quadm (p42, negative)

Percentage Accurate: 52.0% → 86.1%

Time: 17.3s

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: 52.0% 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: 86.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-c}{b}\\ t_1 := c \cdot \left(a \cdot 4\right)\\ t_2 := \sqrt{b \cdot b - t_1}\\ t_3 := \sqrt{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{if}\;b \leq -2.2 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-27}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{t_1}{b - t_2}}{a}\\ \mathbf{elif}\;b \leq -1.55 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-182}:\\ \;\;\;\;-0.5 \cdot \left(t_3 \cdot \frac{t_3}{a}\right)\\ \mathbf{elif}\;b \leq 10^{+128}:\\ \;\;\;\;-0.5 \cdot \frac{b + t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- c) b))
        (t_1 (* c (* a 4.0)))
        (t_2 (sqrt (- (* b b) t_1)))
        (t_3 (sqrt (+ b (hypot b (sqrt (* a (* c -4.0))))))))
   (if (<= b -2.2e+74)
     t_0
     (if (<= b -1.05e-27)
       (* -0.5 (/ (/ t_1 (- b t_2)) a))
       (if (<= b -1.55e-106)
         t_0
         (if (<= b 4.2e-182)
           (* -0.5 (* t_3 (/ t_3 a)))
           (if (<= b 1e+128) (* -0.5 (/ (+ b t_2) a)) (/ (- b) a))))))))

C

double code(double a, double b, double c) {
	double t_0 = -c / b;
	double t_1 = c * (a * 4.0);
	double t_2 = sqrt(((b * b) - t_1));
	double t_3 = sqrt((b + hypot(b, sqrt((a * (c * -4.0))))));
	double tmp;
	if (b <= -2.2e+74) {
		tmp = t_0;
	} else if (b <= -1.05e-27) {
		tmp = -0.5 * ((t_1 / (b - t_2)) / a);
	} else if (b <= -1.55e-106) {
		tmp = t_0;
	} else if (b <= 4.2e-182) {
		tmp = -0.5 * (t_3 * (t_3 / a));
	} else if (b <= 1e+128) {
		tmp = -0.5 * ((b + t_2) / a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = -c / b;
	double t_1 = c * (a * 4.0);
	double t_2 = Math.sqrt(((b * b) - t_1));
	double t_3 = Math.sqrt((b + Math.hypot(b, Math.sqrt((a * (c * -4.0))))));
	double tmp;
	if (b <= -2.2e+74) {
		tmp = t_0;
	} else if (b <= -1.05e-27) {
		tmp = -0.5 * ((t_1 / (b - t_2)) / a);
	} else if (b <= -1.55e-106) {
		tmp = t_0;
	} else if (b <= 4.2e-182) {
		tmp = -0.5 * (t_3 * (t_3 / a));
	} else if (b <= 1e+128) {
		tmp = -0.5 * ((b + t_2) / a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -c / b
	t_1 = c * (a * 4.0)
	t_2 = math.sqrt(((b * b) - t_1))
	t_3 = math.sqrt((b + math.hypot(b, math.sqrt((a * (c * -4.0))))))
	tmp = 0
	if b <= -2.2e+74:
		tmp = t_0
	elif b <= -1.05e-27:
		tmp = -0.5 * ((t_1 / (b - t_2)) / a)
	elif b <= -1.55e-106:
		tmp = t_0
	elif b <= 4.2e-182:
		tmp = -0.5 * (t_3 * (t_3 / a))
	elif b <= 1e+128:
		tmp = -0.5 * ((b + t_2) / a)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-c) / b)
	t_1 = Float64(c * Float64(a * 4.0))
	t_2 = sqrt(Float64(Float64(b * b) - t_1))
	t_3 = sqrt(Float64(b + hypot(b, sqrt(Float64(a * Float64(c * -4.0))))))
	tmp = 0.0
	if (b <= -2.2e+74)
		tmp = t_0;
	elseif (b <= -1.05e-27)
		tmp = Float64(-0.5 * Float64(Float64(t_1 / Float64(b - t_2)) / a));
	elseif (b <= -1.55e-106)
		tmp = t_0;
	elseif (b <= 4.2e-182)
		tmp = Float64(-0.5 * Float64(t_3 * Float64(t_3 / a)));
	elseif (b <= 1e+128)
		tmp = Float64(-0.5 * Float64(Float64(b + t_2) / a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -c / b;
	t_1 = c * (a * 4.0);
	t_2 = sqrt(((b * b) - t_1));
	t_3 = sqrt((b + hypot(b, sqrt((a * (c * -4.0))))));
	tmp = 0.0;
	if (b <= -2.2e+74)
		tmp = t_0;
	elseif (b <= -1.05e-27)
		tmp = -0.5 * ((t_1 / (b - t_2)) / a);
	elseif (b <= -1.55e-106)
		tmp = t_0;
	elseif (b <= 4.2e-182)
		tmp = -0.5 * (t_3 * (t_3 / a));
	elseif (b <= 1e+128)
		tmp = -0.5 * ((b + t_2) / a);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-c) / b), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.2e+74], t$95$0, If[LessEqual[b, -1.05e-27], N[(-0.5 * N[(N[(t$95$1 / N[(b - t$95$2), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.55e-106], t$95$0, If[LessEqual[b, 4.2e-182], N[(-0.5 * N[(t$95$3 * N[(t$95$3 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+128], N[(-0.5 * N[(N[(b + t$95$2), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -2.2000000000000001e74 or -1.05000000000000008e-27 < b < -1.54999999999999993e-106

   1. Initial program 8.4%

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

      1. *-commutative 8.4%

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

      2. sqr-neg 8.4%

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

      3. *-commutative 8.4%

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

      4. sqr-neg 8.4%

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

      5. associate-*r* 8.4%

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

      6. *-commutative 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in b around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. distribute-neg-frac 92.9%

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

   6. Simplified 92.9%

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

   if -2.2000000000000001e74 < b < -1.05000000000000008e-27

   1. Initial program 43.2%

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

      1. sub-neg 43.2%

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

      2. distribute-neg-out 43.2%

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

      3. neg-mul-1 43.2%

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

      4. times-frac 43.2%

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

      5. metadata-eval 43.2%

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

      6. remove-double-neg 43.2%

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

      7. neg-sub0 43.2%

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

      8. associate-+l- 43.2%

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

   3. Simplified 43.2%

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

      1. fma-udef 43.2%

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

      2. *-commutative 43.2%

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

      3. metadata-eval 43.2%

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

      4. cancel-sign-sub-inv 43.2%

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

      5. associate-*l* 43.2%

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

      6. prod-diff 42.9%

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

      7. *-commutative 42.9%

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

      8. fma-neg 42.9%

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

      9. associate-+l- 42.9%

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

   5. Applied egg-rr 42.9%

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

      1. associate--r+ 42.9%

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

      2. +-inverses 43.2%

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

      3. neg-sub0 43.2%

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

      4. associate-*r* 43.2%

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

      5. distribute-rgt-neg-in 43.2%

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

      6. metadata-eval 43.2%

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

      7. *-commutative 43.2%

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

      8. associate-*r* 43.2%

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

   7. Simplified 43.2%

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

      1. flip-+ 42.9%

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

      2. add-sqr-sqrt 42.8%

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

   9. Applied egg-rr 42.8%

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

       1. associate-+l- 83.9%

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

   11. Simplified 83.9%

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

   12. Taylor expanded in b around 0 83.9%

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

   if -1.54999999999999993e-106 < b < 4.2000000000000001e-182

   1. Initial program 73.6%

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

      1. sub-neg 73.6%

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

      2. distribute-neg-out 73.6%

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

      3. neg-mul-1 73.6%

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

      4. times-frac 73.6%

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

      5. metadata-eval 73.6%

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

      6. remove-double-neg 73.6%

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

      7. neg-sub0 73.6%

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

      8. associate-+l- 73.6%

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

   3. Simplified 73.6%

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

      1. add-sqr-sqrt 71.4%

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

      2. *-un-lft-identity 71.4%

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

      3. times-frac 71.5%

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

   5. Applied egg-rr 75.1%

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

   if 4.2000000000000001e-182 < b < 1.0000000000000001e128

   1. Initial program 95.1%

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

      1. sub-neg 95.1%

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

      2. distribute-neg-out 95.1%

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

      3. neg-mul-1 95.1%

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

      4. times-frac 95.1%

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

      5. metadata-eval 95.1%

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

      6. remove-double-neg 95.1%

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

      7. neg-sub0 95.1%

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

      8. associate-+l- 95.1%

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

   3. Simplified 95.1%

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

      1. fma-udef 95.1%

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

      2. *-commutative 95.1%

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

      3. metadata-eval 95.1%

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

      4. cancel-sign-sub-inv 95.1%

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

      5. associate-*l* 95.1%

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

      6. prod-diff 95.0%

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

      7. *-commutative 95.0%

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

      8. fma-neg 95.0%

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

      9. associate-+l- 95.0%

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

   5. Applied egg-rr 95.0%

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

      1. associate--r+ 95.0%

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

      2. +-inverses 95.1%

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

      3. neg-sub0 95.1%

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

      4. associate-*r* 95.1%

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

      5. distribute-rgt-neg-in 95.1%

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

      6. metadata-eval 95.1%

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

      7. *-commutative 95.1%

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

      8. associate-*r* 95.1%

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

   7. Simplified 95.1%

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

   if 1.0000000000000001e128 < b

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. sqr-neg 60.3%

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

      3. *-commutative 60.3%

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

      4. sqr-neg 60.3%

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

      5. associate-*r* 60.3%

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

      6. *-commutative 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in b around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-27}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{c \cdot \left(a \cdot 4\right)}{b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a}\\ \mathbf{elif}\;b \leq -1.55 \cdot 10^{-106}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-182}:\\ \;\;\;\;-0.5 \cdot \left(\sqrt{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{\sqrt{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}{a}\right)\\ \mathbf{elif}\;b \leq 10^{+128}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2: 87.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 4\right)\\ t_1 := \sqrt{b \cdot b - t_0}\\ \mathbf{if}\;b \leq -2.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -1.1 \cdot 10^{-128}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{t_0}{b - t_1}}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+128}:\\ \;\;\;\;-0.5 \cdot \frac{b + t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 4.0))) (t_1 (sqrt (- (* b b) t_0))))
   (if (<= b -2.6e+74)
     (/ (- c) b)
     (if (<= b -1.1e-128)
       (* -0.5 (/ (/ t_0 (- b t_1)) a))
       (if (<= b 2.8e+128) (* -0.5 (/ (+ b t_1) a)) (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	double t_1 = sqrt(((b * b) - t_0));
	double tmp;
	if (b <= -2.6e+74) {
		tmp = -c / b;
	} else if (b <= -1.1e-128) {
		tmp = -0.5 * ((t_0 / (b - t_1)) / a);
	} else if (b <= 2.8e+128) {
		tmp = -0.5 * ((b + t_1) / 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) :: t_1
    real(8) :: tmp
    t_0 = c * (a * 4.0d0)
    t_1 = sqrt(((b * b) - t_0))
    if (b <= (-2.6d+74)) then
        tmp = -c / b
    else if (b <= (-1.1d-128)) then
        tmp = (-0.5d0) * ((t_0 / (b - t_1)) / a)
    else if (b <= 2.8d+128) then
        tmp = (-0.5d0) * ((b + t_1) / 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 = c * (a * 4.0);
	double t_1 = Math.sqrt(((b * b) - t_0));
	double tmp;
	if (b <= -2.6e+74) {
		tmp = -c / b;
	} else if (b <= -1.1e-128) {
		tmp = -0.5 * ((t_0 / (b - t_1)) / a);
	} else if (b <= 2.8e+128) {
		tmp = -0.5 * ((b + t_1) / a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (a * 4.0)
	t_1 = math.sqrt(((b * b) - t_0))
	tmp = 0
	if b <= -2.6e+74:
		tmp = -c / b
	elif b <= -1.1e-128:
		tmp = -0.5 * ((t_0 / (b - t_1)) / a)
	elif b <= 2.8e+128:
		tmp = -0.5 * ((b + t_1) / a)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 4.0))
	t_1 = sqrt(Float64(Float64(b * b) - t_0))
	tmp = 0.0
	if (b <= -2.6e+74)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= -1.1e-128)
		tmp = Float64(-0.5 * Float64(Float64(t_0 / Float64(b - t_1)) / a));
	elseif (b <= 2.8e+128)
		tmp = Float64(-0.5 * Float64(Float64(b + t_1) / a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a * 4.0);
	t_1 = sqrt(((b * b) - t_0));
	tmp = 0.0;
	if (b <= -2.6e+74)
		tmp = -c / b;
	elseif (b <= -1.1e-128)
		tmp = -0.5 * ((t_0 / (b - t_1)) / a);
	elseif (b <= 2.8e+128)
		tmp = -0.5 * ((b + t_1) / a);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.6e+74], N[((-c) / b), $MachinePrecision], If[LessEqual[b, -1.1e-128], N[(-0.5 * N[(N[(t$95$0 / N[(b - t$95$1), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.8e+128], N[(-0.5 * N[(N[(b + t$95$1), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.6000000000000001e74

   1. Initial program 6.1%

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

      1. *-commutative 6.1%

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

      2. sqr-neg 6.1%

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

      3. *-commutative 6.1%

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

      4. sqr-neg 6.1%

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

      5. associate-*r* 6.1%

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

      6. *-commutative 6.1%

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

   3. Simplified 6.1%

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

   4. Taylor expanded in b around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. distribute-neg-frac 94.2%

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

   6. Simplified 94.2%

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

   if -2.6000000000000001e74 < b < -1.10000000000000005e-128

   1. Initial program 37.3%

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

      1. sub-neg 37.3%

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

      2. distribute-neg-out 37.3%

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

      3. neg-mul-1 37.3%

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

      4. times-frac 37.3%

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

      5. metadata-eval 37.3%

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

      6. remove-double-neg 37.3%

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

      7. neg-sub0 37.3%

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

      8. associate-+l- 37.3%

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

   3. Simplified 37.3%

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

      1. fma-udef 37.3%

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

      2. *-commutative 37.3%

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

      3. metadata-eval 37.3%

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

      4. cancel-sign-sub-inv 37.3%

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

      5. associate-*l* 37.3%

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

      6. prod-diff 37.1%

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

      7. *-commutative 37.1%

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

      8. fma-neg 37.1%

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

      9. associate-+l- 37.1%

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

   5. Applied egg-rr 37.1%

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

      1. associate--r+ 37.1%

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

      2. +-inverses 37.3%

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

      3. neg-sub0 37.3%

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

      4. associate-*r* 37.3%

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

      5. distribute-rgt-neg-in 37.3%

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

      6. metadata-eval 37.3%

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

      7. *-commutative 37.3%

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

      8. associate-*r* 37.3%

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

   7. Simplified 37.3%

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

      1. flip-+ 37.1%

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

      2. add-sqr-sqrt 37.0%

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

   9. Applied egg-rr 37.0%

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

       1. associate-+l- 78.2%

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

   11. Simplified 78.2%

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

   12. Taylor expanded in b around 0 78.2%

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

   if -1.10000000000000005e-128 < b < 2.79999999999999983e128

   1. Initial program 86.3%

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

      1. sub-neg 86.3%

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

      2. distribute-neg-out 86.3%

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

      3. neg-mul-1 86.3%

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

      4. times-frac 86.3%

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

      5. metadata-eval 86.3%

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

      6. remove-double-neg 86.3%

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

      7. neg-sub0 86.3%

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

      8. associate-+l- 86.3%

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

   3. Simplified 86.3%

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

      1. fma-udef 86.3%

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

      2. *-commutative 86.3%

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

      3. metadata-eval 86.3%

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

      4. cancel-sign-sub-inv 86.3%

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

      5. associate-*l* 86.4%

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

      6. prod-diff 86.2%

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

      7. *-commutative 86.2%

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

      8. fma-neg 86.2%

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

      9. associate-+l- 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. associate--r+ 86.2%

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

      2. +-inverses 86.4%

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

      3. neg-sub0 86.4%

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

      4. associate-*r* 86.3%

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

      5. distribute-rgt-neg-in 86.3%

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

      6. metadata-eval 86.3%

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

      7. *-commutative 86.3%

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

      8. associate-*r* 86.4%

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

   7. Simplified 86.4%

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

   if 2.79999999999999983e128 < b

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. sqr-neg 60.3%

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

      3. *-commutative 60.3%

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

      4. sqr-neg 60.3%

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

      5. associate-*r* 60.3%

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

      6. *-commutative 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in b around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -1.1 \cdot 10^{-128}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{c \cdot \left(a \cdot 4\right)}{b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+128}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 10^{+128}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.2e-107)
   (/ (- c) b)
   (if (<= b 1e+128)
     (* -0.5 (/ (+ b (sqrt (- (* b b) (* c (* a 4.0))))) a))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e-107) {
		tmp = -c / b;
	} else if (b <= 1e+128) {
		tmp = -0.5 * ((b + sqrt(((b * b) - (c * (a * 4.0))))) / 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) :: tmp
    if (b <= (-7.2d-107)) then
        tmp = -c / b
    else if (b <= 1d+128) then
        tmp = (-0.5d0) * ((b + sqrt(((b * b) - (c * (a * 4.0d0))))) / 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 <= -7.2e-107) {
		tmp = -c / b;
	} else if (b <= 1e+128) {
		tmp = -0.5 * ((b + Math.sqrt(((b * b) - (c * (a * 4.0))))) / a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.2e-107:
		tmp = -c / b
	elif b <= 1e+128:
		tmp = -0.5 * ((b + math.sqrt(((b * b) - (c * (a * 4.0))))) / a)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.2e-107)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1e+128)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))) / a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.2e-107)
		tmp = -c / b;
	elseif (b <= 1e+128)
		tmp = -0.5 * ((b + sqrt(((b * b) - (c * (a * 4.0))))) / a);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.2e-107], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1e+128], N[(-0.5 * N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.19999999999999953e-107

   1. Initial program 19.0%

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

      1. *-commutative 19.0%

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

      2. sqr-neg 19.0%

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

      3. *-commutative 19.0%

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

      4. sqr-neg 19.0%

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

      5. associate-*r* 19.0%

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

      6. *-commutative 19.0%

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

   3. Simplified 19.0%

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

   4. Taylor expanded in b around -inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. distribute-neg-frac 82.2%

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

   6. Simplified 82.2%

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

   if -7.19999999999999953e-107 < b < 1.0000000000000001e128

   1. Initial program 86.0%

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

      1. sub-neg 86.0%

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

      2. distribute-neg-out 86.0%

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

      3. neg-mul-1 86.0%

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

      4. times-frac 86.0%

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

      5. metadata-eval 86.0%

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

      6. remove-double-neg 86.0%

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

      7. neg-sub0 86.0%

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

      8. associate-+l- 86.0%

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

   3. Simplified 86.0%

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

      1. fma-udef 86.0%

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

      2. *-commutative 86.0%

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

      3. metadata-eval 86.0%

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

      4. cancel-sign-sub-inv 86.0%

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

      5. associate-*l* 86.0%

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

      6. prod-diff 85.8%

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

      7. *-commutative 85.8%

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

      8. fma-neg 85.8%

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

      9. associate-+l- 85.8%

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

   5. Applied egg-rr 85.8%

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

      1. associate--r+ 85.8%

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

      2. +-inverses 86.0%

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

      3. neg-sub0 86.0%

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

      4. associate-*r* 86.0%

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

      5. distribute-rgt-neg-in 86.0%

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

      6. metadata-eval 86.0%

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

      7. *-commutative 86.0%

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

      8. associate-*r* 86.0%

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

   7. Simplified 86.0%

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

   if 1.0000000000000001e128 < b

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. sqr-neg 60.3%

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

      3. *-commutative 60.3%

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

      4. sqr-neg 60.3%

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

      5. associate-*r* 60.3%

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

      6. *-commutative 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in b around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 10^{+128}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-106}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-51}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.55e-106)
   (/ (- c) b)
   (if (<= b 1.6e-51)
     (* -0.5 (/ (+ b (sqrt (* a (* c -4.0)))) a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-106) {
		tmp = -c / b;
	} else if (b <= 1.6e-51) {
		tmp = -0.5 * ((b + sqrt((a * (c * -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 <= (-1.55d-106)) then
        tmp = -c / b
    else if (b <= 1.6d-51) then
        tmp = (-0.5d0) * ((b + sqrt((a * (c * (-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 <= -1.55e-106) {
		tmp = -c / b;
	} else if (b <= 1.6e-51) {
		tmp = -0.5 * ((b + Math.sqrt((a * (c * -4.0)))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.55e-106:
		tmp = -c / b
	elif b <= 1.6e-51:
		tmp = -0.5 * ((b + math.sqrt((a * (c * -4.0)))) / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.55e-106)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.6e-51)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(a * Float64(c * -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 <= -1.55e-106)
		tmp = -c / b;
	elseif (b <= 1.6e-51)
		tmp = -0.5 * ((b + sqrt((a * (c * -4.0)))) / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.55e-106], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.6e-51], N[(-0.5 * N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.54999999999999993e-106

   1. Initial program 19.0%

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

      1. *-commutative 19.0%

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

      2. sqr-neg 19.0%

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

      3. *-commutative 19.0%

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

      4. sqr-neg 19.0%

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

      5. associate-*r* 19.0%

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

      6. *-commutative 19.0%

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

   3. Simplified 19.0%

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

   4. Taylor expanded in b around -inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. distribute-neg-frac 82.2%

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

   6. Simplified 82.2%

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

   if -1.54999999999999993e-106 < b < 1.6e-51

   1. Initial program 80.6%

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

      1. sub-neg 80.6%

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

      2. distribute-neg-out 80.6%

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

      3. neg-mul-1 80.6%

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

      4. times-frac 80.6%

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

      5. metadata-eval 80.6%

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

      6. remove-double-neg 80.6%

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

      7. neg-sub0 80.6%

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

      8. associate-+l- 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in b around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. associate-*r* 67.9%

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

   6. Simplified 67.9%

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

   if 1.6e-51 < b

   1. Initial program 75.3%

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

      1. *-commutative 75.3%

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

      2. sqr-neg 75.3%

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

      3. *-commutative 75.3%

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

      4. sqr-neg 75.3%

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

      5. associate-*r* 75.3%

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

      6. *-commutative 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in b around inf 95.0%

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

      1. +-commutative 95.0%

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

      2. mul-1-neg 95.0%

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

      3. unsub-neg 95.0%

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

   6. Simplified 95.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-106}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-51}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5: 68.2% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ (- c) b) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = -c / b
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = -c / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[((-c) / b), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 31.9%

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

      1. *-commutative 31.9%

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

      2. sqr-neg 31.9%

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

      3. *-commutative 31.9%

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

      4. sqr-neg 31.9%

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

      5. associate-*r* 31.9%

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

      6. *-commutative 31.9%

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

   3. Simplified 31.9%

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

   4. Taylor expanded in b around -inf 68.9%

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

      1. mul-1-neg 68.9%

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

      2. distribute-neg-frac 68.9%

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

   6. Simplified 68.9%

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

   if -3.999999999999988e-310 < b

   1. Initial program 76.9%

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

      1. *-commutative 76.9%

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

      2. sqr-neg 76.9%

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

      3. *-commutative 76.9%

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

      4. sqr-neg 76.9%

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

      5. associate-*r* 76.9%

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

      6. *-commutative 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in b around inf 73.2%

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

      1. +-commutative 73.2%

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

      2. mul-1-neg 73.2%

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

      3. unsub-neg 73.2%

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

   6. Simplified 73.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6: 43.2% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{+61}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -2.45e+61) (/ c b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e+61) {
		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 <= (-2.45d+61)) 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 <= -2.45e+61) {
		tmp = c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.45e+61:
		tmp = c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.45e+61)
		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 <= -2.45e+61)
		tmp = c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.45e+61], N[(c / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.45000000000000013e61

   1. Initial program 13.0%

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

      1. *-commutative 13.0%

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

      2. sqr-neg 13.0%

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

      3. *-commutative 13.0%

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

      4. sqr-neg 13.0%

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

      5. associate-*r* 13.0%

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

      6. *-commutative 13.0%

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

   3. Simplified 13.0%

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

      1. clear-num 13.0%

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

      2. associate-/r/ 13.0%

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

      3. *-commutative 13.0%

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

      4. associate-/r* 13.0%

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

      5. metadata-eval 13.0%

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

      6. add-sqr-sqrt 7.3%

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

      7. cancel-sign-sub-inv 7.3%

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

      8. add-sqr-sqrt 12.9%

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

      9. sqrt-unprod 6.2%

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

      10. sqr-neg 6.2%

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

      11. sqrt-prod 0.0%

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

      12. add-sqr-sqrt 5.5%

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

   5. Applied egg-rr 5.2%

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

      1. flip-- 3.7%

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

      2. hypot-udef 3.7%

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

      3. hypot-udef 3.7%

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

      4. add-sqr-sqrt 3.7%

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

      5. add-sqr-sqrt 3.7%

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

      6. fma-def 3.7%

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

   7. Applied egg-rr 3.7%

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

      1. associate-*r* 3.7%

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

      2. metadata-eval 3.7%

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

      3. distribute-rgt-neg-in 3.7%

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

      4. *-commutative 3.7%

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

      5. associate-*r* 3.7%

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

      6. fma-neg 3.7%

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

      7. associate-+l- 3.8%

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

      8. +-inverses 4.4%

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

      9. associate-*r* 4.4%

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

      10. *-commutative 4.4%

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

      11. associate-*l* 4.4%

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

   9. Simplified 4.4%

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

   10. Taylor expanded in a around 0 39.2%

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

   if -2.45000000000000013e61 < b

   1. Initial program 67.3%

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

      1. *-commutative 67.3%

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

      2. sqr-neg 67.3%

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

      3. *-commutative 67.3%

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

      4. sqr-neg 67.3%

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

      5. associate-*r* 67.3%

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

      6. *-commutative 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in b around inf 50.0%

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

      1. associate-*r/ 50.0%

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

      2. mul-1-neg 50.0%

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

   6. Simplified 50.0%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{+61}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7: 68.0% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		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 <= (-4d-310)) 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 <= -4e-310) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		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 <= -4e-310)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 31.9%

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

      1. *-commutative 31.9%

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

      2. sqr-neg 31.9%

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

      3. *-commutative 31.9%

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

      4. sqr-neg 31.9%

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

      5. associate-*r* 31.9%

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

      6. *-commutative 31.9%

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

   3. Simplified 31.9%

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

   4. Taylor expanded in b around -inf 68.9%

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

      1. mul-1-neg 68.9%

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

      2. distribute-neg-frac 68.9%

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

   6. Simplified 68.9%

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

   if -3.999999999999988e-310 < b

   1. Initial program 76.9%

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

      1. *-commutative 76.9%

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

      2. sqr-neg 76.9%

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

      3. *-commutative 76.9%

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

      4. sqr-neg 76.9%

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

      5. associate-*r* 76.9%

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

      6. *-commutative 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in b around inf 72.9%

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

      1. associate-*r/ 72.9%

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

      2. mul-1-neg 72.9%

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

   6. Simplified 72.9%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

   2. sqr-neg 55.8%

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

   3. *-commutative 55.8%

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

   4. sqr-neg 55.8%

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

   5. associate-*r* 55.8%

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

   6. *-commutative 55.8%

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

3. Simplified 55.8%

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

   1. clear-num 55.7%

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

   2. associate-/r/ 55.7%

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

   3. *-commutative 55.7%

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

   4. associate-/r* 55.7%

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

   5. metadata-eval 55.7%

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

   6. add-sqr-sqrt 54.4%

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

   7. cancel-sign-sub-inv 54.4%

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

   8. add-sqr-sqrt 14.9%

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

   9. sqrt-unprod 25.6%

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

   10. sqr-neg 25.6%

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

   11. sqrt-prod 20.5%

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

   12. add-sqr-sqrt 33.2%

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

5. Applied egg-rr 24.4%

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

6. Taylor expanded in b around -inf 2.5%

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

7. Final simplification 2.5%

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

Alternative 9: 10.8% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

   2. sqr-neg 55.8%

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

   3. *-commutative 55.8%

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

   4. sqr-neg 55.8%

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

   5. associate-*r* 55.8%

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

   6. *-commutative 55.8%

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

3. Simplified 55.8%

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

   1. clear-num 55.7%

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

   2. associate-/r/ 55.7%

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

   3. *-commutative 55.7%

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

   4. associate-/r* 55.7%

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

   5. metadata-eval 55.7%

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

   6. add-sqr-sqrt 54.4%

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

   7. cancel-sign-sub-inv 54.4%

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

   8. add-sqr-sqrt 14.9%

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

   9. sqrt-unprod 25.6%

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

   10. sqr-neg 25.6%

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

   11. sqrt-prod 20.5%

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

   12. add-sqr-sqrt 33.2%

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

5. Applied egg-rr 24.4%

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

   1. flip-- 23.3%

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

   2. hypot-udef 23.3%

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

   3. hypot-udef 23.3%

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

   4. add-sqr-sqrt 23.3%

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

   5. add-sqr-sqrt 23.3%

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

   6. fma-def 23.3%

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

7. Applied egg-rr 23.3%

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

   1. associate-*r* 23.3%

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

   2. metadata-eval 23.3%

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

   3. distribute-rgt-neg-in 23.3%

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

   4. *-commutative 23.3%

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

   5. associate-*r* 23.3%

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

   6. fma-neg 23.3%

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

   7. associate-+l- 23.7%

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

   8. +-inverses 24.0%

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

   9. associate-*r* 24.0%

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

   10. *-commutative 24.0%

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

   11. associate-*l* 24.0%

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

9. Simplified 24.0%

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

10. Taylor expanded in a around 0 10.5%

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

11. Final simplification 10.5%

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

Developer target: 70.6% 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 2023277 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

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

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