Quadratic roots, full range

Percentage Accurate: 51.4% → 84.7%

Time: 11.4s

Alternatives: 8

Speedup: 12.9×

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: 51.4% 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: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+19)
   (/ b (- a))
   (if (<= b 1.65e-130)
     (/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+19) {
		tmp = b / -a;
	} else if (b <= 1.65e-130) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+19)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.65e-130)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+19], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.65e-130], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $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 < -1.5e19

   1. Initial program 57.0%

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

      1. *-commutative 57.0%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in b around -inf 94.1%

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

      1. associate-*r/ 94.1%

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

      2. mul-1-neg 94.1%

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

   7. Simplified 94.1%

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

   if -1.5e19 < b < 1.6499999999999999e-130

   1. Initial program 84.1%

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

      1. *-commutative 84.1%

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

   3. Simplified 84.1%

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

   if 1.6499999999999999e-130 < b

   1. Initial program 21.0%

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

      1. *-commutative 21.0%

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

   3. Simplified 21.0%

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

   5. Taylor expanded in a around 0 83.8%

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

      1. associate-*r/ 83.8%

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

      2. mul-1-neg 83.8%

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

   7. Simplified 83.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-122}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+19)
   (/ b (- a))
   (if (<= b 1.45e-122)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+19) {
		tmp = b / -a;
	} else if (b <= 1.45e-122) {
		tmp = (sqrt(((b * b) - (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+19)) then
        tmp = b / -a
    else if (b <= 1.45d-122) then
        tmp = (sqrt(((b * b) - (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+19) {
		tmp = b / -a;
	} else if (b <= 1.45e-122) {
		tmp = (Math.sqrt(((b * b) - (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+19:
		tmp = b / -a
	elif b <= 1.45e-122:
		tmp = (math.sqrt(((b * b) - (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+19)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.45e-122)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e+19)
		tmp = b / -a;
	elseif (b <= 1.45e-122)
		tmp = (sqrt(((b * b) - (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+19], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.45e-122], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $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 < -1e19

   1. Initial program 57.0%

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

      1. *-commutative 57.0%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in b around -inf 94.1%

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

      1. associate-*r/ 94.1%

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

      2. mul-1-neg 94.1%

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

   7. Simplified 94.1%

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

   if -1e19 < b < 1.4500000000000001e-122

   1. Initial program 84.1%

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

   if 1.4500000000000001e-122 < b

   1. Initial program 21.0%

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

      1. *-commutative 21.0%

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

   3. Simplified 21.0%

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

   5. Taylor expanded in a around 0 83.8%

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

      1. associate-*r/ 83.8%

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

      2. mul-1-neg 83.8%

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

   7. Simplified 83.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-122}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-62}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e-62)
   (/ b (- a))
   (if (<= b 5.8e-123)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-62) {
		tmp = b / -a;
	} else if (b <= 5.8e-123) {
		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 <= (-8d-62)) then
        tmp = b / -a
    else if (b <= 5.8d-123) 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 <= -8e-62) {
		tmp = b / -a;
	} else if (b <= 5.8e-123) {
		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 <= -8e-62:
		tmp = b / -a
	elif b <= 5.8e-123:
		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 <= -8e-62)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 5.8e-123)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8e-62)
		tmp = b / -a;
	elseif (b <= 5.8e-123)
		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, -8e-62], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 5.8e-123], 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 < -8.0000000000000003e-62

   1. Initial program 63.1%

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

      1. *-commutative 63.1%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in b around -inf 87.5%

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

      1. associate-*r/ 87.5%

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

      2. mul-1-neg 87.5%

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

   7. Simplified 87.5%

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

   if -8.0000000000000003e-62 < b < 5.80000000000000007e-123

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in a around inf 78.5%

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

      1. *-commutative 78.5%

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

      2. associate-*r* 78.5%

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

   7. Simplified 78.5%

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

   if 5.80000000000000007e-123 < b

   1. Initial program 21.0%

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

      1. *-commutative 21.0%

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

   3. Simplified 21.0%

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

   5. Taylor expanded in a around 0 83.8%

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

      1. associate-*r/ 83.8%

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

      2. mul-1-neg 83.8%

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

   7. Simplified 83.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-62}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-122}:\\ \;\;\;\;\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.4e-58)
   (/ b (- a))
   (if (<= b 1.06e-122)
     (* (- (sqrt (* a (* c -4.0))) b) (/ 0.5 a))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.4e-58) {
		tmp = b / -a;
	} else if (b <= 1.06e-122) {
		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 <= (-9.4d-58)) then
        tmp = b / -a
    else if (b <= 1.06d-122) 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 <= -9.4e-58) {
		tmp = b / -a;
	} else if (b <= 1.06e-122) {
		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 <= -9.4e-58:
		tmp = b / -a
	elif b <= 1.06e-122:
		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 <= -9.4e-58)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.06e-122)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.4e-58)
		tmp = b / -a;
	elseif (b <= 1.06e-122)
		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, -9.4e-58], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.06e-122], 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 < -9.39999999999999989e-58

   1. Initial program 63.1%

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

      1. *-commutative 63.1%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in b around -inf 87.5%

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

      1. associate-*r/ 87.5%

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

      2. mul-1-neg 87.5%

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

   7. Simplified 87.5%

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

   if -9.39999999999999989e-58 < b < 1.05999999999999994e-122

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

   3. Simplified 82.9%

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

      1. div-sub 82.9%

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

      2. sub-neg 82.9%

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

      3. div-inv 82.8%

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

      4. pow2 82.8%

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

      5. *-commutative 82.8%

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

      6. associate-/r* 82.8%

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

      7. metadata-eval 82.8%

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

      8. div-inv 82.8%

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

      9. *-commutative 82.8%

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

      10. associate-/r* 82.8%

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

      11. metadata-eval 82.8%

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

   6. Applied egg-rr 82.8%

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

      1. sub-neg 82.8%

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

      2. distribute-rgt-out-- 82.8%

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

   8. Simplified 82.8%

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

   9. Taylor expanded in a around inf 78.4%

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

       1. *-commutative 78.4%

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

       2. associate-*r* 78.4%

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

   11. Simplified 78.4%

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

   if 1.05999999999999994e-122 < b

   1. Initial program 21.0%

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

      1. *-commutative 21.0%

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

   3. Simplified 21.0%

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

   5. Taylor expanded in a around 0 83.8%

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

      1. associate-*r/ 83.8%

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

      2. mul-1-neg 83.8%

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

   7. Simplified 83.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-122}:\\ \;\;\;\;\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-182}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-146}:\\ \;\;\;\;-0.5 \cdot \left(-\sqrt{c \cdot \frac{-4}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.05e-182)
   (/ b (- a))
   (if (<= b 3e-146) (* -0.5 (- (sqrt (* c (/ -4.0 a))))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.05e-182) {
		tmp = b / -a;
	} else if (b <= 3e-146) {
		tmp = -0.5 * -sqrt((c * (-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 <= (-1.05d-182)) then
        tmp = b / -a
    else if (b <= 3d-146) then
        tmp = (-0.5d0) * -sqrt((c * ((-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 <= -1.05e-182) {
		tmp = b / -a;
	} else if (b <= 3e-146) {
		tmp = -0.5 * -Math.sqrt((c * (-4.0 / a)));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.05e-182:
		tmp = b / -a
	elif b <= 3e-146:
		tmp = -0.5 * -math.sqrt((c * (-4.0 / a)))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.05e-182)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 3e-146)
		tmp = Float64(-0.5 * Float64(-sqrt(Float64(c * Float64(-4.0 / a)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.05e-182)
		tmp = b / -a;
	elseif (b <= 3e-146)
		tmp = -0.5 * -sqrt((c * (-4.0 / a)));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.05e-182], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 3e-146], N[(-0.5 * (-N[Sqrt[N[(c * N[(-4.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05e-182

   1. Initial program 65.7%

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

      1. *-commutative 65.7%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in b around -inf 77.8%

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

      1. associate-*r/ 77.8%

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

      2. mul-1-neg 77.8%

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

   7. Simplified 77.8%

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

   if -1.05e-182 < b < 3.00000000000000019e-146

   1. Initial program 84.9%

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

      1. *-commutative 84.9%

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

   3. Simplified 84.9%

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

      1. add-cube-cbrt 84.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{\left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow3 84.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{\left(4 \cdot a\right) \cdot c}\right)}^{3}}}}{a \cdot 2} \]

      3. associate-*l* 84.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}^{3}}}{a \cdot 2} \]

   6. Applied egg-rr 84.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   7. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      3. rem-square-sqrt 43.7%

         \[\leadsto -0.5 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      4. rem-cube-cbrt 44.1%

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

      5. associate-/l* 44.0%

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

   9. Simplified 44.0%

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

   if 3.00000000000000019e-146 < b

   1. Initial program 21.0%

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

      1. *-commutative 21.0%

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

   3. Simplified 21.0%

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

   5. Taylor expanded in a around 0 83.8%

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

      1. associate-*r/ 83.8%

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

      2. mul-1-neg 83.8%

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

   7. Simplified 83.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-182}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-146}:\\ \;\;\;\;-0.5 \cdot \left(-\sqrt{c \cdot \frac{-4}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.8% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 68.8%

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

      1. *-commutative 68.8%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in b around -inf 67.4%

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

      1. associate-*r/ 67.4%

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

      2. mul-1-neg 67.4%

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

   7. Simplified 67.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 28.3%

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

      1. *-commutative 28.3%

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

   3. Simplified 28.3%

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

   5. Taylor expanded in a around 0 74.9%

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

      1. associate-*r/ 74.9%

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

      2. mul-1-neg 74.9%

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

   7. Simplified 74.9%

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

4. Final simplification 71.6%

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

Alternative 7: 43.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -5e-310) (/ b (- a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b / -a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b / -a
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b / (-a)), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 68.8%

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

      1. *-commutative 68.8%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in b around -inf 67.4%

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

      1. associate-*r/ 67.4%

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

      2. mul-1-neg 67.4%

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

   7. Simplified 67.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 28.3%

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

      1. *-commutative 28.3%

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

   3. Simplified 28.3%

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

      1. neg-sub0 28.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. flip-- 27.8%

         \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - b \cdot b}{0 + b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. metadata-eval 27.8%

         \[\leadsto \frac{\frac{\color{blue}{0} - b \cdot b}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 27.8%

         \[\leadsto \frac{\frac{0 - \color{blue}{{b}^{2}}}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. add-sqr-sqrt 24.6%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. sqrt-prod 19.4%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      7. sqr-neg 19.4%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      8. sqrt-unprod 0.0%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      9. add-sqr-sqrt 21.0%

         \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\left(-b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      10. sub-neg 21.0%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{0 - b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      11. neg-sub0 21.0%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{-b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      12. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      13. sqrt-unprod 19.4%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      14. sqr-neg 19.4%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\sqrt{\color{blue}{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      15. sqrt-prod 24.6%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      16. add-sqr-sqrt 27.8%

          \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 27.8%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{2}}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. neg-sub0 27.8%

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

   8. Simplified 27.8%

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

   9. Taylor expanded in a around 0 15.5%

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

       1. associate-*r/ 15.5%

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

       2. distribute-rgt1-in 15.5%

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

       3. metadata-eval 15.5%

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

       4. mul0-lft 15.5%

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

       5. metadata-eval 15.5%

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

   11. Simplified 15.5%

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

   12. Taylor expanded in a around 0 15.5%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 11.0% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 46.1%

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

   1. *-commutative 46.1%

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

3. Simplified 46.1%

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

   1. neg-sub0 46.1%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. flip-- 45.8%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - b \cdot b}{0 + b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 45.8%

      \[\leadsto \frac{\frac{\color{blue}{0} - b \cdot b}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 45.8%

      \[\leadsto \frac{\frac{0 - \color{blue}{{b}^{2}}}{0 + b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. add-sqr-sqrt 13.7%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. sqrt-prod 16.0%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   7. sqr-neg 16.0%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. sqrt-unprod 11.7%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   9. add-sqr-sqrt 22.8%

      \[\leadsto \frac{\frac{0 - {b}^{2}}{0 + \color{blue}{\left(-b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   10. sub-neg 22.8%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{0 - b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   11. neg-sub0 22.8%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{-b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   12. add-sqr-sqrt 11.7%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   13. sqrt-unprod 16.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   14. sqr-neg 16.0%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\sqrt{\color{blue}{b \cdot b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   15. sqrt-prod 13.7%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   16. add-sqr-sqrt 45.8%

       \[\leadsto \frac{\frac{0 - {b}^{2}}{\color{blue}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 45.8%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{2}}{b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. neg-sub0 45.8%

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

8. Simplified 45.8%

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

9. Taylor expanded in a around 0 9.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
10. 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%

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

11. Simplified 9.9%

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

12. Taylor expanded in a around 0 9.9%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))