Quadratic roots, full range

Percentage Accurate: 51.5% → 85.5%

Time: 15.1s

Alternatives: 8

Speedup: 19.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot 4\right) \cdot c\\ \mathbf{if}\;b \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - t_0} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-61} \lor \neg \left(b \leq 2.9 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{b}^{2} + \mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(a, c \cdot -4, t_0\right)\right)} - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a 4.0) c)))
   (if (<= b -4e+154)
     (/ (- b) a)
     (if (<= b 8.5e-119)
       (/ (- (sqrt (- (* b b) t_0)) b) (* a 2.0))
       (if (or (<= b 6.5e-61) (not (<= b 2.9e-34)))
         (/ (- c) b)
         (/
          (-
           (sqrt (+ (pow b 2.0) (fma a (* c -4.0) (fma a (* c -4.0) t_0))))
           b)
          (* a 2.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = (a * 4.0) * c;
	double tmp;
	if (b <= -4e+154) {
		tmp = -b / a;
	} else if (b <= 8.5e-119) {
		tmp = (sqrt(((b * b) - t_0)) - b) / (a * 2.0);
	} else if ((b <= 6.5e-61) || !(b <= 2.9e-34)) {
		tmp = -c / b;
	} else {
		tmp = (sqrt((pow(b, 2.0) + fma(a, (c * -4.0), fma(a, (c * -4.0), t_0)))) - b) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * 4.0) * c)
	tmp = 0.0
	if (b <= -4e+154)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 8.5e-119)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(a * 2.0));
	elseif ((b <= 6.5e-61) || !(b <= 2.9e-34))
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(sqrt(Float64((b ^ 2.0) + fma(a, Float64(c * -4.0), fma(a, Float64(c * -4.0), t_0)))) - b) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, -4e+154], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 8.5e-119], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 6.5e-61], N[Not[LessEqual[b, 2.9e-34]], $MachinePrecision]], N[((-c) / b), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.00000000000000015e154

   1. Initial program 41.2%

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

      1. *-commutative 41.2%

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

   3. Simplified 41.2%

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

   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%

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

   if -4.00000000000000015e154 < b < 8.49999999999999977e-119

   1. Initial program 82.0%

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

   if 8.49999999999999977e-119 < b < 6.4999999999999994e-61 or 2.9000000000000002e-34 < b

   1. Initial program 17.4%

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

      1. *-commutative 17.4%

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

   3. Simplified 17.4%

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

   4. Taylor expanded in b around inf 86.6%

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

      1. mul-1-neg 86.6%

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

      2. distribute-neg-frac 86.6%

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

   6. Simplified 86.6%

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

   if 6.4999999999999994e-61 < b < 2.9000000000000002e-34

   1. Initial program 86.2%

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

      1. *-commutative 86.2%

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

   3. Simplified 86.2%

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

      1. prod-diff 86.2%

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

      2. *-commutative 86.2%

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

      3. fma-def 86.2%

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

      4. associate-+l+ 86.2%

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

      5. pow2 86.2%

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

      6. distribute-lft-neg-in 86.2%

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

      7. *-commutative 86.2%

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

      8. distribute-rgt-neg-in 86.2%

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

      9. metadata-eval 86.2%

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

      10. associate-*r* 86.2%

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

      11. *-commutative 86.2%

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

      12. *-commutative 86.2%

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

      13. fma-udef 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. fma-def 86.2%

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

      2. fma-def 86.4%

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

      3. associate-*l* 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-61} \lor \neg \left(b \leq 2.9 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{b}^{2} + \mathsf{fma}\left(a, c \cdot -4, \mathsf{fma}\left(a, c \cdot -4, \left(a \cdot 4\right) \cdot c\right)\right)} - b}{a \cdot 2}\\ \end{array} \]

Alternative 2: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+152}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-119} \lor \neg \left(b \leq 5.5 \cdot 10^{-61}\right) \land b \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e+152)
   (/ (- b) a)
   (if (or (<= b 8.5e-119) (and (not (<= b 5.5e-61)) (<= b 3.1e-34)))
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+152) {
		tmp = -b / a;
	} else if ((b <= 8.5e-119) || (!(b <= 5.5e-61) && (b <= 3.1e-34))) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - 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 <= (-4d+152)) then
        tmp = -b / a
    else if ((b <= 8.5d-119) .or. (.not. (b <= 5.5d-61)) .and. (b <= 3.1d-34)) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - 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 <= -4e+152) {
		tmp = -b / a;
	} else if ((b <= 8.5e-119) || (!(b <= 5.5e-61) && (b <= 3.1e-34))) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e+152:
		tmp = -b / a
	elif (b <= 8.5e-119) or (not (b <= 5.5e-61) and (b <= 3.1e-34)):
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e+152)
		tmp = Float64(Float64(-b) / a);
	elseif ((b <= 8.5e-119) || (!(b <= 5.5e-61) && (b <= 3.1e-34)))
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e+152)
		tmp = -b / a;
	elseif ((b <= 8.5e-119) || (~((b <= 5.5e-61)) && (b <= 3.1e-34)))
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e+152], N[((-b) / a), $MachinePrecision], If[Or[LessEqual[b, 8.5e-119], And[N[Not[LessEqual[b, 5.5e-61]], $MachinePrecision], LessEqual[b, 3.1e-34]]], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.0000000000000002e152

   1. Initial program 41.2%

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

      1. *-commutative 41.2%

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

   3. Simplified 41.2%

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

   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%

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

   if -4.0000000000000002e152 < b < 8.49999999999999977e-119 or 5.4999999999999997e-61 < b < 3.0999999999999998e-34

   1. Initial program 82.3%

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

   if 8.49999999999999977e-119 < b < 5.4999999999999997e-61 or 3.0999999999999998e-34 < b

   1. Initial program 17.4%

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

      1. *-commutative 17.4%

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

   3. Simplified 17.4%

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

   4. Taylor expanded in b around inf 86.6%

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

      1. mul-1-neg 86.6%

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

      2. distribute-neg-frac 86.6%

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

   6. Simplified 86.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+152}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-119} \lor \neg \left(b \leq 5.5 \cdot 10^{-61}\right) \land b \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-81}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-81)
   (/ (- b) a)
   (if (<= b 3.6e-119)
     (* (/ -0.5 a) (- b (sqrt (* -4.0 (* a c)))))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-81) {
		tmp = -b / a;
	} else if (b <= 3.6e-119) {
		tmp = (-0.5 / a) * (b - sqrt((-4.0 * (a * c))));
	} 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 <= (-1.15d-81)) then
        tmp = -b / a
    else if (b <= 3.6d-119) then
        tmp = ((-0.5d0) / a) * (b - sqrt(((-4.0d0) * (a * c))))
    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 <= -1.15e-81) {
		tmp = -b / a;
	} else if (b <= 3.6e-119) {
		tmp = (-0.5 / a) * (b - Math.sqrt((-4.0 * (a * c))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e-81:
		tmp = -b / a
	elif b <= 3.6e-119:
		tmp = (-0.5 / a) * (b - math.sqrt((-4.0 * (a * c))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e-81)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 3.6e-119)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(-4.0 * Float64(a * c)))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.15e-81)
		tmp = -b / a;
	elseif (b <= 3.6e-119)
		tmp = (-0.5 / a) * (b - sqrt((-4.0 * (a * c))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e-81], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 3.6e-119], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14999999999999996e-81

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in b around -inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. mul-1-neg 87.4%

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

   6. Simplified 87.4%

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

   if -1.14999999999999996e-81 < b < 3.6e-119

   1. Initial program 73.9%

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

      1. *-commutative 73.9%

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

   3. Simplified 73.9%

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

   5. Applied egg-rr 73.7%

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

      1. *-commutative 73.7%

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

      2. associate-/r* 73.7%

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

      3. metadata-eval 73.7%

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

   7. Simplified 73.7%

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

      1. pow1/2 73.7%

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

      2. metadata-eval 73.7%

         \[\leadsto \left(b - {\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{\left(0.25 \cdot 2\right)}}\right) \cdot \frac{-0.5}{a} \]

      3. pow-to-exp 68.9%

         \[\leadsto \left(b - \color{blue}{e^{\log \left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right) \cdot \left(0.25 \cdot 2\right)}}\right) \cdot \frac{-0.5}{a} \]

      4. metadata-eval 68.9%

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

   9. Applied egg-rr 68.9%

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

   10. Taylor expanded in a around -inf 46.9%

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

       1. mul-1-neg 46.9%

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

       2. unsub-neg 46.9%

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

       3. *-commutative 46.9%

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

   12. Simplified 46.9%

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

       1. expm1-log1p-u 23.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(b - e^{\left(\log \left(c \cdot 4\right) - \log \left(\frac{-1}{a}\right)\right) \cdot 0.5}\right) \cdot \frac{-0.5}{a}\right)\right)} \]

       2. expm1-udef 3.6%

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

   14. Applied egg-rr 21.6%

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

       1. expm1-def 45.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \frac{c}{\frac{1}{a}}}\right)\right)\right)} \]

       2. expm1-log1p 70.9%

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

       3. associate-/r/ 70.8%

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

       4. /-rgt-identity 70.8%

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

       5. *-commutative 70.8%

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

   16. Simplified 70.8%

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

   if 3.6e-119 < b

   1. Initial program 22.1%

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

      1. *-commutative 22.1%

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

   3. Simplified 22.1%

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

   4. Taylor expanded in b around inf 82.3%

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

      1. mul-1-neg 82.3%

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

      2. distribute-neg-frac 82.3%

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

   6. Simplified 82.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-81}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e-81)
   (/ (- b) a)
   (if (<= b 8e-121) (/ (- (sqrt (* -4.0 (* a c))) b) (* a 2.0)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e-81) {
		tmp = -b / a;
	} else if (b <= 8e-121) {
		tmp = (sqrt((-4.0 * (a * c))) - 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 <= (-1.2d-81)) then
        tmp = -b / a
    else if (b <= 8d-121) then
        tmp = (sqrt(((-4.0d0) * (a * c))) - 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 <= -1.2e-81) {
		tmp = -b / a;
	} else if (b <= 8e-121) {
		tmp = (Math.sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.2e-81:
		tmp = -b / a
	elif b <= 8e-121:
		tmp = (math.sqrt((-4.0 * (a * c))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e-81)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 8e-121)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.2e-81)
		tmp = -b / a;
	elseif (b <= 8e-121)
		tmp = (sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e-81], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 8e-121], N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $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 < -1.2e-81

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in b around -inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. mul-1-neg 87.4%

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

   6. Simplified 87.4%

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

   if -1.2e-81 < b < 7.9999999999999998e-121

   1. Initial program 73.9%

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

      1. *-commutative 73.9%

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

   3. Simplified 73.9%

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

   4. Taylor expanded in b around 0 71.0%

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

   if 7.9999999999999998e-121 < b

   1. Initial program 22.1%

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

      1. *-commutative 22.1%

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

   3. Simplified 22.1%

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

   4. Taylor expanded in b around inf 82.3%

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

      1. mul-1-neg 82.3%

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

      2. distribute-neg-frac 82.3%

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

   6. Simplified 82.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 68.2% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 71.2%

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

      1. *-commutative 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in b around -inf 65.6%

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

      1. +-commutative 65.6%

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

      2. mul-1-neg 65.6%

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

      3. unsub-neg 65.6%

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

   6. Simplified 65.6%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.8%

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

      1. *-commutative 31.8%

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

   3. Simplified 31.8%

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

   4. Taylor expanded in b around inf 67.1%

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

      1. mul-1-neg 67.1%

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

      2. distribute-neg-frac 67.1%

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

   6. Simplified 67.1%

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

4. Final simplification 66.4%

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

Alternative 6: 43.3% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-304}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -3.6e-304) (/ (- b) a) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-304) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	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) :: tmp
    if (b <= (-3.6d-304)) then
        tmp = -b / a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-304) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.6e-304:
		tmp = -b / a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e-304)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.6e-304)
		tmp = -b / a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e-304], N[((-b) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -3.6000000000000001e-304

   1. Initial program 70.9%

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

      1. *-commutative 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in b around -inf 65.9%

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

      1. associate-*r/ 65.9%

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

      2. mul-1-neg 65.9%

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

   6. Simplified 65.9%

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

   if -3.6000000000000001e-304 < b

   1. Initial program 32.3%

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

      1. *-commutative 32.3%

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

   3. Simplified 32.3%

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

   5. Applied egg-rr 32.2%

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

      1. *-commutative 32.2%

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

      2. associate-/r* 32.2%

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

      3. metadata-eval 32.2%

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

   7. Simplified 32.2%

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

      1. *-commutative 32.2%

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

      2. sub-neg 32.2%

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

      3. distribute-lft-in 31.9%

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

   9. Applied egg-rr 31.9%

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

   10. Taylor expanded in c around 0 15.1%

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

       1. distribute-rgt-out 15.1%

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

       2. metadata-eval 15.1%

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

       3. mul0-rgt 18.7%

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

   12. Simplified 18.7%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-304}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 68.0% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.5e-299) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.5e-299) {
		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 <= 6.5d-299) 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 <= 6.5e-299) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.5e-299:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.5e-299)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6.5e-299)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.5e-299], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.4999999999999997e-299

   1. Initial program 70.1%

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

      1. *-commutative 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in b around -inf 64.5%

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

      1. associate-*r/ 64.5%

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

      2. mul-1-neg 64.5%

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

   6. Simplified 64.5%

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

   if 6.4999999999999997e-299 < b

   1. Initial program 32.2%

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

      1. *-commutative 32.2%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in b around inf 68.1%

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

      1. mul-1-neg 68.1%

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

      2. distribute-neg-frac 68.1%

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

   6. Simplified 68.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8: 11.7% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 50.9%

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

   1. *-commutative 50.9%

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

3. Simplified 50.9%

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

5. Applied egg-rr 50.8%

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

   1. *-commutative 50.8%

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

   2. associate-/r* 50.8%

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

   3. metadata-eval 50.8%

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

7. Simplified 50.8%

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

   1. *-commutative 50.8%

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

   2. sub-neg 50.8%

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

   3. distribute-lft-in 50.6%

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

9. Applied egg-rr 50.6%

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

10. Taylor expanded in c around 0 9.0%

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

    1. distribute-rgt-out 9.0%

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

    2. metadata-eval 9.0%

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

    3. mul0-rgt 11.0%

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

12. Simplified 11.0%

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

13. Final simplification 11.0%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023332 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))