2. inv-pow 33.1%

          \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}\right)}^{-1}} \]

       3. neg-mul-1 33.1%

          \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}\right)}^{-1} \]

       4. fma-def 33.1%

          \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}\right)}^{-1} \]

   12. Applied egg-rr 33.1%

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

       1. unpow-1 33.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       2. *-lft-identity 33.1%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{1 \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       3. times-frac 33.1%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{1} \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       4. /-rgt-identity 33.1%

          \[\leadsto \frac{1}{\color{blue}{a} \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}} \]

       5. fma-udef 33.1%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{-1 \cdot b + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}}} \]

       6. *-commutative 33.1%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{b \cdot -1} + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}} \]

       7. fma-def 33.1%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{\mathsf{fma}\left(b, -1, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       8. fma-def 33.2%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{a \cdot \left(c \cdot -8\right) + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}\right)}} \]

       9. associate-*r* 33.2%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -8} + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       10. *-commutative 33.2%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{-8 \cdot \left(a \cdot c\right)} + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       11. fma-udef 33.1%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot 4\right)\right)}}\right)}} \]

       12. unpow2 33.1%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left(\color{blue}{{b}^{2}} + c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       13. associate-*r* 33.1%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}\right)}} \]

       14. *-commutative 33.1%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{\left(a \cdot c\right)} \cdot 4\right)}\right)}} \]

       15. *-commutative 33.1%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}\right)}} \]

       16. +-commutative 33.1%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \color{blue}{\left(4 \cdot \left(a \cdot c\right) + {b}^{2}\right)}}\right)}} \]

   14. Simplified 33.5%

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

   15. Taylor expanded in b around inf 63.4%

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

       1. +-commutative 63.4%

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

       2. mul-1-neg 63.4%

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

       3. unsub-neg 63.4%

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

   17. Simplified 63.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

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

Alternative 5: 68.6% 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 74.5%

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

      1. *-commutative 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in b around -inf 62.3%

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

      1. +-commutative 62.3%

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

      2. mul-1-neg 62.3%

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

      3. unsub-neg 62.3%

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

   7. Simplified 62.3%

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

   if -4.999999999999985e-310 < b

   1. Initial program 33.6%

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

      1. *-commutative 33.6%

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

   3. Simplified 33.6%

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

   5. Taylor expanded in b around inf 63.2%

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

      1. mul-1-neg 63.2%

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

      2. distribute-neg-frac 63.2%

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

   7. Simplified 63.2%

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

4. Final simplification 62.8%

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

Alternative 6: 45.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -5e-310) (/ (- b) a) (/ 0.0 a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = -b / a
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -b / a
	else:
		tmp = 0.0 / a
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[((-b) / a), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.5%

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

      1. *-commutative 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in b around -inf 62.0%

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

      1. associate-*r/ 62.0%

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

      2. mul-1-neg 62.0%

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

   7. Simplified 62.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 33.6%

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

      1. *-commutative 33.6%

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

   3. Simplified 33.6%

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

      1. prod-diff 33.2%

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

      2. *-commutative 33.2%

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

      3. fma-def 33.2%

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

      4. associate-+l+ 33.2%

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

      5. pow2 33.2%

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

      6. distribute-lft-neg-in 33.2%

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

      7. *-commutative 33.2%

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

      8. distribute-rgt-neg-in 33.2%

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

      9. metadata-eval 33.2%

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

      10. associate-*r* 33.2%

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

      11. *-commutative 33.2%

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

      12. *-commutative 33.2%

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

      13. fma-udef 33.2%

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

   6. Applied egg-rr 33.2%

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

      1. +-commutative 33.2%

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

      2. associate-+r+ 33.2%

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

      3. associate-+l+ 33.2%

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

      4. associate-*r* 33.2%

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

      5. associate-*r* 33.2%

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

      6. distribute-lft-out 33.2%

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

      7. metadata-eval 33.2%

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

      8. +-commutative 33.2%

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

      9. unpow2 33.2%

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

      10. fma-udef 33.2%

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

      11. associate-*l* 33.2%

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

      12. *-commutative 33.2%

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

   8. Simplified 33.2%

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

      1. associate-*l* 33.2%

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

      2. fma-def 33.2%

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

   10. Applied egg-rr 33.2%

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

       1. clear-num 33.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}}} \]

       2. inv-pow 33.1%

          \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}\right)}^{-1}} \]

       3. neg-mul-1 33.1%

          \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}\right)}^{-1} \]

       4. fma-def 33.1%

          \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}\right)}^{-1} \]

   12. Applied egg-rr 33.1%

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

       1. unpow-1 33.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       2. *-lft-identity 33.1%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{1 \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       3. times-frac 33.1%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{1} \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       4. /-rgt-identity 33.1%

          \[\leadsto \frac{1}{\color{blue}{a} \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}} \]

       5. fma-udef 33.1%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{-1 \cdot b + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}}} \]

       6. *-commutative 33.1%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{b \cdot -1} + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}} \]

       7. fma-def 33.1%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{\mathsf{fma}\left(b, -1, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

       8. fma-def 33.2%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{a \cdot \left(c \cdot -8\right) + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}\right)}} \]

       9. associate-*r* 33.2%

          \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -8} + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       10. *-commutative 33.2%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{-8 \cdot \left(a \cdot c\right)} + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       11. fma-udef 33.1%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot 4\right)\right)}}\right)}} \]

       12. unpow2 33.1%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left(\color{blue}{{b}^{2}} + c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

       13. associate-*r* 33.1%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}\right)}} \]

       14. *-commutative 33.1%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{\left(a \cdot c\right)} \cdot 4\right)}\right)}} \]

       15. *-commutative 33.1%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}\right)}} \]

       16. +-commutative 33.1%

           \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \color{blue}{\left(4 \cdot \left(a \cdot c\right) + {b}^{2}\right)}}\right)}} \]

   14. Simplified 33.5%

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

   15. Taylor expanded in a around 0 16.1%

       \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
   16. Step-by-step derivation

       1. associate-*r/ 16.1%

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

       2. distribute-rgt1-in 16.1%

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

       3. metadata-eval 16.1%

          \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

       4. mul0-lft 16.1%

          \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

       5. metadata-eval 16.1%

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

   17. Simplified 16.1%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-290}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.35e-290) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.35e-290) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.35d-290) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.35e-290) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.35e-290:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.35e-290)
		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 <= 1.35e-290)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.35e-290], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.34999999999999999e-290

   1. Initial program 74.3%

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

      1. *-commutative 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in b around -inf 60.5%

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

      1. associate-*r/ 60.5%

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

      2. mul-1-neg 60.5%

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

   7. Simplified 60.5%

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

   if 1.34999999999999999e-290 < b

   1. Initial program 32.9%

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

      1. *-commutative 32.9%

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

   3. Simplified 32.9%

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

   5. Taylor expanded in b around inf 64.5%

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

      1. mul-1-neg 64.5%

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

      2. distribute-neg-frac 64.5%

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

   7. Simplified 64.5%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-290}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 11.1% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

   1. *-commutative 51.2%

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

3. Simplified 51.2%

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

   1. prod-diff 50.9%

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

   2. *-commutative 50.9%

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

   3. fma-def 50.9%

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

   4. associate-+l+ 50.9%

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

   5. pow2 50.9%

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

   6. distribute-lft-neg-in 50.9%

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

   7. *-commutative 50.9%

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

   8. distribute-rgt-neg-in 50.9%

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

   9. metadata-eval 50.9%

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

   10. associate-*r* 50.9%

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

   11. *-commutative 50.9%

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

   12. *-commutative 50.9%

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

   13. fma-udef 50.9%

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

6. Applied egg-rr 50.9%

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

   1. +-commutative 50.9%

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

   2. associate-+r+ 50.9%

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

   3. associate-+l+ 50.9%

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

   4. associate-*r* 50.9%

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

   5. associate-*r* 50.9%

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

   6. distribute-lft-out 50.9%

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

   7. metadata-eval 50.9%

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

   8. +-commutative 50.9%

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

   9. unpow2 50.9%

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

   10. fma-udef 50.9%

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

   11. associate-*l* 50.9%

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

   12. *-commutative 50.9%

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

8. Simplified 50.9%

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

   1. associate-*l* 50.9%

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

   2. fma-def 50.9%

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

10. Applied egg-rr 50.9%

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

    1. clear-num 50.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}}} \]

    2. inv-pow 50.7%

       \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}\right)}^{-1}} \]

    3. neg-mul-1 50.7%

       \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}\right)}^{-1} \]

    4. fma-def 50.7%

       \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}\right)}^{-1} \]

12. Applied egg-rr 50.7%

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

    1. unpow-1 50.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

    2. *-lft-identity 50.7%

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{1 \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

    3. times-frac 50.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{1} \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

    4. /-rgt-identity 50.7%

       \[\leadsto \frac{1}{\color{blue}{a} \cdot \frac{2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}} \]

    5. fma-udef 50.7%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{-1 \cdot b + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}}} \]

    6. *-commutative 50.7%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{b \cdot -1} + \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}} \]

    7. fma-def 50.7%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\color{blue}{\mathsf{fma}\left(b, -1, \sqrt{\mathsf{fma}\left(a, c \cdot -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}\right)}}} \]

    8. fma-def 50.7%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{a \cdot \left(c \cdot -8\right) + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}\right)}} \]

    9. associate-*r* 50.7%

       \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -8} + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

    10. *-commutative 50.7%

        \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{\color{blue}{-8 \cdot \left(a \cdot c\right)} + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

    11. fma-udef 50.7%

        \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot 4\right)\right)}}\right)}} \]

    12. unpow2 50.7%

        \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left(\color{blue}{{b}^{2}} + c \cdot \left(a \cdot 4\right)\right)}\right)}} \]

    13. associate-*r* 50.7%

        \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}\right)}} \]

    14. *-commutative 50.7%

        \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{\left(a \cdot c\right)} \cdot 4\right)}\right)}} \]

    15. *-commutative 50.7%

        \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \left({b}^{2} + \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}\right)}} \]

    16. +-commutative 50.7%

        \[\leadsto \frac{1}{a \cdot \frac{2}{\mathsf{fma}\left(b, -1, \sqrt{-8 \cdot \left(a \cdot c\right) + \color{blue}{\left(4 \cdot \left(a \cdot c\right) + {b}^{2}\right)}}\right)}} \]

14. Simplified 51.0%

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

15. Taylor expanded in a around 0 10.3%

    \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
16. Step-by-step derivation

    1. associate-*r/ 10.3%

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

    2. distribute-rgt1-in 10.3%

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

    3. metadata-eval 10.3%

       \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 10.3%

       \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 10.3%

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

17. Simplified 10.3%

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

18. Final simplification 10.3%

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

Developer target: 71.4% 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 2024019 
(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)))