quadm (p42, negative)

Percentage Accurate: 53.2% → 90.5%

Time: 12.2s

Alternatives: 9

Speedup: 19.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{+130}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -2.5 \cdot 10^{-297}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - t_0}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+65}:\\ \;\;\;\;-0.5 \cdot \frac{b + t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* c 4.0) a)))))
   (if (<= b -1.1e+130)
     (/ (- c) b)
     (if (<= b -2.5e-297)
       (* -0.5 (/ (* c 4.0) (- b t_0)))
       (if (<= b 4e+65) (* -0.5 (/ (+ b t_0) a)) (- (/ c b) (/ b a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((c * 4.0) * a)));
	double tmp;
	if (b <= -1.1e+130) {
		tmp = -c / b;
	} else if (b <= -2.5e-297) {
		tmp = -0.5 * ((c * 4.0) / (b - t_0));
	} else if (b <= 4e+65) {
		tmp = -0.5 * ((b + t_0) / a);
	} else {
		tmp = (c / b) - (b / 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) - ((c * 4.0d0) * a)))
    if (b <= (-1.1d+130)) then
        tmp = -c / b
    else if (b <= (-2.5d-297)) then
        tmp = (-0.5d0) * ((c * 4.0d0) / (b - t_0))
    else if (b <= 4d+65) then
        tmp = (-0.5d0) * ((b + t_0) / a)
    else
        tmp = (c / b) - (b / 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) - ((c * 4.0) * a)));
	double tmp;
	if (b <= -1.1e+130) {
		tmp = -c / b;
	} else if (b <= -2.5e-297) {
		tmp = -0.5 * ((c * 4.0) / (b - t_0));
	} else if (b <= 4e+65) {
		tmp = -0.5 * ((b + t_0) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((c * 4.0) * a)))
	tmp = 0
	if b <= -1.1e+130:
		tmp = -c / b
	elif b <= -2.5e-297:
		tmp = -0.5 * ((c * 4.0) / (b - t_0))
	elif b <= 4e+65:
		tmp = -0.5 * ((b + t_0) / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(c * 4.0) * a)))
	tmp = 0.0
	if (b <= -1.1e+130)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= -2.5e-297)
		tmp = Float64(-0.5 * Float64(Float64(c * 4.0) / Float64(b - t_0)));
	elseif (b <= 4e+65)
		tmp = Float64(-0.5 * Float64(Float64(b + t_0) / a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((c * 4.0) * a)));
	tmp = 0.0;
	if (b <= -1.1e+130)
		tmp = -c / b;
	elseif (b <= -2.5e-297)
		tmp = -0.5 * ((c * 4.0) / (b - t_0));
	elseif (b <= 4e+65)
		tmp = -0.5 * ((b + t_0) / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(c * 4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.1e+130], N[((-c) / b), $MachinePrecision], If[LessEqual[b, -2.5e-297], N[(-0.5 * N[(N[(c * 4.0), $MachinePrecision] / N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e+65], N[(-0.5 * N[(N[(b + t$95$0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.09999999999999997e130

   1. Initial program 4.3%

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

   2. Taylor expanded in b around -inf 95.9%

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

      1. associate-*r/ 95.9%

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

      2. neg-mul-1 95.9%

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

   4. Simplified 95.9%

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

   if -1.09999999999999997e130 < b < -2.5e-297

   1. Initial program 44.5%

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

   3. Simplified 44.5%

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

      1. fma-udef 44.5%

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

      2. associate-*r* 44.5%

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

      3. metadata-eval 44.5%

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

      4. distribute-rgt-neg-in 44.5%

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

      5. *-commutative 44.5%

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

      6. +-commutative 44.5%

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

      7. sub-neg 44.5%

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

      8. *-commutative 44.5%

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

      9. associate-*l* 44.5%

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

   5. Applied egg-rr 44.5%

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

      1. flip-+ 44.4%

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

      2. add-sqr-sqrt 44.5%

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

   7. Applied egg-rr 44.5%

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

      1. associate--r- 78.7%

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

      2. +-inverses 78.7%

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

      3. *-commutative 78.7%

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

      4. associate-*l* 78.7%

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

      5. sub-neg 78.7%

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

      6. +-commutative 78.7%

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

      7. distribute-rgt-neg-in 78.7%

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

      8. distribute-rgt-neg-in 78.7%

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

      9. metadata-eval 78.7%

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

      10. fma-udef 78.7%

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

   9. Simplified 78.7%

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

       1. expm1-log1p-u 60.0%

          \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0 + c \cdot \left(4 \cdot a\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a}\right)\right)} \]

       2. expm1-udef 16.7%

          \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{0 + c \cdot \left(4 \cdot a\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a}\right)} - 1\right)} \]

       3. associate-/l/ 15.4%

          \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{0 + c \cdot \left(4 \cdot a\right)}{a \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}}\right)} - 1\right) \]

       4. +-lft-identity 15.4%

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

   11. Applied egg-rr 15.4%

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

       1. expm1-def 51.4%

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

       2. expm1-log1p 67.7%

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

       3. associate-*r* 67.7%

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

       4. *-commutative 67.7%

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

       5. times-frac 91.1%

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

       6. *-inverses 91.1%

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

   13. Simplified 91.1%

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

       1. fma-udef 44.5%

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

       2. associate-*r* 44.5%

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

       3. metadata-eval 44.5%

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

       4. distribute-rgt-neg-in 44.5%

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

       5. *-commutative 44.5%

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

       6. +-commutative 44.5%

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

       7. sub-neg 44.5%

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

       8. *-commutative 44.5%

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

       9. associate-*l* 44.5%

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

   15. Applied egg-rr 91.1%

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

   if -2.5e-297 < b < 4e65

   1. Initial program 87.9%

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

   3. Simplified 87.9%

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

      1. fma-udef 87.9%

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

      2. associate-*r* 87.9%

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

      3. metadata-eval 87.9%

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

      4. distribute-rgt-neg-in 87.9%

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

      5. *-commutative 87.9%

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

      6. +-commutative 87.9%

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

      7. sub-neg 87.9%

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

      8. *-commutative 87.9%

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

      9. associate-*l* 87.9%

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

   5. Applied egg-rr 87.9%

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

   if 4e65 < b

   1. Initial program 54.8%

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

   2. Taylor expanded in b around inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. unsub-neg 98.3%

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

   4. Simplified 98.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+130}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -2.5 \cdot 10^{-297}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+65}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-95}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{elif}\;b \leq 7.4 \cdot 10^{+68}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e-95)
   (* -0.5 (/ (* c 4.0) (+ b (- b (* (/ c (/ b a)) 2.0)))))
   (if (<= b 7.4e+68)
     (* -0.5 (/ (+ b (sqrt (- (* b b) (* (* c 4.0) a)))) a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-95) {
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	} else if (b <= 7.4e+68) {
		tmp = -0.5 * ((b + sqrt(((b * b) - ((c * 4.0) * a)))) / a);
	} else {
		tmp = (c / b) - (b / 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 <= (-3.5d-95)) then
        tmp = (-0.5d0) * ((c * 4.0d0) / (b + (b - ((c / (b / a)) * 2.0d0))))
    else if (b <= 7.4d+68) then
        tmp = (-0.5d0) * ((b + sqrt(((b * b) - ((c * 4.0d0) * a)))) / a)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-95) {
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	} else if (b <= 7.4e+68) {
		tmp = -0.5 * ((b + Math.sqrt(((b * b) - ((c * 4.0) * a)))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.5e-95:
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))))
	elif b <= 7.4e+68:
		tmp = -0.5 * ((b + math.sqrt(((b * b) - ((c * 4.0) * a)))) / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e-95)
		tmp = Float64(-0.5 * Float64(Float64(c * 4.0) / Float64(b + Float64(b - Float64(Float64(c / Float64(b / a)) * 2.0)))));
	elseif (b <= 7.4e+68)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(Float64(c * 4.0) * a)))) / a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.5e-95)
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	elseif (b <= 7.4e+68)
		tmp = -0.5 * ((b + sqrt(((b * b) - ((c * 4.0) * a)))) / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.5e-95], N[(-0.5 * N[(N[(c * 4.0), $MachinePrecision] / N[(b + N[(b - N[(N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.4e+68], N[(-0.5 * N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(c * 4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.4999999999999997e-95

   1. Initial program 18.5%

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

   3. Simplified 18.5%

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

      1. fma-udef 18.5%

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

      2. associate-*r* 18.5%

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

      3. metadata-eval 18.5%

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

      4. distribute-rgt-neg-in 18.5%

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

      5. *-commutative 18.5%

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

      6. +-commutative 18.5%

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

      7. sub-neg 18.5%

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

      8. *-commutative 18.5%

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

      9. associate-*l* 18.5%

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

   5. Applied egg-rr 18.5%

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

      1. flip-+ 17.8%

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

      2. add-sqr-sqrt 17.8%

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

   7. Applied egg-rr 17.8%

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

      1. associate--r- 49.3%

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

      2. +-inverses 65.6%

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

      3. *-commutative 65.6%

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

      4. associate-*l* 65.6%

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

      5. sub-neg 65.6%

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

      6. +-commutative 65.6%

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

      7. distribute-rgt-neg-in 65.6%

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

      8. distribute-rgt-neg-in 65.6%

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

      9. metadata-eval 65.6%

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

      10. fma-udef 65.6%

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

   9. Simplified 65.6%

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

       1. expm1-log1p-u 57.6%

          \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0 + c \cdot \left(4 \cdot a\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a}\right)\right)} \]

       2. expm1-udef 27.0%

          \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{0 + c \cdot \left(4 \cdot a\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a}\right)} - 1\right)} \]

       3. associate-/l/ 27.0%

          \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{0 + c \cdot \left(4 \cdot a\right)}{a \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}}\right)} - 1\right) \]

       4. +-lft-identity 27.0%

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

   11. Applied egg-rr 27.0%

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

       1. expm1-def 53.6%

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

       2. expm1-log1p 60.8%

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

       3. associate-*r* 60.8%

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

       4. *-commutative 60.8%

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

       5. times-frac 74.4%

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

       6. *-inverses 74.4%

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

   13. Simplified 74.4%

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

   14. Taylor expanded in b around -inf 84.3%

       \[\leadsto -0.5 \cdot \left(\frac{c \cdot 4}{b - \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} + -1 \cdot b\right)}} \cdot 1\right) \]
   15. Step-by-step derivation

       1. mul-1-neg 84.3%

          \[\leadsto -0.5 \cdot \left(\frac{c \cdot 4}{b - \left(2 \cdot \frac{c \cdot a}{b} + \color{blue}{\left(-b\right)}\right)} \cdot 1\right) \]

       2. unsub-neg 84.3%

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

       3. *-commutative 84.3%

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

       4. associate-/l* 85.3%

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

   16. Simplified 85.3%

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

   if -3.4999999999999997e-95 < b < 7.39999999999999996e68

   1. Initial program 85.0%

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

   3. Simplified 85.0%

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

      1. fma-udef 85.0%

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

      2. associate-*r* 85.0%

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

      3. metadata-eval 85.0%

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

      4. distribute-rgt-neg-in 85.0%

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

      5. *-commutative 85.0%

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

      6. +-commutative 85.0%

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

      7. sub-neg 85.0%

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

      8. *-commutative 85.0%

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

      9. associate-*l* 85.0%

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

   5. Applied egg-rr 85.0%

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

   if 7.39999999999999996e68 < b

   1. Initial program 54.8%

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

   2. Taylor expanded in b around inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. unsub-neg 98.3%

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

   4. Simplified 98.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-95}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{elif}\;b \leq 7.4 \cdot 10^{+68}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-94}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{+20}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.4e-94)
   (* -0.5 (/ (* c 4.0) (+ b (- b (* (/ c (/ b a)) 2.0)))))
   (if (<= b 2.05e+20)
     (* -0.5 (/ (+ b (sqrt (* (* c a) -4.0))) a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4e-94) {
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	} else if (b <= 2.05e+20) {
		tmp = -0.5 * ((b + sqrt(((c * a) * -4.0))) / a);
	} else {
		tmp = (c / b) - (b / 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 <= (-3.4d-94)) then
        tmp = (-0.5d0) * ((c * 4.0d0) / (b + (b - ((c / (b / a)) * 2.0d0))))
    else if (b <= 2.05d+20) then
        tmp = (-0.5d0) * ((b + sqrt(((c * a) * (-4.0d0)))) / a)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4e-94) {
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	} else if (b <= 2.05e+20) {
		tmp = -0.5 * ((b + Math.sqrt(((c * a) * -4.0))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.4e-94:
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))))
	elif b <= 2.05e+20:
		tmp = -0.5 * ((b + math.sqrt(((c * a) * -4.0))) / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.4e-94)
		tmp = Float64(-0.5 * Float64(Float64(c * 4.0) / Float64(b + Float64(b - Float64(Float64(c / Float64(b / a)) * 2.0)))));
	elseif (b <= 2.05e+20)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(c * a) * -4.0))) / a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.4e-94)
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	elseif (b <= 2.05e+20)
		tmp = -0.5 * ((b + sqrt(((c * a) * -4.0))) / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.4e-94], N[(-0.5 * N[(N[(c * 4.0), $MachinePrecision] / N[(b + N[(b - N[(N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.05e+20], N[(-0.5 * N[(N[(b + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.3999999999999998e-94

   1. Initial program 18.5%

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

   3. Simplified 18.5%

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

      1. fma-udef 18.5%

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

      2. associate-*r* 18.5%

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

      3. metadata-eval 18.5%

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

      4. distribute-rgt-neg-in 18.5%

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

      5. *-commutative 18.5%

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

      6. +-commutative 18.5%

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

      7. sub-neg 18.5%

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

      8. *-commutative 18.5%

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

      9. associate-*l* 18.5%

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

   5. Applied egg-rr 18.5%

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

      1. flip-+ 17.8%

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

      2. add-sqr-sqrt 17.8%

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

   7. Applied egg-rr 17.8%

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

      1. associate--r- 49.3%

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

      2. +-inverses 65.6%

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

      3. *-commutative 65.6%

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

      4. associate-*l* 65.6%

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

      5. sub-neg 65.6%

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

      6. +-commutative 65.6%

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

      7. distribute-rgt-neg-in 65.6%

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

      8. distribute-rgt-neg-in 65.6%

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

      9. metadata-eval 65.6%

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

      10. fma-udef 65.6%

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

   9. Simplified 65.6%

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

       1. expm1-log1p-u 57.6%

          \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0 + c \cdot \left(4 \cdot a\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a}\right)\right)} \]

       2. expm1-udef 27.0%

          \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{0 + c \cdot \left(4 \cdot a\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a}\right)} - 1\right)} \]

       3. associate-/l/ 27.0%

          \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{0 + c \cdot \left(4 \cdot a\right)}{a \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}}\right)} - 1\right) \]

       4. +-lft-identity 27.0%

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

   11. Applied egg-rr 27.0%

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

       1. expm1-def 53.6%

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

       2. expm1-log1p 60.8%

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

       3. associate-*r* 60.8%

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

       4. *-commutative 60.8%

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

       5. times-frac 74.4%

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

       6. *-inverses 74.4%

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

   13. Simplified 74.4%

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

   14. Taylor expanded in b around -inf 84.3%

       \[\leadsto -0.5 \cdot \left(\frac{c \cdot 4}{b - \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} + -1 \cdot b\right)}} \cdot 1\right) \]
   15. Step-by-step derivation

       1. mul-1-neg 84.3%

          \[\leadsto -0.5 \cdot \left(\frac{c \cdot 4}{b - \left(2 \cdot \frac{c \cdot a}{b} + \color{blue}{\left(-b\right)}\right)} \cdot 1\right) \]

       2. unsub-neg 84.3%

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

       3. *-commutative 84.3%

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

       4. associate-/l* 85.3%

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

   16. Simplified 85.3%

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

   if -3.3999999999999998e-94 < b < 2.05e20

   1. Initial program 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in a around inf 67.5%

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

      1. *-commutative 67.5%

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

   6. Simplified 67.5%

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

   if 2.05e20 < b

   1. Initial program 62.8%

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

   2. Taylor expanded in b around inf 93.2%

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

      1. mul-1-neg 93.2%

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

      2. unsub-neg 93.2%

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

   4. Simplified 93.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-94}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{+20}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4: 67.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.16 \cdot 10^{-296}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.16e-296)
   (* -0.5 (/ (* c 4.0) (+ b (- b (* (/ c (/ b a)) 2.0)))))
   (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.16e-296) {
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	} else {
		tmp = (c / b) - (b / 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 <= 2.16d-296) then
        tmp = (-0.5d0) * ((c * 4.0d0) / (b + (b - ((c / (b / a)) * 2.0d0))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.16e-296) {
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.16e-296:
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.16e-296)
		tmp = Float64(-0.5 * Float64(Float64(c * 4.0) / Float64(b + Float64(b - Float64(Float64(c / Float64(b / a)) * 2.0)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.16e-296)
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.16e-296], N[(-0.5 * N[(N[(c * 4.0), $MachinePrecision] / N[(b + N[(b - N[(N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1600000000000001e-296

   1. Initial program 31.0%

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

   3. Simplified 31.0%

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

      1. fma-udef 31.0%

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

      2. associate-*r* 31.0%

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

      3. metadata-eval 31.0%

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

      4. distribute-rgt-neg-in 31.0%

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

      5. *-commutative 31.0%

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

      6. +-commutative 31.0%

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

      7. sub-neg 31.0%

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

      8. *-commutative 31.0%

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

      9. associate-*l* 31.0%

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

   5. Applied egg-rr 31.0%

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

      1. flip-+ 30.4%

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

      2. add-sqr-sqrt 30.5%

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

   7. Applied egg-rr 30.5%

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

      1. associate--r- 54.8%

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

      2. +-inverses 67.5%

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

      3. *-commutative 67.5%

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

      4. associate-*l* 67.5%

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

      5. sub-neg 67.5%

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

      6. +-commutative 67.5%

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

      7. distribute-rgt-neg-in 67.5%

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

      8. distribute-rgt-neg-in 67.5%

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

      9. metadata-eval 67.5%

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

      10. fma-udef 67.5%

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

   9. Simplified 67.5%

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

       1. expm1-log1p-u 55.7%

          \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0 + c \cdot \left(4 \cdot a\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a}\right)\right)} \]

       2. expm1-udef 24.2%

          \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{0 + c \cdot \left(4 \cdot a\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a}\right)} - 1\right)} \]

       3. associate-/l/ 23.4%

          \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{0 + c \cdot \left(4 \cdot a\right)}{a \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}}\right)} - 1\right) \]

       4. +-lft-identity 23.4%

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

   11. Applied egg-rr 23.4%

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

       1. expm1-def 50.5%

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

       2. expm1-log1p 60.7%

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

       3. associate-*r* 60.7%

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

       4. *-commutative 60.7%

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

       5. times-frac 76.1%

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

       6. *-inverses 76.1%

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

   13. Simplified 76.1%

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

   14. Taylor expanded in b around -inf 69.8%

       \[\leadsto -0.5 \cdot \left(\frac{c \cdot 4}{b - \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} + -1 \cdot b\right)}} \cdot 1\right) \]
   15. Step-by-step derivation

       1. mul-1-neg 69.8%

          \[\leadsto -0.5 \cdot \left(\frac{c \cdot 4}{b - \left(2 \cdot \frac{c \cdot a}{b} + \color{blue}{\left(-b\right)}\right)} \cdot 1\right) \]

       2. unsub-neg 69.8%

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

       3. *-commutative 69.8%

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

       4. associate-/l* 70.5%

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

   16. Simplified 70.5%

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

   if 2.1600000000000001e-296 < b

   1. Initial program 74.4%

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

   2. Taylor expanded in b around inf 67.0%

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

      1. mul-1-neg 67.0%

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

      2. unsub-neg 67.0%

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

   4. Simplified 67.0%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.16 \cdot 10^{-296}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5: 67.4% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 30.5%

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

   2. Taylor expanded in b around -inf 70.8%

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

      1. associate-*r/ 70.8%

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

      2. neg-mul-1 70.8%

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

   4. Simplified 70.8%

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

   if -4.999999999999985e-310 < b

   1. Initial program 74.6%

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

   2. Taylor expanded in b around inf 66.4%

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

      1. mul-1-neg 66.4%

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

      2. unsub-neg 66.4%

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

   4. Simplified 66.4%

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

4. Final simplification 68.7%

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

Alternative 6: 42.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -5.5e+90) (/ c b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e+90) {
		tmp = c / b;
	} else {
		tmp = -b / 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 <= (-5.5d+90)) then
        tmp = c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e+90) {
		tmp = c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.5e+90:
		tmp = c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.5e+90)
		tmp = Float64(c / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.5e+90)
		tmp = c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.5e+90], N[(c / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.49999999999999999e90

   1. Initial program 12.0%

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

      1. clear-num 12.0%

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

      2. associate-/r/ 12.0%

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

      3. associate-/r* 12.0%

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

      4. metadata-eval 12.0%

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

      5. add-sqr-sqrt 5.4%

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

      6. cancel-sign-sub-inv 5.4%

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

      7. add-sqr-sqrt 12.0%

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

      8. sqrt-unprod 4.4%

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

      9. sqr-neg 4.4%

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

      10. sqrt-prod 0.0%

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

      11. add-sqr-sqrt 2.2%

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

   3. Applied egg-rr 2.2%

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

   4. Taylor expanded in a around 0 35.1%

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

   if -5.49999999999999999e90 < b

   1. Initial program 65.1%

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

   2. Taylor expanded in b around inf 44.1%

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

      1. associate-*r/ 44.1%

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

      2. mul-1-neg 44.1%

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

   4. Simplified 44.1%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7: 67.3% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.04 \cdot 10^{-303}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.04e-303) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.04e-303) {
		tmp = -c / b;
	} else {
		tmp = -b / 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 <= (-1.04d-303)) then
        tmp = -c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.04e-303) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.04e-303:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.04e-303)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.04e-303)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.04e-303], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.03999999999999999e-303

   1. Initial program 29.9%

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

   2. Taylor expanded in b around -inf 71.3%

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

      1. associate-*r/ 71.3%

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

      2. neg-mul-1 71.3%

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

   4. Simplified 71.3%

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

   if -1.03999999999999999e-303 < b

   1. Initial program 74.8%

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

   2. Taylor expanded in b around inf 65.9%

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

      1. associate-*r/ 65.9%

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

      2. mul-1-neg 65.9%

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

   4. Simplified 65.9%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.04 \cdot 10^{-303}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8: 2.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

   1. clear-num 51.9%

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

   2. associate-/r/ 51.9%

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

   3. associate-/r* 51.9%

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

   4. metadata-eval 51.9%

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

   5. add-sqr-sqrt 50.1%

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

   6. cancel-sign-sub-inv 50.1%

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

   7. add-sqr-sqrt 15.5%

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

   8. sqrt-unprod 29.4%

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

   9. sqr-neg 29.4%

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

   10. sqrt-prod 17.9%

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

   11. add-sqr-sqrt 31.2%

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

3. Applied egg-rr 30.4%

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

4. Taylor expanded in b around -inf 2.6%

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

5. Final simplification 2.6%

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

Alternative 9: 10.8% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

   1. clear-num 51.9%

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

   2. associate-/r/ 51.9%

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

   3. associate-/r* 51.9%

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

   4. metadata-eval 51.9%

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

   5. add-sqr-sqrt 50.1%

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

   6. cancel-sign-sub-inv 50.1%

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

   7. add-sqr-sqrt 15.5%

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

   8. sqrt-unprod 29.4%

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

   9. sqr-neg 29.4%

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

   10. sqrt-prod 17.9%

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

   11. add-sqr-sqrt 31.2%

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

3. Applied egg-rr 30.4%

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

4. Taylor expanded in a around 0 11.1%

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

5. Final simplification 11.1%

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

Developer target: 71.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 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 = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-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 = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-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 = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-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 = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a))));
	else
		tmp = 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 = c / (a * ((-b + t_0) / (2.0 * a)));
	else
		tmp = (-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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023222 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

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

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