Quadratic roots, full range

Percentage Accurate: 52.0% → 85.5%

Time: 14.3s

Alternatives: 10

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: 52.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+125}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+125)
   (- (/ b a))
   (if (<= b 5e-73)
     (/ (- (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 <= -5e+125) {
		tmp = -(b / a);
	} else if (b <= 5e-73) {
		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 <= -5e+125)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 5e-73)
		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, -5e+125], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 5e-73], 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 < -4.99999999999999962e125

   1. Initial program 39.6%

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

      1. *-commutative 39.6%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in b around -inf 95.8%

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

      1. associate-*r/ 95.8%

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

      2. mul-1-neg 95.8%

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

   7. Simplified 95.8%

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

   if -4.99999999999999962e125 < b < 4.9999999999999998e-73

   1. Initial program 84.6%

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

      1. *-commutative 84.6%

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

   3. Simplified 84.6%

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

   if 4.9999999999999998e-73 < b

   1. Initial program 15.5%

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

      1. *-commutative 15.5%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in a around 0 85.9%

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

      1. associate-*r/ 85.9%

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

      2. mul-1-neg 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+125}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{+126}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.15e+126)
   (- (/ b a))
   (if (<= b 9.5e-73)
     (/ (- (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 <= -2.15e+126) {
		tmp = -(b / a);
	} else if (b <= 9.5e-73) {
		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 <= (-2.15d+126)) then
        tmp = -(b / a)
    else if (b <= 9.5d-73) 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 <= -2.15e+126) {
		tmp = -(b / a);
	} else if (b <= 9.5e-73) {
		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 <= -2.15e+126:
		tmp = -(b / a)
	elif b <= 9.5e-73:
		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 <= -2.15e+126)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 9.5e-73)
		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 <= -2.15e+126)
		tmp = -(b / a);
	elseif (b <= 9.5e-73)
		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, -2.15e+126], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 9.5e-73], 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 < -2.1500000000000001e126

   1. Initial program 39.6%

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

      1. *-commutative 39.6%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in b around -inf 95.8%

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

      1. associate-*r/ 95.8%

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

      2. mul-1-neg 95.8%

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

   7. Simplified 95.8%

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

   if -2.1500000000000001e126 < b < 9.50000000000000005e-73

   1. Initial program 84.6%

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

   if 9.50000000000000005e-73 < b

   1. Initial program 15.5%

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

      1. *-commutative 15.5%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in a around 0 85.9%

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

      1. associate-*r/ 85.9%

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

      2. mul-1-neg 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{+126}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{-39}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.65e-39)
   (- (/ b a))
   (if (<= b 5.6e-73)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.65e-39) {
		tmp = -(b / a);
	} else if (b <= 5.6e-73) {
		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 <= (-2.65d-39)) then
        tmp = -(b / a)
    else if (b <= 5.6d-73) 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 <= -2.65e-39) {
		tmp = -(b / a);
	} else if (b <= 5.6e-73) {
		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 <= -2.65e-39:
		tmp = -(b / a)
	elif b <= 5.6e-73:
		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 <= -2.65e-39)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 5.6e-73)
		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 <= -2.65e-39)
		tmp = -(b / a);
	elseif (b <= 5.6e-73)
		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, -2.65e-39], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 5.6e-73], 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 < -2.65000000000000002e-39

   1. Initial program 63.0%

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

      1. *-commutative 63.0%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in b around -inf 86.6%

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

      1. associate-*r/ 86.6%

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

      2. mul-1-neg 86.6%

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

   7. Simplified 86.6%

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

   if -2.65000000000000002e-39 < b < 5.60000000000000023e-73

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in a around inf 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-*r* 70.3%

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

   7. Simplified 70.3%

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

   if 5.60000000000000023e-73 < b

   1. Initial program 15.5%

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

      1. *-commutative 15.5%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in a around 0 85.9%

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

      1. associate-*r/ 85.9%

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

      2. mul-1-neg 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{-39}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-42}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-73}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e-42)
   (- (/ b a))
   (if (<= b 8e-73) (/ 0.5 (/ a (- (sqrt (* -4.0 (* a c))) b))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-42) {
		tmp = -(b / a);
	} else if (b <= 8e-73) {
		tmp = 0.5 / (a / (sqrt((-4.0 * (a * c))) - b));
	} 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-42)) then
        tmp = -(b / a)
    else if (b <= 8d-73) then
        tmp = 0.5d0 / (a / (sqrt(((-4.0d0) * (a * c))) - b))
    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-42) {
		tmp = -(b / a);
	} else if (b <= 8e-73) {
		tmp = 0.5 / (a / (Math.sqrt((-4.0 * (a * c))) - b));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e-42:
		tmp = -(b / a)
	elif b <= 8e-73:
		tmp = 0.5 / (a / (math.sqrt((-4.0 * (a * c))) - b))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e-42)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 8e-73)
		tmp = Float64(0.5 / Float64(a / Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b)));
	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-42)
		tmp = -(b / a);
	elseif (b <= 8e-73)
		tmp = 0.5 / (a / (sqrt((-4.0 * (a * c))) - b));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e-42], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 8e-73], N[(0.5 / N[(a / N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.5e-42

   1. Initial program 63.0%

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

      1. *-commutative 63.0%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in b around -inf 86.6%

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

      1. associate-*r/ 86.6%

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

      2. mul-1-neg 86.6%

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

   7. Simplified 86.6%

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

   if -4.5e-42 < b < 7.99999999999999998e-73

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in a around inf 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-*r* 70.3%

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

   7. Simplified 70.3%

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

      1. div-sub 70.3%

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

      2. *-un-lft-identity 70.3%

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

      3. *-commutative 70.3%

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

      4. times-frac 70.3%

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

      5. metadata-eval 70.3%

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

      6. *-un-lft-identity 70.3%

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

      7. *-commutative 70.3%

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

      8. times-frac 70.3%

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

      9. metadata-eval 70.3%

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

   9. Applied egg-rr 70.3%

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

       1. distribute-lft-out-- 70.3%

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

       2. associate-*r* 70.3%

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

   11. Simplified 70.3%

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

       1. *-commutative 70.3%

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

       2. sub-div 70.3%

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

       3. pow1/2 70.3%

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

       4. metadata-eval 70.3%

          \[\leadsto \frac{{\left(\left(a \cdot c\right) \cdot -4\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} - b}{a} \cdot 0.5 \]

       5. pow-pow 55.3%

          \[\leadsto \frac{\color{blue}{{\left({\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{a} \cdot 0.5 \]

       6. pow1/3 59.1%

          \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}}} - b}{a} \cdot 0.5 \]

       7. metadata-eval 59.1%

          \[\leadsto \frac{\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}} - b}{a} \cdot \color{blue}{\frac{1}{2}} \]

       8. div-inv 59.1%

          \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}} - b}{a}}{2}} \]

       9. associate-/r* 59.1%

          \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}} - b}{a \cdot 2}} \]

       10. clear-num 59.1%

           \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}} - b}}} \]

       11. *-commutative 59.1%

           \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}} - b}} \]

       12. *-un-lft-identity 59.1%

           \[\leadsto \frac{1}{\frac{2 \cdot a}{\color{blue}{1 \cdot \left(\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}} - b\right)}}} \]

       13. times-frac 59.1%

           \[\leadsto \frac{1}{\color{blue}{\frac{2}{1} \cdot \frac{a}{\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}} - b}}} \]

       14. metadata-eval 59.1%

           \[\leadsto \frac{1}{\color{blue}{2} \cdot \frac{a}{\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}} - b}} \]

   13. Applied egg-rr 70.2%

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

       1. associate-/r* 70.2%

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

       2. metadata-eval 70.2%

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

       3. associate-*r* 70.2%

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

       4. *-commutative 70.2%

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

       5. *-commutative 70.2%

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

   15. Simplified 70.2%

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

   if 7.99999999999999998e-73 < b

   1. Initial program 15.5%

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

      1. *-commutative 15.5%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in a around 0 85.9%

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

      1. associate-*r/ 85.9%

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

      2. mul-1-neg 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-42}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-73}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-39}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-73}:\\ \;\;\;\;\left(\sqrt{-4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.8e-39)
   (- (/ b a))
   (if (<= b 4.2e-73)
     (* (- (sqrt (* -4.0 (* a c))) b) (/ 0.5 a))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.8e-39) {
		tmp = -(b / a);
	} else if (b <= 4.2e-73) {
		tmp = (sqrt((-4.0 * (a * c))) - 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 <= (-5.8d-39)) then
        tmp = -(b / a)
    else if (b <= 4.2d-73) then
        tmp = (sqrt(((-4.0d0) * (a * c))) - 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 <= -5.8e-39) {
		tmp = -(b / a);
	} else if (b <= 4.2e-73) {
		tmp = (Math.sqrt((-4.0 * (a * c))) - b) * (0.5 / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.8e-39:
		tmp = -(b / a)
	elif b <= 4.2e-73:
		tmp = (math.sqrt((-4.0 * (a * c))) - b) * (0.5 / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.8e-39)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 4.2e-73)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) - 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 <= -5.8e-39)
		tmp = -(b / a);
	elseif (b <= 4.2e-73)
		tmp = (sqrt((-4.0 * (a * c))) - b) * (0.5 / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.8e-39], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 4.2e-73], N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $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 < -5.79999999999999975e-39

   1. Initial program 63.0%

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

      1. *-commutative 63.0%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in b around -inf 86.6%

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

      1. associate-*r/ 86.6%

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

      2. mul-1-neg 86.6%

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

   7. Simplified 86.6%

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

   if -5.79999999999999975e-39 < b < 4.1999999999999997e-73

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in a around inf 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-*r* 70.3%

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

   7. Simplified 70.3%

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

      1. div-sub 70.3%

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

      2. *-un-lft-identity 70.3%

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

      3. *-commutative 70.3%

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

      4. times-frac 70.3%

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

      5. metadata-eval 70.3%

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

      6. *-un-lft-identity 70.3%

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

      7. *-commutative 70.3%

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

      8. times-frac 70.3%

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

      9. metadata-eval 70.3%

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

   9. Applied egg-rr 70.3%

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

       1. distribute-lft-out-- 70.3%

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

       2. associate-*r* 70.3%

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

   11. Simplified 70.3%

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

       1. *-commutative 70.3%

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

       2. sub-div 70.3%

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

       3. pow1/2 70.3%

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

       4. metadata-eval 70.3%

          \[\leadsto \frac{{\left(\left(a \cdot c\right) \cdot -4\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} - b}{a} \cdot 0.5 \]

       5. pow-pow 55.3%

          \[\leadsto \frac{\color{blue}{{\left({\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{a} \cdot 0.5 \]

       6. pow1/3 59.1%

          \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}}} - b}{a} \cdot 0.5 \]

       7. metadata-eval 59.1%

          \[\leadsto \frac{\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}} - b}{a} \cdot \color{blue}{\frac{1}{2}} \]

       8. div-inv 59.1%

          \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}} - b}{a}}{2}} \]

       9. associate-/r* 59.1%

          \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}} - b}{a \cdot 2}} \]

       10. div-sub 59.1%

           \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}}}{a \cdot 2} - \frac{b}{a \cdot 2}} \]

       11. sub-neg 59.1%

           \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(\left(a \cdot c\right) \cdot -4\right)}^{1.5}}}{a \cdot 2} + \left(-\frac{b}{a \cdot 2}\right)} \]

   13. Applied egg-rr 70.3%

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

       1. sub-neg 70.3%

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

       2. distribute-lft-out-- 70.3%

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

       3. div-sub 70.3%

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

       4. *-commutative 70.3%

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

       5. associate-*l/ 70.3%

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

       6. associate-*r/ 70.1%

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

       7. *-commutative 70.1%

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

       8. associate-*r* 70.1%

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

       9. *-commutative 70.1%

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

       10. *-commutative 70.1%

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

   15. Simplified 70.1%

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

   if 4.1999999999999997e-73 < b

   1. Initial program 15.5%

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

      1. *-commutative 15.5%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in a around 0 85.9%

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

      1. associate-*r/ 85.9%

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

      2. mul-1-neg 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-39}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-73}:\\ \;\;\;\;\left(\sqrt{-4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.38 \cdot 10^{-177}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-152}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.38e-177)
   (- (/ b a))
   (if (<= b 1.35e-152) (* -0.5 (sqrt (* -4.0 (/ c a)))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.38e-177) {
		tmp = -(b / a);
	} else if (b <= 1.35e-152) {
		tmp = -0.5 * sqrt((-4.0 * (c / 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 <= (-1.38d-177)) then
        tmp = -(b / a)
    else if (b <= 1.35d-152) then
        tmp = (-0.5d0) * sqrt(((-4.0d0) * (c / 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 <= -1.38e-177) {
		tmp = -(b / a);
	} else if (b <= 1.35e-152) {
		tmp = -0.5 * Math.sqrt((-4.0 * (c / a)));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.38e-177:
		tmp = -(b / a)
	elif b <= 1.35e-152:
		tmp = -0.5 * math.sqrt((-4.0 * (c / a)))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.38e-177)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 1.35e-152)
		tmp = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.38e-177)
		tmp = -(b / a);
	elseif (b <= 1.35e-152)
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.38e-177], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 1.35e-152], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.37999999999999991e-177

   1. Initial program 65.9%

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

      1. *-commutative 65.9%

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

   3. Simplified 66.0%

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

   5. Taylor expanded in b around -inf 76.6%

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

      1. associate-*r/ 76.6%

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

      2. mul-1-neg 76.6%

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

   7. Simplified 76.6%

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

   if -1.37999999999999991e-177 < b < 1.34999999999999999e-152

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

   3. Simplified 83.4%

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

      1. add-cube-cbrt 82.5%

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

      2. pow3 82.4%

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

   6. Applied egg-rr 82.4%

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

   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. rem-cube-cbrt 0.0%

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

      2. associate-/l* 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 26.5%

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

   9. Simplified 26.5%

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

       1. add-sqr-sqrt 1.4%

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

       2. sqrt-unprod 36.5%

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

       3. pow2 36.5%

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

   11. Applied egg-rr 36.5%

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

       1. unpow2 36.5%

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

       2. *-commutative 36.5%

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

       3. *-commutative 36.5%

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

       4. swap-sqr 36.5%

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

       5. metadata-eval 36.5%

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

       6. metadata-eval 36.5%

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

       7. swap-sqr 36.5%

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

       8. *-lft-identity 36.5%

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

       9. *-lft-identity 36.5%

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

       10. rem-square-sqrt 36.5%

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

       11. associate-*r/ 36.5%

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

       12. *-commutative 36.5%

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

       13. associate-*r/ 36.5%

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

   13. Simplified 36.5%

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

   if 1.34999999999999999e-152 < b

   1. Initial program 23.9%

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

      1. *-commutative 23.9%

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

   3. Simplified 23.9%

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

   5. Taylor expanded in a around 0 78.4%

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

      1. associate-*r/ 78.4%

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

      2. mul-1-neg 78.4%

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

   7. Simplified 78.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.38 \cdot 10^{-177}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-152}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.9% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-255}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.85e-255) (- (/ b a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-255) {
		tmp = -(b / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.85d-255) then
        tmp = -(b / a)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e-255) {
		tmp = -(b / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.85e-255:
		tmp = -(b / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.85e-255)
		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 <= 1.85e-255)
		tmp = -(b / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.85e-255], (-N[(b / a), $MachinePrecision]), N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8500000000000001e-255

   1. Initial program 69.1%

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

      1. *-commutative 69.1%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in b around -inf 64.0%

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

      1. associate-*r/ 64.0%

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

      2. mul-1-neg 64.0%

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

   7. Simplified 64.0%

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

   if 1.8500000000000001e-255 < b

   1. Initial program 32.1%

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

      1. *-commutative 32.1%

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

   3. Simplified 32.1%

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

   5. Taylor expanded in a around 0 69.7%

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

      1. associate-*r/ 69.7%

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

      2. mul-1-neg 69.7%

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

   7. Simplified 69.7%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-255}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.24 \cdot 10^{+33}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1.24e+33) (- (/ b a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.24e+33) {
		tmp = -(b / a);
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.24d+33) then
        tmp = -(b / a)
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.24e+33) {
		tmp = -(b / a);
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.24e+33:
		tmp = -(b / a)
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.24e+33)
		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 <= 1.24e+33)
		tmp = -(b / a);
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.24e+33], (-N[(b / a), $MachinePrecision]), N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.23999999999999997e33

   1. Initial program 67.5%

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

      1. *-commutative 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in b around -inf 48.3%

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

      1. associate-*r/ 48.3%

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

      2. mul-1-neg 48.3%

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

   7. Simplified 48.3%

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

   if 1.23999999999999997e33 < b

   1. Initial program 8.7%

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

      1. *-commutative 8.7%

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

   3. Simplified 8.7%

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

   5. Taylor expanded in a around 0 92.8%

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

      1. associate-*r/ 92.8%

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

      2. mul-1-neg 92.8%

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

   7. Simplified 92.8%

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

      1. add-sqr-sqrt 37.7%

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

      2. sqrt-unprod 39.5%

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

      3. sqr-neg 39.5%

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

      4. sqrt-unprod 10.9%

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

      5. add-sqr-sqrt 19.7%

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

      6. *-un-lft-identity 19.7%

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

   9. Applied egg-rr 19.7%

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

       1. *-lft-identity 19.7%

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

   11. Simplified 19.7%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.24 \cdot 10^{+33}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 10.5% 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 52.8%

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

   1. *-commutative 52.8%

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

3. Simplified 52.8%

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

5. Taylor expanded in a around 0 32.1%

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

   1. associate-*r/ 32.1%

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

   2. mul-1-neg 32.1%

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

7. Simplified 32.1%

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

   1. add-sqr-sqrt 12.5%

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

   2. sqrt-unprod 13.6%

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

   3. sqr-neg 13.6%

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

   4. sqrt-unprod 3.8%

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

   5. add-sqr-sqrt 7.3%

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

   6. *-un-lft-identity 7.3%

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

9. Applied egg-rr 7.3%

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

    1. *-lft-identity 7.3%

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

11. Simplified 7.3%

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

Alternative 10: 2.6% 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 52.8%

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

   1. *-commutative 52.8%

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

3. Simplified 52.8%

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

5. Taylor expanded in b around -inf 36.9%

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

   1. associate-*r/ 36.9%

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

   2. mul-1-neg 36.9%

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

7. Simplified 36.9%

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

   1. neg-mul-1 36.9%

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

   2. add-sqr-sqrt 20.6%

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

   3. times-frac 20.5%

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

9. Applied egg-rr 20.5%

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

    1. frac-times 20.6%

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

    2. neg-mul-1 20.6%

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

    3. add-sqr-sqrt 36.9%

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

    4. add-sqr-sqrt 35.4%

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

    5. sqrt-unprod 25.0%

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

    6. sqr-neg 25.0%

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

    7. sqrt-prod 1.8%

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

    8. add-sqr-sqrt 2.6%

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

11. Applied egg-rr 2.6%

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

Reproduce

herbie shell --seed 2024181 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))