quadp (p42, positive)

Percentage Accurate: 51.7% → 84.2%

Time: 12.4s

Alternatives: 7

Speedup: 12.9×

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: 51.7% 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: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{-b}\\ t_1 := \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \frac{0.5}{a}, -0.5 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 10^{-58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;\left(b - t\_1\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+98}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} + \frac{-1}{{b}^{3}}\right)\right) + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (- b))) (t_1 (sqrt (fma a (* c -4.0) (pow b 2.0)))))
   (if (<= b -1.5e+87)
     (/ b (- a))
     (if (<= b 4e-127)
       (fma t_1 (/ 0.5 a) (* -0.5 (/ b a)))
       (if (<= b 1e-58)
         t_0
         (if (<= b 3.5e-15)
           (* (- b t_1) (/ -0.5 a))
           (if (<= b 5.2e+98)
             (*
              c
              (+
               (*
                c
                (*
                 a
                 (+ (* -2.0 (/ (* a c) (pow b 5.0))) (/ -1.0 (pow b 3.0)))))
               (/ -1.0 b)))
             t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = c / -b;
	double t_1 = sqrt(fma(a, (c * -4.0), pow(b, 2.0)));
	double tmp;
	if (b <= -1.5e+87) {
		tmp = b / -a;
	} else if (b <= 4e-127) {
		tmp = fma(t_1, (0.5 / a), (-0.5 * (b / a)));
	} else if (b <= 1e-58) {
		tmp = t_0;
	} else if (b <= 3.5e-15) {
		tmp = (b - t_1) * (-0.5 / a);
	} else if (b <= 5.2e+98) {
		tmp = c * ((c * (a * ((-2.0 * ((a * c) / pow(b, 5.0))) + (-1.0 / pow(b, 3.0))))) + (-1.0 / b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / Float64(-b))
	t_1 = sqrt(fma(a, Float64(c * -4.0), (b ^ 2.0)))
	tmp = 0.0
	if (b <= -1.5e+87)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 4e-127)
		tmp = fma(t_1, Float64(0.5 / a), Float64(-0.5 * Float64(b / a)));
	elseif (b <= 1e-58)
		tmp = t_0;
	elseif (b <= 3.5e-15)
		tmp = Float64(Float64(b - t_1) * Float64(-0.5 / a));
	elseif (b <= 5.2e+98)
		tmp = Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * c) / (b ^ 5.0))) + Float64(-1.0 / (b ^ 3.0))))) + Float64(-1.0 / b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / (-b)), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.5e+87], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 4e-127], N[(t$95$1 * N[(0.5 / a), $MachinePrecision] + N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-58], t$95$0, If[LessEqual[b, 3.5e-15], N[(N[(b - t$95$1), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e+98], N[(c * N[(N[(c * N[(a * N[(N[(-2.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.4999999999999999e87

   1. Initial program 49.0%

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

   3. Taylor expanded in b around -inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. distribute-neg-frac2 91.5%

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

   5. Simplified 91.5%

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

   if -1.4999999999999999e87 < b < 4.0000000000000001e-127

   1. Initial program 84.5%

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

      1. *-un-lft-identity 84.5%

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

      2. *-commutative 84.5%

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

   4. Applied egg-rr 84.6%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} \cdot \frac{0.5}{a} + \left(-\frac{b}{a \cdot 2}\right)} \]
   5. Step-by-step derivation

      1. fma-define 84.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}, \frac{0.5}{a}, -\frac{b}{a \cdot 2}\right)} \]

      2. distribute-neg-frac 84.5%

         \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}, \frac{0.5}{a}, \color{blue}{\frac{-b}{a \cdot 2}}\right) \]

      3. mul-1-neg 84.5%

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

      4. *-commutative 84.5%

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

      5. times-frac 84.5%

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

      6. metadata-eval 84.5%

         \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}, \frac{0.5}{a}, \color{blue}{-0.5} \cdot \frac{b}{a}\right) \]

   6. Simplified 84.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}, \frac{0.5}{a}, -0.5 \cdot \frac{b}{a}\right)} \]

   if 4.0000000000000001e-127 < b < 1e-58 or 5.1999999999999999e98 < b

   1. Initial program 14.4%

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

   3. Taylor expanded in b around inf 85.7%

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

      1. associate-*r/ 85.7%

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

      2. neg-mul-1 85.7%

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

   5. Simplified 85.7%

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

   if 1e-58 < b < 3.5000000000000001e-15

   1. Initial program 75.5%

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

      1. associate-/r* 75.5%

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

      2. div-inv 75.7%

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

   4. Applied egg-rr 75.7%

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

      1. div-inv 75.7%

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

      2. metadata-eval 75.7%

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

   6. Applied egg-rr 75.7%

      \[\leadsto \color{blue}{\left(\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot -0.5\right) \cdot \frac{1}{a}} \]
   7. Step-by-step derivation

      1. associate-*r/ 75.5%

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

      2. *-rgt-identity 75.5%

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot -0.5}}{a} \]

      3. associate-/l* 75.7%

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}} \]

   8. Simplified 75.7%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}} \]

   if 3.5000000000000001e-15 < b < 5.1999999999999999e98

   1. Initial program 27.5%

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

   3. Taylor expanded in c around 0 68.2%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]

   4. Taylor expanded in a around 0 82.8%

      \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right)} - \frac{1}{b}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}, \frac{0.5}{a}, -0.5 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 10^{-58}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+98}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} + \frac{-1}{{b}^{3}}\right)\right) + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{-b}\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+87}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 10^{-58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-14}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+98}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} + \frac{-1}{{b}^{3}}\right)\right) + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (- b))))
   (if (<= b -1.35e+87)
     (/ b (- a))
     (if (<= b 4e-127)
       (/ (- (sqrt (- (* b b) (* (* a c) 4.0))) b) (* a 2.0))
       (if (<= b 1e-58)
         t_0
         (if (<= b 2.15e-14)
           (* (- b (sqrt (fma a (* c -4.0) (pow b 2.0)))) (/ -0.5 a))
           (if (<= b 5.2e+98)
             (*
              c
              (+
               (*
                c
                (*
                 a
                 (+ (* -2.0 (/ (* a c) (pow b 5.0))) (/ -1.0 (pow b 3.0)))))
               (/ -1.0 b)))
             t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = c / -b;
	double tmp;
	if (b <= -1.35e+87) {
		tmp = b / -a;
	} else if (b <= 4e-127) {
		tmp = (sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0);
	} else if (b <= 1e-58) {
		tmp = t_0;
	} else if (b <= 2.15e-14) {
		tmp = (b - sqrt(fma(a, (c * -4.0), pow(b, 2.0)))) * (-0.5 / a);
	} else if (b <= 5.2e+98) {
		tmp = c * ((c * (a * ((-2.0 * ((a * c) / pow(b, 5.0))) + (-1.0 / pow(b, 3.0))))) + (-1.0 / b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / Float64(-b))
	tmp = 0.0
	if (b <= -1.35e+87)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 4e-127)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * c) * 4.0))) - b) / Float64(a * 2.0));
	elseif (b <= 1e-58)
		tmp = t_0;
	elseif (b <= 2.15e-14)
		tmp = Float64(Float64(b - sqrt(fma(a, Float64(c * -4.0), (b ^ 2.0)))) * Float64(-0.5 / a));
	elseif (b <= 5.2e+98)
		tmp = Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * c) / (b ^ 5.0))) + Float64(-1.0 / (b ^ 3.0))))) + Float64(-1.0 / b)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / (-b)), $MachinePrecision]}, If[LessEqual[b, -1.35e+87], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 4e-127], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-58], t$95$0, If[LessEqual[b, 2.15e-14], N[(N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e+98], N[(c * N[(N[(c * N[(a * N[(N[(-2.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.35000000000000003e87

   1. Initial program 49.0%

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

   3. Taylor expanded in b around -inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. distribute-neg-frac2 91.5%

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

   5. Simplified 91.5%

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

   if -1.35000000000000003e87 < b < 4.0000000000000001e-127

   1. Initial program 84.5%

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

   if 4.0000000000000001e-127 < b < 1e-58 or 5.1999999999999999e98 < b

   1. Initial program 14.4%

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

   3. Taylor expanded in b around inf 85.7%

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

      1. associate-*r/ 85.7%

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

      2. neg-mul-1 85.7%

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

   5. Simplified 85.7%

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

   if 1e-58 < b < 2.14999999999999999e-14

   1. Initial program 75.5%

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

      1. associate-/r* 75.5%

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

      2. div-inv 75.7%

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

   4. Applied egg-rr 75.7%

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

      1. div-inv 75.7%

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

      2. metadata-eval 75.7%

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

   6. Applied egg-rr 75.7%

      \[\leadsto \color{blue}{\left(\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot -0.5\right) \cdot \frac{1}{a}} \]
   7. Step-by-step derivation

      1. associate-*r/ 75.5%

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

      2. *-rgt-identity 75.5%

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot -0.5}}{a} \]

      3. associate-/l* 75.7%

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}} \]

   8. Simplified 75.7%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}} \]

   if 2.14999999999999999e-14 < b < 5.1999999999999999e98

   1. Initial program 27.5%

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

   3. Taylor expanded in c around 0 68.2%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]

   4. Taylor expanded in a around 0 82.8%

      \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right)} - \frac{1}{b}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+87}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 10^{-58}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-14}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+98}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} + \frac{-1}{{b}^{3}}\right)\right) + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{-b}\\ t_1 := \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+87}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+98}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} + \frac{-1}{{b}^{3}}\right)\right) + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (- b)))
        (t_1 (/ (- (sqrt (- (* b b) (* (* a c) 4.0))) b) (* a 2.0))))
   (if (<= b -1.35e+87)
     (/ b (- a))
     (if (<= b 4e-127)
       t_1
       (if (<= b 1.02e-58)
         t_0
         (if (<= b 2.9e-15)
           t_1
           (if (<= b 6e+98)
             (*
              c
              (+
               (*
                c
                (*
                 a
                 (+ (* -2.0 (/ (* a c) (pow b 5.0))) (/ -1.0 (pow b 3.0)))))
               (/ -1.0 b)))
             t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = c / -b;
	double t_1 = (sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -1.35e+87) {
		tmp = b / -a;
	} else if (b <= 4e-127) {
		tmp = t_1;
	} else if (b <= 1.02e-58) {
		tmp = t_0;
	} else if (b <= 2.9e-15) {
		tmp = t_1;
	} else if (b <= 6e+98) {
		tmp = c * ((c * (a * ((-2.0 * ((a * c) / pow(b, 5.0))) + (-1.0 / pow(b, 3.0))))) + (-1.0 / b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c / -b
    t_1 = (sqrt(((b * b) - ((a * c) * 4.0d0))) - b) / (a * 2.0d0)
    if (b <= (-1.35d+87)) then
        tmp = b / -a
    else if (b <= 4d-127) then
        tmp = t_1
    else if (b <= 1.02d-58) then
        tmp = t_0
    else if (b <= 2.9d-15) then
        tmp = t_1
    else if (b <= 6d+98) then
        tmp = c * ((c * (a * (((-2.0d0) * ((a * c) / (b ** 5.0d0))) + ((-1.0d0) / (b ** 3.0d0))))) + ((-1.0d0) / b))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c / -b;
	double t_1 = (Math.sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -1.35e+87) {
		tmp = b / -a;
	} else if (b <= 4e-127) {
		tmp = t_1;
	} else if (b <= 1.02e-58) {
		tmp = t_0;
	} else if (b <= 2.9e-15) {
		tmp = t_1;
	} else if (b <= 6e+98) {
		tmp = c * ((c * (a * ((-2.0 * ((a * c) / Math.pow(b, 5.0))) + (-1.0 / Math.pow(b, 3.0))))) + (-1.0 / b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c / -b
	t_1 = (math.sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0)
	tmp = 0
	if b <= -1.35e+87:
		tmp = b / -a
	elif b <= 4e-127:
		tmp = t_1
	elif b <= 1.02e-58:
		tmp = t_0
	elif b <= 2.9e-15:
		tmp = t_1
	elif b <= 6e+98:
		tmp = c * ((c * (a * ((-2.0 * ((a * c) / math.pow(b, 5.0))) + (-1.0 / math.pow(b, 3.0))))) + (-1.0 / b))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c / Float64(-b))
	t_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * c) * 4.0))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (b <= -1.35e+87)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 4e-127)
		tmp = t_1;
	elseif (b <= 1.02e-58)
		tmp = t_0;
	elseif (b <= 2.9e-15)
		tmp = t_1;
	elseif (b <= 6e+98)
		tmp = Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * c) / (b ^ 5.0))) + Float64(-1.0 / (b ^ 3.0))))) + Float64(-1.0 / b)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c / -b;
	t_1 = (sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0);
	tmp = 0.0;
	if (b <= -1.35e+87)
		tmp = b / -a;
	elseif (b <= 4e-127)
		tmp = t_1;
	elseif (b <= 1.02e-58)
		tmp = t_0;
	elseif (b <= 2.9e-15)
		tmp = t_1;
	elseif (b <= 6e+98)
		tmp = c * ((c * (a * ((-2.0 * ((a * c) / (b ^ 5.0))) + (-1.0 / (b ^ 3.0))))) + (-1.0 / b));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / (-b)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.35e+87], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 4e-127], t$95$1, If[LessEqual[b, 1.02e-58], t$95$0, If[LessEqual[b, 2.9e-15], t$95$1, If[LessEqual[b, 6e+98], N[(c * N[(N[(c * N[(a * N[(N[(-2.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.35000000000000003e87

   1. Initial program 49.0%

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

   3. Taylor expanded in b around -inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. distribute-neg-frac2 91.5%

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

   5. Simplified 91.5%

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

   if -1.35000000000000003e87 < b < 4.0000000000000001e-127 or 1.0199999999999999e-58 < b < 2.90000000000000019e-15

   1. Initial program 83.8%

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

   if 4.0000000000000001e-127 < b < 1.0199999999999999e-58 or 6.0000000000000003e98 < b

   1. Initial program 14.4%

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

   3. Taylor expanded in b around inf 85.7%

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

      1. associate-*r/ 85.7%

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

      2. neg-mul-1 85.7%

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

   5. Simplified 85.7%

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

   if 2.90000000000000019e-15 < b < 6.0000000000000003e98

   1. Initial program 27.5%

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

   3. Taylor expanded in c around 0 68.2%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]

   4. Taylor expanded in a around 0 82.8%

      \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right)} - \frac{1}{b}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+87}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-58}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+98}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} + \frac{-1}{{b}^{3}}\right)\right) + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+87)
   (/ b (- a))
   (if (<= b 3.5e-127)
     (/ (- (sqrt (- (* b b) (* (* a c) 4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+87) {
		tmp = b / -a;
	} else if (b <= 3.5e-127) {
		tmp = (sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.5d+87)) then
        tmp = b / -a
    else if (b <= 3.5d-127) then
        tmp = (sqrt(((b * b) - ((a * c) * 4.0d0))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+87) {
		tmp = b / -a;
	} else if (b <= 3.5e-127) {
		tmp = (Math.sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.5e+87:
		tmp = b / -a
	elif b <= 3.5e-127:
		tmp = (math.sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+87)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 3.5e-127)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * c) * 4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.5e+87)
		tmp = b / -a;
	elseif (b <= 3.5e-127)
		tmp = (sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+87], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 3.5e-127], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.4999999999999999e87

   1. Initial program 49.0%

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

   3. Taylor expanded in b around -inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. distribute-neg-frac2 91.5%

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

   5. Simplified 91.5%

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

   if -1.4999999999999999e87 < b < 3.49999999999999989e-127

   1. Initial program 84.5%

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

   if 3.49999999999999989e-127 < b

   1. Initial program 22.4%

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

   3. Taylor expanded in b around inf 80.0%

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

      1. associate-*r/ 80.0%

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

      2. neg-mul-1 80.0%

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

   5. Simplified 80.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0085:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -0.0085)
   (/ b (- a))
   (if (<= b 4e-127)
     (* (/ -0.5 a) (- b (sqrt (* a (* c -4.0)))))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -0.0085) {
		tmp = b / -a;
	} else if (b <= 4e-127) {
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-0.0085d0)) then
        tmp = b / -a
    else if (b <= 4d-127) then
        tmp = ((-0.5d0) / a) * (b - sqrt((a * (c * (-4.0d0)))))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -0.0085) {
		tmp = b / -a;
	} else if (b <= 4e-127) {
		tmp = (-0.5 / a) * (b - Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -0.0085:
		tmp = b / -a
	elif b <= 4e-127:
		tmp = (-0.5 / a) * (b - math.sqrt((a * (c * -4.0))))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -0.0085)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 4e-127)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -0.0085)
		tmp = b / -a;
	elseif (b <= 4e-127)
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -0.0085], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 4e-127], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.0085000000000000006

   1. Initial program 57.5%

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

   3. Taylor expanded in b around -inf 87.8%

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

      1. mul-1-neg 87.8%

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

      2. distribute-neg-frac2 87.8%

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

   5. Simplified 87.8%

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

   if -0.0085000000000000006 < b < 4.0000000000000001e-127

   1. Initial program 83.3%

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

      1. associate-/r* 83.3%

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

      2. div-inv 83.3%

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

   4. Applied egg-rr 83.3%

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

      1. div-inv 83.3%

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

      2. metadata-eval 83.3%

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

   6. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{\left(\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot -0.5\right) \cdot \frac{1}{a}} \]
   7. Step-by-step derivation

      1. associate-*r/ 83.3%

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

      2. *-rgt-identity 83.3%

         \[\leadsto \frac{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot -0.5}}{a} \]

      3. associate-/l* 83.3%

         \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}} \]

   8. Simplified 83.3%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}} \]

   9. Taylor expanded in a around inf 76.1%

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

       1. *-commutative 76.1%

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

       2. associate-*r* 76.1%

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

   11. Simplified 76.1%

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

   if 4.0000000000000001e-127 < b

   1. Initial program 22.4%

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

   3. Taylor expanded in b around inf 80.0%

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

      1. associate-*r/ 80.0%

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

      2. neg-mul-1 80.0%

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

   5. Simplified 80.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0085:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-295}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.1e-295) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1e-295) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.1d-295) then
        tmp = b / -a
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1e-295) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.1e-295:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.1e-295)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.1e-295)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.1e-295], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.09999999999999993e-295

   1. Initial program 67.0%

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

   3. Taylor expanded in b around -inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. distribute-neg-frac2 66.7%

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

   5. Simplified 66.7%

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

   if 2.09999999999999993e-295 < b

   1. Initial program 34.9%

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

   3. Taylor expanded in b around inf 65.1%

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

      1. associate-*r/ 65.1%

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

      2. neg-mul-1 65.1%

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

   5. Simplified 65.1%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-295}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.6% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(b / Float64(-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 50.5%

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

3. Taylor expanded in b around -inf 33.6%

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

   1. mul-1-neg 33.6%

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

   2. distribute-neg-frac2 33.6%

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

5. Simplified 33.6%

   \[\leadsto \color{blue}{\frac{b}{-a}} \]
6. Add Preprocessing

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024095 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))))))

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