Quadratic roots, full range

Percentage Accurate: 51.8% → 85.7%

Time: 42.2s

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: 51.8% 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: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+151}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e+151)
   (- (/ b a))
   (if (<= b 7.6e-34)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e+151) {
		tmp = -(b / a);
	} else if (b <= 7.6e-34) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3d+151)) then
        tmp = -(b / a)
    else if (b <= 7.6d-34) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e+151) {
		tmp = -(b / a);
	} else if (b <= 7.6e-34) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e+151:
		tmp = -(b / a)
	elif b <= 7.6e-34:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e+151)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 7.6e-34)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - 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 <= -3e+151)
		tmp = -(b / a);
	elseif (b <= 7.6e-34)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e+151], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 7.6e-34], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.9999999999999999e151

   1. Initial program 52.3%

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

      1. *-commutative 52.3%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in b around -inf 96.3%

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

      1. associate-*r/ 96.3%

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

      2. mul-1-neg 96.3%

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

   7. Simplified 96.3%

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

   if -2.9999999999999999e151 < b < 7.6000000000000002e-34

   1. Initial program 83.1%

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

   if 7.6000000000000002e-34 < b

   1. Initial program 15.2%

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

      1. *-commutative 15.2%

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

   3. Simplified 15.2%

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

   5. Taylor expanded in b around inf 89.5%

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

      1. associate-*r/ 89.5%

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

      2. mul-1-neg 89.5%

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

   7. Simplified 89.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+151}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-70}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e-70)
   (- (/ b a))
   (if (<= b 1.3e-38)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-70) {
		tmp = -(b / a);
	} else if (b <= 1.3e-38) {
		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 <= (-5.5d-70)) then
        tmp = -(b / a)
    else if (b <= 1.3d-38) 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 <= -5.5e-70) {
		tmp = -(b / a);
	} else if (b <= 1.3e-38) {
		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 <= -5.5e-70:
		tmp = -(b / a)
	elif b <= 1.3e-38:
		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 <= -5.5e-70)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 1.3e-38)
		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 <= -5.5e-70)
		tmp = -(b / a);
	elseif (b <= 1.3e-38)
		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, -5.5e-70], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 1.3e-38], 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 < -5.5000000000000001e-70

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in b around -inf 88.1%

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

      1. associate-*r/ 88.1%

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

      2. mul-1-neg 88.1%

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

   7. Simplified 88.1%

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

   if -5.5000000000000001e-70 < b < 1.30000000000000005e-38

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in b around 0 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-*r* 72.6%

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

   7. Simplified 72.6%

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

      1. add-sqr-sqrt 72.2%

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

      2. pow2 72.2%

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

      3. pow1/2 72.2%

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

      4. sqrt-pow1 72.2%

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

      5. metadata-eval 72.2%

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

   9. Applied egg-rr 72.2%

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

       1. pow-pow 72.6%

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

       2. metadata-eval 72.6%

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

       3. pow1/2 72.6%

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

       4. +-commutative 72.6%

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

       5. unsub-neg 72.6%

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

   11. Applied egg-rr 72.6%

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

   if 1.30000000000000005e-38 < b

   1. Initial program 15.2%

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

      1. *-commutative 15.2%

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

   3. Simplified 15.2%

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

   5. Taylor expanded in b around inf 89.5%

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

      1. associate-*r/ 89.5%

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

      2. mul-1-neg 89.5%

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

   7. Simplified 89.5%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-70}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-68}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-42}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-68)
   (- (/ b a))
   (if (<= b 1.95e-42)
     (/ (+ b (sqrt (* a (* c -4.0)))) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-68) {
		tmp = -(b / a);
	} else if (b <= 1.95e-42) {
		tmp = (b + sqrt((a * (c * -4.0)))) / (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 <= (-2.4d-68)) then
        tmp = -(b / a)
    else if (b <= 1.95d-42) then
        tmp = (b + sqrt((a * (c * (-4.0d0))))) / (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 <= -2.4e-68) {
		tmp = -(b / a);
	} else if (b <= 1.95e-42) {
		tmp = (b + Math.sqrt((a * (c * -4.0)))) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e-68:
		tmp = -(b / a)
	elif b <= 1.95e-42:
		tmp = (b + math.sqrt((a * (c * -4.0)))) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e-68)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 1.95e-42)
		tmp = Float64(Float64(b + sqrt(Float64(a * Float64(c * -4.0)))) / 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 <= -2.4e-68)
		tmp = -(b / a);
	elseif (b <= 1.95e-42)
		tmp = (b + sqrt((a * (c * -4.0)))) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e-68], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 1.95e-42], N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.39999999999999991e-68

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in b around -inf 88.1%

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

      1. associate-*r/ 88.1%

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

      2. mul-1-neg 88.1%

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

   7. Simplified 88.1%

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

   if -2.39999999999999991e-68 < b < 1.9500000000000001e-42

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in b around 0 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-*r* 72.6%

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

   7. Simplified 72.6%

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

      1. *-un-lft-identity 72.6%

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

      2. *-commutative 72.6%

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

      3. times-frac 72.6%

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

      4. metadata-eval 72.6%

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

      5. add-sqr-sqrt 32.3%

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

      6. sqrt-unprod 72.1%

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

      7. sqr-neg 72.1%

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

      8. sqrt-prod 40.6%

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

      9. add-sqr-sqrt 72.2%

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

   9. Applied egg-rr 72.2%

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

       1. metadata-eval 72.2%

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

       2. times-frac 72.2%

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

       3. *-lft-identity 72.2%

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

       4. *-commutative 72.2%

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

   11. Simplified 72.2%

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

   if 1.9500000000000001e-42 < b

   1. Initial program 15.2%

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

      1. *-commutative 15.2%

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

   3. Simplified 15.2%

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

   5. Taylor expanded in b around inf 89.5%

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

      1. associate-*r/ 89.5%

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

      2. mul-1-neg 89.5%

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

   7. Simplified 89.5%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-68}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-42}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-70}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e-70)
   (- (/ b a))
   (if (<= b 1.1e-41)
     (* (/ 0.5 a) (+ b (sqrt (* c (* a -4.0)))))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-70) {
		tmp = -(b / a);
	} else if (b <= 1.1e-41) {
		tmp = (0.5 / a) * (b + sqrt((c * (a * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.5d-70)) then
        tmp = -(b / a)
    else if (b <= 1.1d-41) then
        tmp = (0.5d0 / a) * (b + sqrt((c * (a * (-4.0d0)))))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-70) {
		tmp = -(b / a);
	} else if (b <= 1.1e-41) {
		tmp = (0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e-70:
		tmp = -(b / a)
	elif b <= 1.1e-41:
		tmp = (0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e-70)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 1.1e-41)
		tmp = Float64(Float64(0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.5e-70)
		tmp = -(b / a);
	elseif (b <= 1.1e-41)
		tmp = (0.5 / a) * (b + sqrt((c * (a * -4.0))));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e-70], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 1.1e-41], N[(N[(0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.50000000000000022e-70

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in b around -inf 88.1%

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

      1. associate-*r/ 88.1%

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

      2. mul-1-neg 88.1%

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

   7. Simplified 88.1%

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

   if -4.50000000000000022e-70 < b < 1.1e-41

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in b around 0 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-*r* 72.6%

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

   7. Simplified 72.6%

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

      1. clear-num 72.5%

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

      2. inv-pow 72.5%

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

      3. *-commutative 72.5%

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

      4. *-un-lft-identity 72.5%

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

      5. times-frac 72.5%

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

      6. metadata-eval 72.5%

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

      7. add-sqr-sqrt 32.3%

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

      8. sqrt-unprod 72.0%

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

      9. sqr-neg 72.0%

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

      10. sqrt-prod 40.5%

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

      11. add-sqr-sqrt 72.1%

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

   9. Applied egg-rr 72.1%

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

       1. unpow-1 72.1%

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

       2. associate-*r/ 72.1%

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

       3. *-commutative 72.1%

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

   11. Simplified 72.1%

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

       1. associate-/r/ 72.0%

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

       2. *-commutative 72.0%

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

       3. associate-/r* 72.0%

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

       4. metadata-eval 72.0%

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

       5. *-commutative 72.0%

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

       6. associate-*l* 72.0%

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

   13. Applied egg-rr 72.0%

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

   if 1.1e-41 < b

   1. Initial program 15.2%

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

      1. *-commutative 15.2%

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

   3. Simplified 15.2%

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

   5. Taylor expanded in b around inf 89.5%

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

      1. associate-*r/ 89.5%

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

      2. mul-1-neg 89.5%

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

   7. Simplified 89.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-70}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{-297}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.15e-297) (- (/ b a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.15e-297) {
		tmp = -(b / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.15d-297) then
        tmp = -(b / a)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.15e-297) {
		tmp = -(b / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.15e-297:
		tmp = -(b / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.15e-297)
		tmp = Float64(-Float64(b / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.15e-297)
		tmp = -(b / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.15e-297], (-N[(b / a), $MachinePrecision]), N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1500000000000002e-297

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in b around -inf 69.5%

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

      1. associate-*r/ 69.5%

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

      2. mul-1-neg 69.5%

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

   7. Simplified 69.5%

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

   if 2.1500000000000002e-297 < b

   1. Initial program 33.0%

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

      1. *-commutative 33.0%

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

   3. Simplified 33.0%

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

   5. Taylor expanded in b around inf 64.1%

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

      1. associate-*r/ 64.1%

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

      2. mul-1-neg 64.1%

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

   7. Simplified 64.1%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{-297}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+27}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 5.8e+27) (- (/ b a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.8e+27) {
		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 <= 5.8d+27) 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 <= 5.8e+27) {
		tmp = -(b / a);
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.8e+27:
		tmp = -(b / a)
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.8e+27)
		tmp = Float64(-Float64(b / a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.8e+27)
		tmp = -(b / a);
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.8e+27], (-N[(b / a), $MachinePrecision]), N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.8000000000000002e27

   1. Initial program 70.4%

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

      1. *-commutative 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in b around -inf 51.6%

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

      1. associate-*r/ 51.6%

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

      2. mul-1-neg 51.6%

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

   7. Simplified 51.6%

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

   if 5.8000000000000002e27 < b

   1. Initial program 13.1%

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

      1. *-commutative 13.1%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in b around inf 82.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. associate-*r/ 82.3%

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

      2. frac-2neg 82.3%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 41.0%

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

      5. sqr-neg 41.0%

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

      6. sqrt-prod 40.7%

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

      7. add-sqr-sqrt 40.7%

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

   7. Applied egg-rr 40.7%

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

      1. distribute-frac-neg 40.7%

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

      2. associate-*r/ 40.7%

         \[\leadsto \frac{-\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]

      3. associate-*r/ 40.8%

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

      4. distribute-lft-neg-in 40.8%

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

      5. metadata-eval 40.8%

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

   9. Simplified 40.8%

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

   10. Taylor expanded in a around 0 40.7%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+27}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 10.9% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in b around inf 26.0%

   \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. associate-*r/ 26.0%

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

   2. frac-2neg 26.0%

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

   3. add-sqr-sqrt 1.2%

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

   4. sqrt-unprod 11.8%

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

   5. sqr-neg 11.8%

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

   6. sqrt-prod 10.7%

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

   7. add-sqr-sqrt 12.5%

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

7. Applied egg-rr 12.5%

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

   1. distribute-frac-neg 12.5%

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

   2. associate-*r/ 12.5%

      \[\leadsto \frac{-\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]

   3. associate-*r/ 12.6%

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

   4. distribute-lft-neg-in 12.6%

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

   5. metadata-eval 12.6%

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

9. Simplified 12.6%

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

10. Taylor expanded in a around 0 12.6%

    \[\leadsto \color{blue}{\frac{c}{b}} \]
11. Add Preprocessing

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 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in b around -inf 39.1%

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

   1. associate-*r/ 39.1%

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

   2. mul-1-neg 39.1%

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

7. Simplified 39.1%

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

   1. div-inv 39.0%

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

   2. add-sqr-sqrt 37.7%

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

   3. sqrt-unprod 29.9%

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

   4. sqr-neg 29.9%

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

   5. sqrt-prod 1.8%

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

   6. add-sqr-sqrt 2.5%

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

9. Applied egg-rr 2.5%

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

    1. associate-*r/ 2.5%

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

    2. *-rgt-identity 2.5%

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

11. Simplified 2.5%

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

Reproduce

herbie shell --seed 2024116 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))