Quadratic roots, full range

Percentage Accurate: 51.3% → 85.8%

Time: 11.3s

Alternatives: 7

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: 51.3% 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: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.76 \cdot 10^{+99}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.76e+99)
   (/ (- b) a)
   (if (<= b 1.6e-37)
     (/ (- (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 <= -1.76e+99) {
		tmp = -b / a;
	} else if (b <= 1.6e-37) {
		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 <= (-1.76d+99)) then
        tmp = -b / a
    else if (b <= 1.6d-37) 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 <= -1.76e+99) {
		tmp = -b / a;
	} else if (b <= 1.6e-37) {
		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 <= -1.76e+99:
		tmp = -b / a
	elif b <= 1.6e-37:
		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 <= -1.76e+99)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.6e-37)
		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 <= -1.76e+99)
		tmp = -b / a;
	elseif (b <= 1.6e-37)
		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, -1.76e+99], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.6e-37], 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 < -1.76000000000000007e99

   1. Initial program 54.2%

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

      1. *-commutative 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in b around -inf 96.1%

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

      1. associate-*r/ 96.1%

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

      2. mul-1-neg 96.1%

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

   6. Simplified 96.1%

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

   if -1.76000000000000007e99 < b < 1.5999999999999999e-37

   1. Initial program 76.3%

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

   if 1.5999999999999999e-37 < b

   1. Initial program 12.8%

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

      1. *-commutative 12.8%

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

   3. Simplified 12.8%

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

   4. Taylor expanded in b around inf 92.2%

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

      1. mul-1-neg 92.2%

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

      2. distribute-neg-frac 92.2%

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

   6. Simplified 92.2%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.76 \cdot 10^{+99}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-29}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.1 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-29)
   (- (/ c b) (/ b a))
   (if (<= b 7.1e-37)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-29) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7.1e-37) {
		tmp = (sqrt((c * (a * -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 <= (-1d-29)) then
        tmp = (c / b) - (b / a)
    else if (b <= 7.1d-37) then
        tmp = (sqrt((c * (a * (-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 <= -1e-29) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7.1e-37) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-29:
		tmp = (c / b) - (b / a)
	elif b <= 7.1e-37:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-29)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 7.1e-37)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - 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 <= -1e-29)
		tmp = (c / b) - (b / a);
	elseif (b <= 7.1e-37)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-29], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.1e-37], N[(N[(N[Sqrt[N[(c * N[(a * -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 < -9.99999999999999943e-30

   1. Initial program 62.9%

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

      1. *-commutative 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in b around -inf 83.1%

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

      1. +-commutative 83.1%

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

      2. mul-1-neg 83.1%

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

      3. unsub-neg 83.1%

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

   6. Simplified 83.1%

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

   if -9.99999999999999943e-30 < b < 7.09999999999999978e-37

   1. Initial program 75.8%

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

      1. *-commutative 75.8%

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

   3. Simplified 75.8%

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

      1. prod-diff 75.4%

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

      2. *-commutative 75.4%

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

      3. fma-def 75.4%

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

      4. associate-+l+ 75.4%

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

      5. pow2 75.4%

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

      6. *-commutative 75.4%

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

      7. distribute-rgt-neg-in 75.4%

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

      8. *-commutative 75.4%

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

      9. distribute-rgt-neg-in 75.4%

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

      10. metadata-eval 75.4%

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

      11. *-commutative 75.4%

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

      12. fma-udef 75.4%

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

      13. distribute-lft-neg-in 75.4%

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

      14. distribute-rgt-neg-in 75.4%

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

      15. *-commutative 75.4%

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

      16. distribute-rgt-neg-in 75.4%

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

      17. metadata-eval 75.4%

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

      18. associate-*l* 75.4%

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

   5. Applied egg-rr 75.4%

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

      1. fma-def 75.4%

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

      2. fma-def 75.4%

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

      3. *-commutative 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in b around 0 69.8%

      \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. neg-mul-1 69.8%

         \[\leadsto \frac{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} + \color{blue}{\left(-b\right)}}{a \cdot 2} \]

      2. unsub-neg 69.8%

         \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} - b}}{a \cdot 2} \]

      3. distribute-rgt-out 70.2%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 + 4\right)}} - b}{a \cdot 2} \]

      4. *-commutative 70.2%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot \left(-8 + 4\right)} - b}{a \cdot 2} \]

      5. metadata-eval 70.2%

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

      6. associate-*r* 70.2%

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

      7. *-commutative 70.2%

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

   10. Simplified 70.2%

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

   if 7.09999999999999978e-37 < b

   1. Initial program 12.8%

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

      1. *-commutative 12.8%

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

   3. Simplified 12.8%

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

   4. Taylor expanded in b around inf 92.2%

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

      1. mul-1-neg 92.2%

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

      2. distribute-neg-frac 92.2%

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

   6. Simplified 92.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-29}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.1 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.8e-30)
   (- (/ c b) (/ b a))
   (if (<= b 1.55e-35) (* 0.5 (/ (sqrt (* c (* a -4.0))) a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.8e-30) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.55e-35) {
		tmp = 0.5 * (sqrt((c * (a * -4.0))) / 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 <= (-9.8d-30)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.55d-35) then
        tmp = 0.5d0 * (sqrt((c * (a * (-4.0d0)))) / 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 <= -9.8e-30) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.55e-35) {
		tmp = 0.5 * (Math.sqrt((c * (a * -4.0))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.8e-30:
		tmp = (c / b) - (b / a)
	elif b <= 1.55e-35:
		tmp = 0.5 * (math.sqrt((c * (a * -4.0))) / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.8e-30)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.55e-35)
		tmp = Float64(0.5 * Float64(sqrt(Float64(c * Float64(a * -4.0))) / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.8e-30)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.55e-35)
		tmp = 0.5 * (sqrt((c * (a * -4.0))) / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.8e-30], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.55e-35], N[(0.5 * N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.79999999999999942e-30

   1. Initial program 62.9%

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

      1. *-commutative 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in b around -inf 83.1%

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

      1. +-commutative 83.1%

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

      2. mul-1-neg 83.1%

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

      3. unsub-neg 83.1%

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

   6. Simplified 83.1%

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

   if -9.79999999999999942e-30 < b < 1.55000000000000006e-35

   1. Initial program 75.8%

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

      1. *-commutative 75.8%

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

   3. Simplified 75.8%

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

      1. prod-diff 75.4%

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

      2. *-commutative 75.4%

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

      3. fma-def 75.4%

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

      4. associate-+l+ 75.4%

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

      5. pow2 75.4%

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

      6. *-commutative 75.4%

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

      7. distribute-rgt-neg-in 75.4%

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

      8. *-commutative 75.4%

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

      9. distribute-rgt-neg-in 75.4%

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

      10. metadata-eval 75.4%

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

      11. *-commutative 75.4%

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

      12. fma-udef 75.4%

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

      13. distribute-lft-neg-in 75.4%

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

      14. distribute-rgt-neg-in 75.4%

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

      15. *-commutative 75.4%

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

      16. distribute-rgt-neg-in 75.4%

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

      17. metadata-eval 75.4%

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

      18. associate-*l* 75.4%

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

   5. Applied egg-rr 75.4%

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

      1. fma-def 75.4%

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

      2. fma-def 75.4%

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

      3. *-commutative 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in b around 0 69.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{a} \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 69.5%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}}{a}} \]

      2. *-lft-identity 69.5%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}}}{a} \]

      3. distribute-rgt-out 69.9%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 + 4\right)}}}{a} \]

      4. *-commutative 69.9%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot \left(-8 + 4\right)}}{a} \]

      5. metadata-eval 69.9%

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

      6. associate-*r* 69.9%

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

      7. *-commutative 69.9%

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

   10. Simplified 69.9%

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

   if 1.55000000000000006e-35 < b

   1. Initial program 12.8%

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

      1. *-commutative 12.8%

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

   3. Simplified 12.8%

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

   4. Taylor expanded in b around inf 92.2%

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

      1. mul-1-neg 92.2%

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

      2. distribute-neg-frac 92.2%

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

   6. Simplified 92.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 68.4% 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 69.3%

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

      1. *-commutative 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in b around -inf 58.4%

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

      1. +-commutative 58.4%

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

      2. mul-1-neg 58.4%

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

      3. unsub-neg 58.4%

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

   6. Simplified 58.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 36.6%

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

      1. *-commutative 36.6%

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

   3. Simplified 36.6%

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

   4. Taylor expanded in b around inf 62.4%

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

      1. mul-1-neg 62.4%

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

      2. distribute-neg-frac 62.4%

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

   6. Simplified 62.4%

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

4. Final simplification 60.5%

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

Alternative 5: 43.9% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-301}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.6e-301) (/ (- b) a) (/ 0.0 a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.6e-301) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.6d-301)) then
        tmp = -b / a
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.6e-301) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.6e-301:
		tmp = -b / a
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.6e-301)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.6e-301)
		tmp = -b / a;
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.6e-301], N[((-b) / a), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.6000000000000001e-301

   1. Initial program 68.6%

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

      1. *-commutative 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in b around -inf 59.6%

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

      1. associate-*r/ 59.6%

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

      2. mul-1-neg 59.6%

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

   6. Simplified 59.6%

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

   if -6.6000000000000001e-301 < b

   1. Initial program 38.0%

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

      1. *-commutative 38.0%

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

   3. Simplified 38.0%

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

      1. prod-diff 37.7%

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

      2. *-commutative 37.7%

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

      3. fma-def 37.7%

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

      4. associate-+l+ 37.7%

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

      5. pow2 37.7%

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

      6. *-commutative 37.7%

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

      7. distribute-rgt-neg-in 37.7%

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

      8. *-commutative 37.7%

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

      9. distribute-rgt-neg-in 37.7%

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

      10. metadata-eval 37.7%

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

      11. *-commutative 37.7%

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

      12. fma-udef 37.7%

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

      13. distribute-lft-neg-in 37.7%

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

      14. distribute-rgt-neg-in 37.7%

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

      15. *-commutative 37.7%

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

      16. distribute-rgt-neg-in 37.7%

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

      17. metadata-eval 37.7%

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

      18. associate-*l* 37.7%

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

   5. Applied egg-rr 37.7%

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

      1. fma-def 37.7%

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

      2. fma-def 37.7%

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

      3. *-commutative 37.7%

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

   7. Simplified 37.7%

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

      1. clear-num 37.7%

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

      2. inv-pow 37.7%

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

   9. Applied egg-rr 40.7%

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

       1. unpow-1 40.7%

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

       2. *-commutative 40.7%

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

       3. metadata-eval 40.7%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \mathsf{fma}\left(c, a \cdot \color{blue}{\left(-8 + 4\right)}, a \cdot \left(c \cdot 4\right)\right)\right)}\right)\right)}} \]

       4. distribute-rgt-out 40.7%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \mathsf{fma}\left(c, \color{blue}{-8 \cdot a + 4 \cdot a}, a \cdot \left(c \cdot 4\right)\right)\right)}\right)\right)}} \]

       5. associate-*r* 40.7%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \mathsf{fma}\left(c, -8 \cdot a + 4 \cdot a, \color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right)}\right)\right)}} \]

       6. *-commutative 40.7%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \mathsf{fma}\left(c, -8 \cdot a + 4 \cdot a, \color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right)}\right)\right)}} \]

       7. fma-def 40.7%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \color{blue}{c \cdot \left(-8 \cdot a + 4 \cdot a\right) + 4 \cdot \left(a \cdot c\right)}\right)}\right)\right)}} \]

       8. *-commutative 40.7%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \color{blue}{\left(-8 \cdot a + 4 \cdot a\right) \cdot c} + 4 \cdot \left(a \cdot c\right)\right)}\right)\right)}} \]

       9. associate-*r* 40.7%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \left(-8 \cdot a + 4 \cdot a\right) \cdot c + \color{blue}{\left(4 \cdot a\right) \cdot c}\right)}\right)\right)}} \]

       10. distribute-rgt-in 41.0%

           \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \color{blue}{c \cdot \left(\left(-8 \cdot a + 4 \cdot a\right) + 4 \cdot a\right)}\right)}\right)\right)}} \]

       11. distribute-rgt-out 41.0%

           \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, c \cdot \left(\color{blue}{a \cdot \left(-8 + 4\right)} + 4 \cdot a\right)\right)}\right)\right)}} \]

       12. metadata-eval 41.0%

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

       13. *-commutative 41.0%

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

       14. distribute-rgt-out 41.0%

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

       15. metadata-eval 41.0%

           \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, c \cdot \left(a \cdot \color{blue}{0}\right)\right)}\right)\right)}} \]

       16. mul0-rgt 41.0%

           \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, c \cdot \color{blue}{0}\right)}\right)\right)}} \]

       17. mul0-rgt 41.0%

           \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \color{blue}{0}\right)}\right)\right)}} \]

   11. Simplified 41.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}\right)\right)}}} \]

   12. Taylor expanded in a around 0 15.8%

       \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
   13. Step-by-step derivation

       1. associate-*r/ 15.8%

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

       2. distribute-rgt1-in 15.8%

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

       3. metadata-eval 15.8%

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

       4. mul0-lft 15.8%

          \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

       5. metadata-eval 15.8%

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

   14. Simplified 15.8%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-301}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 6: 68.2% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-292}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.65e-292) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.65e-292) {
		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 <= 1.65d-292) 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 <= 1.65e-292) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.65e-292:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.65e-292)
		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 <= 1.65e-292)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.65e-292], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.64999999999999997e-292

   1. Initial program 69.9%

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

      1. *-commutative 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in b around -inf 57.3%

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

      1. associate-*r/ 57.3%

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

      2. mul-1-neg 57.3%

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

   6. Simplified 57.3%

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

   if 1.64999999999999997e-292 < b

   1. Initial program 35.7%

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

      1. *-commutative 35.7%

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

   3. Simplified 35.7%

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

   4. Taylor expanded in b around inf 63.2%

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

      1. mul-1-neg 63.2%

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

      2. distribute-neg-frac 63.2%

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

   6. Simplified 63.2%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-292}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7: 10.9% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 51.8%

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

   1. *-commutative 51.8%

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

3. Simplified 51.8%

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

   1. prod-diff 51.5%

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

   2. *-commutative 51.5%

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

   3. fma-def 51.5%

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

   4. associate-+l+ 51.5%

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

   5. pow2 51.5%

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

   6. *-commutative 51.5%

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

   7. distribute-rgt-neg-in 51.5%

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

   8. *-commutative 51.5%

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

   9. distribute-rgt-neg-in 51.5%

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

   10. metadata-eval 51.5%

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

   11. *-commutative 51.5%

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

   12. fma-udef 51.5%

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

   13. distribute-lft-neg-in 51.5%

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

   14. distribute-rgt-neg-in 51.5%

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

   15. *-commutative 51.5%

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

   16. distribute-rgt-neg-in 51.5%

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

   17. metadata-eval 51.5%

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

   18. associate-*l* 51.5%

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

5. Applied egg-rr 51.5%

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

   1. fma-def 51.5%

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

   2. fma-def 51.5%

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

   3. *-commutative 51.5%

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

7. Simplified 51.5%

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

   1. clear-num 51.4%

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

   2. inv-pow 51.4%

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

9. Applied egg-rr 50.0%

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

    1. unpow-1 50.0%

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

    2. *-commutative 50.0%

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

    3. metadata-eval 50.0%

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \mathsf{fma}\left(c, a \cdot \color{blue}{\left(-8 + 4\right)}, a \cdot \left(c \cdot 4\right)\right)\right)}\right)\right)}} \]

    4. distribute-rgt-out 50.0%

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \mathsf{fma}\left(c, \color{blue}{-8 \cdot a + 4 \cdot a}, a \cdot \left(c \cdot 4\right)\right)\right)}\right)\right)}} \]

    5. associate-*r* 50.0%

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \mathsf{fma}\left(c, -8 \cdot a + 4 \cdot a, \color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right)}\right)\right)}} \]

    6. *-commutative 50.0%

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \mathsf{fma}\left(c, -8 \cdot a + 4 \cdot a, \color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right)}\right)\right)}} \]

    7. fma-def 50.0%

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \color{blue}{c \cdot \left(-8 \cdot a + 4 \cdot a\right) + 4 \cdot \left(a \cdot c\right)}\right)}\right)\right)}} \]

    8. *-commutative 50.0%

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \color{blue}{\left(-8 \cdot a + 4 \cdot a\right) \cdot c} + 4 \cdot \left(a \cdot c\right)\right)}\right)\right)}} \]

    9. associate-*r* 50.0%

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \left(-8 \cdot a + 4 \cdot a\right) \cdot c + \color{blue}{\left(4 \cdot a\right) \cdot c}\right)}\right)\right)}} \]

    10. distribute-rgt-in 50.3%

        \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \color{blue}{c \cdot \left(\left(-8 \cdot a + 4 \cdot a\right) + 4 \cdot a\right)}\right)}\right)\right)}} \]

    11. distribute-rgt-out 50.3%

        \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, c \cdot \left(\color{blue}{a \cdot \left(-8 + 4\right)} + 4 \cdot a\right)\right)}\right)\right)}} \]

    12. metadata-eval 50.3%

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

    13. *-commutative 50.3%

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

    14. distribute-rgt-out 50.3%

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

    15. metadata-eval 50.3%

        \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, c \cdot \left(a \cdot \color{blue}{0}\right)\right)}\right)\right)}} \]

    16. mul0-rgt 50.3%

        \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, c \cdot \color{blue}{0}\right)}\right)\right)}} \]

    17. mul0-rgt 50.3%

        \[\leadsto \frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, \color{blue}{0}\right)}\right)\right)}} \]

11. Simplified 50.3%

    \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}\right)\right)}}} \]

12. Taylor expanded in a around 0 9.9%

    \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
13. Step-by-step derivation

    1. associate-*r/ 9.9%

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

    2. distribute-rgt1-in 9.9%

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

    3. metadata-eval 9.9%

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

    4. mul0-lft 9.9%

       \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 9.9%

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

14. Simplified 9.9%

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

15. Final simplification 9.9%

    \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023310 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))