Quadratic roots, full range

Percentage Accurate: 52.2% → 86.2%

Time: 16.5s

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: 52.2% 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: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+120}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e+120)
   (/ (- b) a)
   (if (<= b 1.55e-53)
     (/ (- (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 <= -3e+120) {
		tmp = -b / a;
	} else if (b <= 1.55e-53) {
		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 <= (-3d+120)) then
        tmp = -b / a
    else if (b <= 1.55d-53) 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 <= -3e+120) {
		tmp = -b / a;
	} else if (b <= 1.55e-53) {
		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 <= -3e+120:
		tmp = -b / a
	elif b <= 1.55e-53:
		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 <= -3e+120)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.55e-53)
		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 <= -3e+120)
		tmp = -b / a;
	elseif (b <= 1.55e-53)
		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, -3e+120], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.55e-53], 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 < -3e120

   1. Initial program 45.5%

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

   2. Taylor expanded in b around -inf 98.1%

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

      1. associate-*r/ 98.1%

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

      2. mul-1-neg 98.1%

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

   4. Simplified 98.1%

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

   if -3e120 < b < 1.55000000000000008e-53

   1. Initial program 87.5%

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

   if 1.55000000000000008e-53 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. neg-mul-1 85.0%

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

   4. Simplified 85.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+120}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-59}:\\ \;\;\;\;\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e-41)
   (- (/ c b) (/ b a))
   (if (<= b 4.4e-59)
     (* (- (sqrt (* a (* c -4.0))) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-41) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.4e-59) {
		tmp = (sqrt((a * (c * -4.0))) - b) * (0.5 / 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 <= (-4.2d-41)) then
        tmp = (c / b) - (b / a)
    else if (b <= 4.4d-59) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) * (0.5d0 / 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 <= -4.2e-41) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.4e-59) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.2e-41:
		tmp = (c / b) - (b / a)
	elif b <= 4.4e-59:
		tmp = (math.sqrt((a * (c * -4.0))) - b) * (0.5 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e-41)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4.4e-59)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.2e-41)
		tmp = (c / b) - (b / a);
	elseif (b <= 4.4e-59)
		tmp = (sqrt((a * (c * -4.0))) - b) * (0.5 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e-41], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-59], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.20000000000000025e-41

   1. Initial program 69.2%

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

   2. Taylor expanded in b around -inf 88.5%

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

      1. +-commutative 88.5%

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

      2. mul-1-neg 88.5%

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

      3. unsub-neg 88.5%

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

   4. Simplified 88.5%

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

   if -4.20000000000000025e-41 < b < 4.3999999999999998e-59

   1. Initial program 82.5%

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

      1. add-sqr-sqrt 82.2%

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

      2. pow2 82.2%

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

      3. pow1/2 82.2%

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

      4. sqrt-pow1 82.2%

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

      5. fma-neg 82.2%

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

      6. *-commutative 82.2%

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

      7. distribute-rgt-neg-in 82.2%

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

      8. *-commutative 82.2%

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

      9. distribute-rgt-neg-in 82.2%

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

      10. metadata-eval 82.2%

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

      11. metadata-eval 82.2%

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

   3. Applied egg-rr 82.2%

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

   4. Taylor expanded in b around 0 77.2%

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

   5. Taylor expanded in a around 0 30.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{\left(e^{0.25 \cdot \left(\log a + \log \left(-4 \cdot c\right)\right)}\right)}^{2} - b}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 30.4%

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

   7. Simplified 77.3%

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

   if 4.3999999999999998e-59 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. neg-mul-1 85.0%

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

   4. Simplified 85.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-59}:\\ \;\;\;\;\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-41}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e-41)
   (- (/ c b) (/ b a))
   (if (<= b 1.8e-55)
     (/ (- (sqrt (* -4.0 (* a c))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e-41) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.8e-55) {
		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 <= (-7d-41)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.8d-55) 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 <= -7e-41) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.8e-55) {
		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 <= -7e-41:
		tmp = (c / b) - (b / a)
	elif b <= 1.8e-55:
		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 <= -7e-41)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.8e-55)
		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 <= -7e-41)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.8e-55)
		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, -7e-41], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-55], 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 < -6.9999999999999999e-41

   1. Initial program 69.2%

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

   2. Taylor expanded in b around -inf 88.5%

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

      1. +-commutative 88.5%

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

      2. mul-1-neg 88.5%

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

      3. unsub-neg 88.5%

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

   4. Simplified 88.5%

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

   if -6.9999999999999999e-41 < b < 1.8e-55

   1. Initial program 82.5%

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

      1. add-sqr-sqrt 82.2%

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

      2. pow2 82.2%

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

      3. pow1/2 82.2%

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

      4. sqrt-pow1 82.2%

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

      5. fma-neg 82.2%

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

      6. *-commutative 82.2%

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

      7. distribute-rgt-neg-in 82.2%

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

      8. *-commutative 82.2%

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

      9. distribute-rgt-neg-in 82.2%

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

      10. metadata-eval 82.2%

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

      11. metadata-eval 82.2%

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

   3. Applied egg-rr 82.2%

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

   4. Taylor expanded in c around inf 50.8%

      \[\leadsto \frac{\color{blue}{{\left(e^{0.25 \cdot \left(\log \left(-4 \cdot a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right)}\right)}^{2} - b}}{2 \cdot a} \]
   5. Step-by-step derivation

   6. Simplified 77.5%

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

   if 1.8e-55 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. neg-mul-1 85.0%

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

   4. Simplified 85.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-41}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e-41)
   (- (/ c b) (/ b a))
   (if (<= b 2.8e-49)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e-41) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.8e-49) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.4d-41)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.8d-49) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e-41) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.8e-49) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.4e-41:
		tmp = (c / b) - (b / a)
	elif b <= 2.8e-49:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.4e-41)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.8e-49)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -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 <= -4.4e-41)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.8e-49)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.4e-41], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.8e-49], N[(N[(N[Sqrt[N[(a * N[(c * -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 < -4.4e-41

   1. Initial program 69.2%

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

   2. Taylor expanded in b around -inf 88.5%

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

      1. +-commutative 88.5%

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

      2. mul-1-neg 88.5%

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

      3. unsub-neg 88.5%

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

   4. Simplified 88.5%

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

   if -4.4e-41 < b < 2.79999999999999997e-49

   1. Initial program 82.5%

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

      1. add-sqr-sqrt 82.2%

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

      2. pow2 82.2%

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

      3. pow1/2 82.2%

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

      4. sqrt-pow1 82.2%

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

      5. fma-neg 82.2%

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

      6. *-commutative 82.2%

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

      7. distribute-rgt-neg-in 82.2%

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

      8. *-commutative 82.2%

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

      9. distribute-rgt-neg-in 82.2%

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

      10. metadata-eval 82.2%

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

      11. metadata-eval 82.2%

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

   3. Applied egg-rr 82.2%

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

   4. Taylor expanded in b around 0 77.2%

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

   5. Taylor expanded in a around 0 30.4%

      \[\leadsto \frac{\color{blue}{{\left(e^{0.25 \cdot \left(\log a + \log \left(-4 \cdot c\right)\right)}\right)}^{2} - b}}{2 \cdot a} \]
   6. Step-by-step derivation

      1. *-commutative 30.4%

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

      2. log-prod 72.5%

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

      3. log-pow 72.5%

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

      4. rem-exp-log 77.3%

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

      5. unpow2 77.3%

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

      6. pow-sqr 77.6%

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

      7. metadata-eval 77.6%

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

      8. unpow1/2 77.6%

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

   7. Simplified 77.6%

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

   if 2.79999999999999997e-49 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. neg-mul-1 85.0%

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

   4. Simplified 85.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 68.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-311) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-311) {
		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 <= (-1d-311)) 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 <= -1e-311) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-311:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-311)
		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 <= -1e-311)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-311], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.99999999999948e-312

   1. Initial program 73.8%

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

   2. Taylor expanded in b around -inf 69.7%

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

      1. +-commutative 69.7%

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

      2. mul-1-neg 69.7%

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

      3. unsub-neg 69.7%

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

   4. Simplified 69.7%

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

   if -9.99999999999948e-312 < b

   1. Initial program 33.5%

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

   2. Taylor expanded in b around inf 65.2%

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

      1. associate-*r/ 65.2%

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

      2. neg-mul-1 65.2%

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

   4. Simplified 65.2%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 68.2% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-311) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-311) {
		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 <= (-1d-311)) 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 <= -1e-311) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-311:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-311)
		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 <= -1e-311)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-311], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.99999999999948e-312

   1. Initial program 73.8%

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

   2. Taylor expanded in b around -inf 69.5%

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

      1. associate-*r/ 69.5%

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

      2. mul-1-neg 69.5%

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

   4. Simplified 69.5%

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

   if -9.99999999999948e-312 < b

   1. Initial program 33.5%

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

   2. Taylor expanded in b around inf 65.2%

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

      1. associate-*r/ 65.2%

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

      2. neg-mul-1 65.2%

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

   4. Simplified 65.2%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7: 35.6% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

2. Taylor expanded in b around -inf 35.6%

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

   1. associate-*r/ 35.6%

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

   2. mul-1-neg 35.6%

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

4. Simplified 35.6%

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

5. Final simplification 35.6%

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

Alternative 8: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

   1. clear-num 53.3%

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

   2. associate-/r/ 53.2%

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

   3. associate-/r* 53.2%

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

   4. metadata-eval 53.2%

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

   5. add-sqr-sqrt 36.2%

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

   6. sqrt-unprod 51.3%

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

   7. sqr-neg 51.3%

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

   8. sqrt-prod 15.2%

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

   9. add-sqr-sqrt 33.9%

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

   10. sub-neg 33.9%

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

   11. add-sqr-sqrt 30.9%

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

   12. hypot-def 28.2%

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

   13. *-commutative 28.2%

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

   14. distribute-rgt-neg-in 28.2%

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

   15. *-commutative 28.2%

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

   16. distribute-rgt-neg-in 28.2%

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

   17. metadata-eval 28.2%

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

3. Applied egg-rr 28.2%

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

4. Taylor expanded in a around 0 2.5%

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

5. Final simplification 2.5%

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

Reproduce

herbie shell --seed 2023283 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))