The quadratic formula (r2)

Percentage Accurate: 53.1% → 91.2%

Time: 13.6s

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: 53.1% 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: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+146}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{elif}\;b \leq -1.1 \cdot 10^{-308}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - t_0}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+120}:\\ \;\;\;\;-0.5 \cdot \frac{b + t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* c 4.0) a)))))
   (if (<= b -2e+146)
     (* -0.5 (/ (* c 4.0) (+ b (- b (* (/ c (/ b a)) 2.0)))))
     (if (<= b -1.1e-308)
       (* -0.5 (/ (* c 4.0) (- b t_0)))
       (if (<= b 2.6e+120) (* -0.5 (/ (+ b t_0) a)) (/ (- 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 <= -2e+146) {
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	} else if (b <= -1.1e-308) {
		tmp = -0.5 * ((c * 4.0) / (b - t_0));
	} else if (b <= 2.6e+120) {
		tmp = -0.5 * ((b + t_0) / a);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((c * 4.0d0) * a)))
    if (b <= (-2d+146)) then
        tmp = (-0.5d0) * ((c * 4.0d0) / (b + (b - ((c / (b / a)) * 2.0d0))))
    else if (b <= (-1.1d-308)) then
        tmp = (-0.5d0) * ((c * 4.0d0) / (b - t_0))
    else if (b <= 2.6d+120) then
        tmp = (-0.5d0) * ((b + t_0) / a)
    else
        tmp = -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 <= -2e+146) {
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	} else if (b <= -1.1e-308) {
		tmp = -0.5 * ((c * 4.0) / (b - t_0));
	} else if (b <= 2.6e+120) {
		tmp = -0.5 * ((b + t_0) / a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((c * 4.0) * a)))
	tmp = 0
	if b <= -2e+146:
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))))
	elif b <= -1.1e-308:
		tmp = -0.5 * ((c * 4.0) / (b - t_0))
	elif b <= 2.6e+120:
		tmp = -0.5 * ((b + t_0) / a)
	else:
		tmp = -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 <= -2e+146)
		tmp = Float64(-0.5 * Float64(Float64(c * 4.0) / Float64(b + Float64(b - Float64(Float64(c / Float64(b / a)) * 2.0)))));
	elseif (b <= -1.1e-308)
		tmp = Float64(-0.5 * Float64(Float64(c * 4.0) / Float64(b - t_0)));
	elseif (b <= 2.6e+120)
		tmp = Float64(-0.5 * Float64(Float64(b + t_0) / a));
	else
		tmp = Float64(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 <= -2e+146)
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	elseif (b <= -1.1e-308)
		tmp = -0.5 * ((c * 4.0) / (b - t_0));
	elseif (b <= 2.6e+120)
		tmp = -0.5 * ((b + t_0) / a);
	else
		tmp = -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, -2e+146], 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, -1.1e-308], N[(-0.5 * N[(N[(c * 4.0), $MachinePrecision] / N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.6e+120], N[(-0.5 * N[(N[(b + t$95$0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.99999999999999987e146

   1. Initial program 2.0%

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

   3. Simplified 2.0%

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

      1. fma-udef 2.0%

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

      2. associate-*r* 2.0%

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

      3. metadata-eval 2.0%

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

      4. distribute-rgt-neg-in 2.0%

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

      5. *-commutative 2.0%

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

      6. +-commutative 2.0%

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

      7. sub-neg 2.0%

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

      8. *-commutative 2.0%

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

      9. associate-*l* 2.0%

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

   5. Applied egg-rr 2.0%

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

      1. flip-+ 0.3%

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

      2. add-sqr-sqrt 0.3%

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

   7. Applied egg-rr 0.3%

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

      1. associate--r- 5.2%

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

      2. +-inverses 44.6%

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

      3. sub-neg 44.6%

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

      4. +-commutative 44.6%

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

      5. distribute-rgt-neg-in 44.6%

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

      6. fma-udef 44.6%

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

      7. distribute-rgt-neg-in 44.6%

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

      8. metadata-eval 44.6%

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

   9. Simplified 44.6%

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

       1. expm1-log1p-u 44.6%

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

       2. expm1-udef 39.7%

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

       3. +-lft-identity 39.7%

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

   11. Applied egg-rr 39.7%

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

       1. expm1-def 44.6%

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

       2. expm1-log1p 44.6%

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

       3. associate-/l/ 44.6%

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

       4. times-frac 44.8%

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

       5. *-inverses 44.8%

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

   13. Simplified 44.8%

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

   14. Taylor expanded in b around -inf 94.9%

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

       1. mul-1-neg 94.9%

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

       2. unsub-neg 94.9%

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

       3. *-commutative 94.9%

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

       4. associate-/l* 95.1%

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

   16. Simplified 95.1%

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

   if -1.99999999999999987e146 < b < -1.1000000000000001e-308

   1. Initial program 42.6%

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

   3. Simplified 42.6%

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

      1. fma-udef 42.6%

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

      2. associate-*r* 42.6%

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

      3. metadata-eval 42.6%

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

      4. distribute-rgt-neg-in 42.6%

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

      5. *-commutative 42.6%

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

      6. +-commutative 42.6%

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

      7. sub-neg 42.6%

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

      8. *-commutative 42.6%

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

      9. associate-*l* 42.6%

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

   5. Applied egg-rr 42.6%

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

      1. flip-+ 42.3%

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

      2. add-sqr-sqrt 42.1%

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

   7. Applied egg-rr 42.1%

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

      1. associate--r- 68.4%

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

      2. +-inverses 68.4%

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

      3. sub-neg 68.4%

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

      4. +-commutative 68.4%

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

      5. distribute-rgt-neg-in 68.4%

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

      6. fma-udef 68.4%

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

      7. distribute-rgt-neg-in 68.4%

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

      8. metadata-eval 68.4%

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

   9. Simplified 68.4%

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

       1. expm1-log1p-u 47.8%

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

       2. expm1-udef 26.8%

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

       3. +-lft-identity 26.8%

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

   11. Applied egg-rr 26.8%

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

       1. expm1-def 47.8%

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

       2. expm1-log1p 68.4%

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

       3. associate-/l/ 64.8%

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

       4. times-frac 81.8%

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

       5. *-inverses 81.8%

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

   13. Simplified 81.8%

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

       1. fma-udef 42.6%

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

       2. associate-*r* 42.6%

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

       3. metadata-eval 42.6%

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

       4. distribute-rgt-neg-in 42.6%

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

       5. *-commutative 42.6%

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

       6. +-commutative 42.6%

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

       7. sub-neg 42.6%

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

       8. *-commutative 42.6%

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

       9. associate-*l* 42.6%

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

   15. Applied egg-rr 81.8%

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

   if -1.1000000000000001e-308 < b < 2.5999999999999999e120

   1. Initial program 81.5%

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

   3. Simplified 81.5%

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

      1. fma-udef 81.5%

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

      2. associate-*r* 81.5%

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

      3. metadata-eval 81.5%

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

      4. distribute-rgt-neg-in 81.5%

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

      5. *-commutative 81.5%

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

      6. +-commutative 81.5%

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

      7. sub-neg 81.5%

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

      8. *-commutative 81.5%

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

      9. associate-*l* 81.5%

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

   5. Applied egg-rr 81.5%

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

   if 2.5999999999999999e120 < b

   1. Initial program 52.1%

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

   2. Taylor expanded in b around inf 96.4%

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

      1. associate-*r/ 96.4%

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

      2. mul-1-neg 96.4%

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

   4. Simplified 96.4%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+146}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{elif}\;b \leq -1.1 \cdot 10^{-308}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+120}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(c \cdot a\right) \cdot -4}\\ \mathbf{if}\;b \leq -2.2 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-121}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - t_0}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-17} \lor \neg \left(b \leq 2.2\right):\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + t_0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* c a) -4.0))))
   (if (<= b -2.2e-59)
     (* -0.5 (/ (* c 4.0) (+ b (- b (* (/ c (/ b a)) 2.0)))))
     (if (<= b 1.3e-121)
       (* -0.5 (/ (* c 4.0) (- b t_0)))
       (if (or (<= b 1.7e-17) (not (<= b 2.2)))
         (- (/ c b) (/ b a))
         (* -0.5 (/ (+ b t_0) a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((c * a) * -4.0));
	double tmp;
	if (b <= -2.2e-59) {
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	} else if (b <= 1.3e-121) {
		tmp = -0.5 * ((c * 4.0) / (b - t_0));
	} else if ((b <= 1.7e-17) || !(b <= 2.2)) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -0.5 * ((b + t_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(((c * a) * (-4.0d0)))
    if (b <= (-2.2d-59)) then
        tmp = (-0.5d0) * ((c * 4.0d0) / (b + (b - ((c / (b / a)) * 2.0d0))))
    else if (b <= 1.3d-121) then
        tmp = (-0.5d0) * ((c * 4.0d0) / (b - t_0))
    else if ((b <= 1.7d-17) .or. (.not. (b <= 2.2d0))) then
        tmp = (c / b) - (b / a)
    else
        tmp = (-0.5d0) * ((b + t_0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((c * a) * -4.0));
	double tmp;
	if (b <= -2.2e-59) {
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	} else if (b <= 1.3e-121) {
		tmp = -0.5 * ((c * 4.0) / (b - t_0));
	} else if ((b <= 1.7e-17) || !(b <= 2.2)) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -0.5 * ((b + t_0) / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((c * a) * -4.0))
	tmp = 0
	if b <= -2.2e-59:
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))))
	elif b <= 1.3e-121:
		tmp = -0.5 * ((c * 4.0) / (b - t_0))
	elif (b <= 1.7e-17) or not (b <= 2.2):
		tmp = (c / b) - (b / a)
	else:
		tmp = -0.5 * ((b + t_0) / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(c * a) * -4.0))
	tmp = 0.0
	if (b <= -2.2e-59)
		tmp = Float64(-0.5 * Float64(Float64(c * 4.0) / Float64(b + Float64(b - Float64(Float64(c / Float64(b / a)) * 2.0)))));
	elseif (b <= 1.3e-121)
		tmp = Float64(-0.5 * Float64(Float64(c * 4.0) / Float64(b - t_0)));
	elseif ((b <= 1.7e-17) || !(b <= 2.2))
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(-0.5 * Float64(Float64(b + t_0) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((c * a) * -4.0));
	tmp = 0.0;
	if (b <= -2.2e-59)
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	elseif (b <= 1.3e-121)
		tmp = -0.5 * ((c * 4.0) / (b - t_0));
	elseif ((b <= 1.7e-17) || ~((b <= 2.2)))
		tmp = (c / b) - (b / a);
	else
		tmp = -0.5 * ((b + t_0) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.2e-59], 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, 1.3e-121], N[(-0.5 * N[(N[(c * 4.0), $MachinePrecision] / N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 1.7e-17], N[Not[LessEqual[b, 2.2]], $MachinePrecision]], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(b + t$95$0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.1999999999999999e-59

   1. Initial program 18.3%

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

   3. Simplified 18.3%

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

      1. fma-udef 18.3%

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

      2. associate-*r* 18.3%

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

      3. metadata-eval 18.3%

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

      4. distribute-rgt-neg-in 18.3%

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

      5. *-commutative 18.3%

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

      6. +-commutative 18.3%

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

      7. sub-neg 18.3%

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

      8. *-commutative 18.3%

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

      9. associate-*l* 18.3%

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

   5. Applied egg-rr 18.3%

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

      1. flip-+ 17.3%

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

      2. add-sqr-sqrt 17.3%

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

   7. Applied egg-rr 17.3%

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

      1. associate--r- 44.2%

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

      2. +-inverses 61.4%

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

      3. sub-neg 61.4%

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

      4. +-commutative 61.4%

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

      5. distribute-rgt-neg-in 61.4%

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

      6. fma-udef 61.4%

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

      7. distribute-rgt-neg-in 61.4%

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

      8. metadata-eval 61.4%

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

   9. Simplified 61.4%

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

       1. expm1-log1p-u 53.0%

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

       2. expm1-udef 36.4%

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

       3. +-lft-identity 36.4%

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

   11. Applied egg-rr 36.4%

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

       1. expm1-def 53.0%

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

       2. expm1-log1p 61.4%

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

       3. associate-/l/ 62.8%

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

       4. times-frac 69.1%

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

       5. *-inverses 69.1%

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

   13. Simplified 69.1%

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

   14. Taylor expanded in b around -inf 84.0%

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

       1. mul-1-neg 84.0%

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

       2. unsub-neg 84.0%

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

       3. *-commutative 84.0%

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

       4. associate-/l* 84.2%

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

   16. Simplified 84.2%

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

   if -2.1999999999999999e-59 < b < 1.29999999999999993e-121

   1. Initial program 65.5%

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

   3. Simplified 65.5%

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

      1. fma-udef 65.5%

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

      2. associate-*r* 65.5%

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

      3. metadata-eval 65.5%

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

      4. distribute-rgt-neg-in 65.5%

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

      5. *-commutative 65.5%

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

      6. +-commutative 65.5%

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

      7. sub-neg 65.5%

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

      8. *-commutative 65.5%

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

      9. associate-*l* 65.5%

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

   5. Applied egg-rr 65.5%

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

      1. flip-+ 64.5%

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

      2. add-sqr-sqrt 64.3%

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

   7. Applied egg-rr 64.3%

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

      1. associate--r- 65.7%

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

      2. +-inverses 65.7%

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

      3. sub-neg 65.7%

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

      4. +-commutative 65.7%

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

      5. distribute-rgt-neg-in 65.7%

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

      6. fma-udef 65.7%

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

      7. distribute-rgt-neg-in 65.7%

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

      8. metadata-eval 65.7%

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

   9. Simplified 65.7%

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

       1. expm1-log1p-u 43.3%

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

       2. expm1-udef 23.4%

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

       3. +-lft-identity 23.4%

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

   11. Applied egg-rr 23.4%

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

       1. expm1-def 43.3%

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

       2. expm1-log1p 65.7%

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

       3. associate-/l/ 56.0%

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

       4. times-frac 72.7%

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

       5. *-inverses 72.7%

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

   13. Simplified 72.7%

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

   14. Taylor expanded in a around inf 66.8%

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

       1. *-commutative 65.5%

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

   16. Simplified 66.8%

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

   if 1.29999999999999993e-121 < b < 1.6999999999999999e-17 or 2.2000000000000002 < b

   1. Initial program 68.3%

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

   2. Taylor expanded in b around inf 84.6%

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

      1. mul-1-neg 84.6%

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

      2. unsub-neg 84.6%

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

   4. Simplified 84.6%

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

   if 1.6999999999999999e-17 < b < 2.2000000000000002

   1. Initial program 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in a around inf 83.3%

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

      1. *-commutative 83.3%

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

   6. Simplified 83.3%

      \[\leadsto -0.5 \cdot \frac{b + \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -4}}}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-121}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b - \sqrt{\left(c \cdot a\right) \cdot -4}}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-17} \lor \neg \left(b \leq 2.2\right):\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \end{array} \]

Alternative 3: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-109}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+119}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.1e-109)
   (* -0.5 (/ (* c 4.0) (+ b (- b (* (/ c (/ b a)) 2.0)))))
   (if (<= b 5e+119)
     (* -0.5 (/ (+ b (sqrt (- (* b b) (* (* c 4.0) a)))) a))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.1e-109) {
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	} else if (b <= 5e+119) {
		tmp = -0.5 * ((b + sqrt(((b * b) - ((c * 4.0) * a)))) / a);
	} 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 <= (-3.1d-109)) then
        tmp = (-0.5d0) * ((c * 4.0d0) / (b + (b - ((c / (b / a)) * 2.0d0))))
    else if (b <= 5d+119) then
        tmp = (-0.5d0) * ((b + sqrt(((b * b) - ((c * 4.0d0) * a)))) / a)
    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 <= -3.1e-109) {
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	} else if (b <= 5e+119) {
		tmp = -0.5 * ((b + Math.sqrt(((b * b) - ((c * 4.0) * a)))) / a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.1e-109:
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))))
	elif b <= 5e+119:
		tmp = -0.5 * ((b + math.sqrt(((b * b) - ((c * 4.0) * a)))) / a)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.1e-109)
		tmp = Float64(-0.5 * Float64(Float64(c * 4.0) / Float64(b + Float64(b - Float64(Float64(c / Float64(b / a)) * 2.0)))));
	elseif (b <= 5e+119)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(Float64(c * 4.0) * a)))) / a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.1e-109)
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	elseif (b <= 5e+119)
		tmp = -0.5 * ((b + sqrt(((b * b) - ((c * 4.0) * a)))) / a);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.1e-109], 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, 5e+119], 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[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.1e-109

   1. Initial program 20.1%

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

   3. Simplified 20.1%

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

      1. fma-udef 20.1%

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

      2. associate-*r* 20.1%

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

      3. metadata-eval 20.1%

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

      4. distribute-rgt-neg-in 20.1%

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

      5. *-commutative 20.1%

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

      6. +-commutative 20.1%

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

      7. sub-neg 20.1%

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

      8. *-commutative 20.1%

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

      9. associate-*l* 20.1%

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

   5. Applied egg-rr 20.1%

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

      1. flip-+ 19.1%

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

      2. add-sqr-sqrt 19.1%

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

   7. Applied egg-rr 19.1%

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

      1. associate--r- 44.5%

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

      2. +-inverses 60.3%

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

      3. sub-neg 60.3%

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

      4. +-commutative 60.3%

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

      5. distribute-rgt-neg-in 60.3%

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

      6. fma-udef 60.3%

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

      7. distribute-rgt-neg-in 60.3%

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

      8. metadata-eval 60.3%

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

   9. Simplified 60.3%

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

       1. expm1-log1p-u 51.5%

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

       2. expm1-udef 35.2%

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

       3. +-lft-identity 35.2%

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

   11. Applied egg-rr 35.2%

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

       1. expm1-def 51.5%

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

       2. expm1-log1p 60.3%

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

       3. associate-/l/ 61.5%

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

       4. times-frac 69.7%

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

       5. *-inverses 69.7%

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

   13. Simplified 69.7%

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

   14. Taylor expanded in b around -inf 80.3%

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

       1. mul-1-neg 80.3%

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

       2. unsub-neg 80.3%

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

       3. *-commutative 80.3%

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

       4. associate-/l* 80.5%

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

   16. Simplified 80.5%

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

   if -3.1e-109 < b < 4.9999999999999999e119

   1. Initial program 76.7%

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

   3. Simplified 76.7%

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

      1. fma-udef 76.7%

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

      2. associate-*r* 76.7%

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

      3. metadata-eval 76.7%

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

      4. distribute-rgt-neg-in 76.7%

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

      5. *-commutative 76.7%

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

      6. +-commutative 76.7%

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

      7. sub-neg 76.7%

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

      8. *-commutative 76.7%

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

      9. associate-*l* 76.7%

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

   5. Applied egg-rr 76.7%

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

   if 4.9999999999999999e119 < b

   1. Initial program 52.1%

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

   2. Taylor expanded in b around inf 96.4%

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

      1. associate-*r/ 96.4%

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

      2. mul-1-neg 96.4%

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

   4. Simplified 96.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-109}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+119}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-121}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{elif}\;b \leq 2.2:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e-121)
   (* -0.5 (/ (* c 4.0) (+ b (- b (* (/ c (/ b a)) 2.0)))))
   (if (<= b 2.2)
     (* -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 <= -7e-121) {
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	} else if (b <= 2.2) {
		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 <= (-7d-121)) then
        tmp = (-0.5d0) * ((c * 4.0d0) / (b + (b - ((c / (b / a)) * 2.0d0))))
    else if (b <= 2.2d0) 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 <= -7e-121) {
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	} else if (b <= 2.2) {
		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 <= -7e-121:
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))))
	elif b <= 2.2:
		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 <= -7e-121)
		tmp = Float64(-0.5 * Float64(Float64(c * 4.0) / Float64(b + Float64(b - Float64(Float64(c / Float64(b / a)) * 2.0)))));
	elseif (b <= 2.2)
		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 <= -7e-121)
		tmp = -0.5 * ((c * 4.0) / (b + (b - ((c / (b / a)) * 2.0))));
	elseif (b <= 2.2)
		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, -7e-121], 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, 2.2], 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 < -6.99999999999999985e-121

   1. Initial program 20.1%

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

   3. Simplified 20.1%

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

      1. fma-udef 20.1%

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

      2. associate-*r* 20.1%

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

      3. metadata-eval 20.1%

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

      4. distribute-rgt-neg-in 20.1%

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

      5. *-commutative 20.1%

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

      6. +-commutative 20.1%

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

      7. sub-neg 20.1%

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

      8. *-commutative 20.1%

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

      9. associate-*l* 20.1%

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

   5. Applied egg-rr 20.1%

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

      1. flip-+ 19.1%

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

      2. add-sqr-sqrt 19.1%

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

   7. Applied egg-rr 19.1%

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

      1. associate--r- 44.5%

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

      2. +-inverses 60.3%

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

      3. sub-neg 60.3%

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

      4. +-commutative 60.3%

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

      5. distribute-rgt-neg-in 60.3%

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

      6. fma-udef 60.3%

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

      7. distribute-rgt-neg-in 60.3%

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

      8. metadata-eval 60.3%

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

   9. Simplified 60.3%

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

       1. expm1-log1p-u 51.5%

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

       2. expm1-udef 35.2%

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

       3. +-lft-identity 35.2%

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

   11. Applied egg-rr 35.2%

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

       1. expm1-def 51.5%

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

       2. expm1-log1p 60.3%

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

       3. associate-/l/ 61.5%

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

       4. times-frac 69.7%

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

       5. *-inverses 69.7%

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

   13. Simplified 69.7%

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

   14. Taylor expanded in b around -inf 80.3%

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

       1. mul-1-neg 80.3%

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

       2. unsub-neg 80.3%

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

       3. *-commutative 80.3%

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

       4. associate-/l* 80.5%

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

   16. Simplified 80.5%

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

   if -6.99999999999999985e-121 < b < 2.2000000000000002

   1. Initial program 73.0%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in a around inf 64.4%

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

      1. *-commutative 64.4%

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

   6. Simplified 64.4%

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

   if 2.2000000000000002 < b

   1. Initial program 64.6%

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

   2. Taylor expanded in b around inf 89.1%

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

      1. mul-1-neg 89.1%

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

      2. unsub-neg 89.1%

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

   4. Simplified 89.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-121}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{elif}\;b \leq 2.2:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5: 67.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-309)
   (* -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 <= -1e-309) {
		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 <= (-1d-309)) 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 <= -1e-309) {
		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 <= -1e-309:
		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 <= -1e-309)
		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 <= -1e-309)
		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, -1e-309], 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 < -1.000000000000002e-309

   1. Initial program 30.2%

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

   3. Simplified 30.2%

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

      1. fma-udef 30.2%

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

      2. associate-*r* 30.2%

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

      3. metadata-eval 30.2%

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

      4. distribute-rgt-neg-in 30.2%

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

      5. *-commutative 30.2%

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

      6. +-commutative 30.2%

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

      7. sub-neg 30.2%

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

      8. *-commutative 30.2%

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

      9. associate-*l* 30.2%

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

   5. Applied egg-rr 30.2%

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

      1. flip-+ 29.4%

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

      2. add-sqr-sqrt 29.3%

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

   7. Applied egg-rr 29.3%

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

      1. associate--r- 49.0%

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

      2. +-inverses 61.1%

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

      3. sub-neg 61.1%

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

      4. +-commutative 61.1%

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

      5. distribute-rgt-neg-in 61.1%

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

      6. fma-udef 61.1%

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

      7. distribute-rgt-neg-in 61.1%

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

      8. metadata-eval 61.1%

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

   9. Simplified 61.1%

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

       1. expm1-log1p-u 46.8%

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

       2. expm1-udef 30.8%

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

       3. +-lft-identity 30.8%

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

   11. Applied egg-rr 30.8%

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

       1. expm1-def 46.8%

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

       2. expm1-log1p 61.1%

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

       3. associate-/l/ 58.6%

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

       4. times-frac 70.4%

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

       5. *-inverses 70.4%

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

   13. Simplified 70.4%

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

   14. Taylor expanded in b around -inf 68.0%

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

       1. mul-1-neg 68.0%

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

       2. unsub-neg 68.0%

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

       3. *-commutative 68.0%

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

       4. associate-/l* 68.3%

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

   16. Simplified 68.3%

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

   if -1.000000000000002e-309 < b

   1. Initial program 70.3%

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

   2. Taylor expanded in b around inf 61.4%

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

      1. mul-1-neg 61.4%

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

      2. unsub-neg 61.4%

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

   4. Simplified 61.4%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot 4}{b + \left(b - \frac{c}{\frac{b}{a}} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6: 67.2% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-309) (/ (- c) b) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-309) {
		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 <= (-1d-309)) 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 <= -1e-309) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-309:
		tmp = -c / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-309)
		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 <= -1e-309)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-309], N[((-c) / b), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.000000000000002e-309

   1. Initial program 30.2%

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

   2. Taylor expanded in b around -inf 68.0%

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

      1. associate-*r/ 68.0%

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

      2. neg-mul-1 68.0%

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

   4. Simplified 68.0%

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

   if -1.000000000000002e-309 < b

   1. Initial program 70.3%

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

   2. Taylor expanded in b around inf 61.4%

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

      1. mul-1-neg 61.4%

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

      2. unsub-neg 61.4%

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

   4. Simplified 61.4%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 7: 42.8% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -75000000:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -75000000.0) (/ c b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -75000000.0) {
		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 <= (-75000000.0d0)) 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 <= -75000000.0) {
		tmp = c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -75000000.0:
		tmp = c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -75000000.0)
		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 <= -75000000.0)
		tmp = c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -75000000.0], N[(c / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.5e7

   1. Initial program 17.1%

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

   2. Taylor expanded in b around inf 2.1%

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

      1. mul-1-neg 2.1%

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

      2. unsub-neg 2.1%

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

   4. Simplified 2.1%

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

   5. Taylor expanded in c around inf 29.6%

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

   if -7.5e7 < b

   1. Initial program 64.4%

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

   2. Taylor expanded in b around inf 45.0%

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

      1. associate-*r/ 45.0%

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

      2. mul-1-neg 45.0%

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

   4. Simplified 45.0%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -75000000:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8: 67.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-309) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-309) {
		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 <= (-1d-309)) 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 <= -1e-309) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-309:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-309)
		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 <= -1e-309)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-309], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.000000000000002e-309

   1. Initial program 30.2%

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

   2. Taylor expanded in b around -inf 68.0%

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

      1. associate-*r/ 68.0%

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

      2. neg-mul-1 68.0%

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

   4. Simplified 68.0%

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

   if -1.000000000000002e-309 < b

   1. Initial program 70.3%

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

   2. Taylor expanded in b around inf 61.3%

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

      1. associate-*r/ 61.3%

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

      2. mul-1-neg 61.3%

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

   4. Simplified 61.3%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 9: 10.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.9%

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

2. Taylor expanded in b around inf 32.7%

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

   1. mul-1-neg 32.7%

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

   2. unsub-neg 32.7%

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

4. Simplified 32.7%

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

5. Taylor expanded in c around inf 10.8%

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

6. Final simplification 10.8%

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

Developer target: 71.5% 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 2023224 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))