quadp (p42, positive)

Percentage Accurate: 52.4% → 85.8%

Time: 11.5s

Alternatives: 8

Speedup: 11.6×

Specification

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.4% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+125}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+125)
   (/ b (- 0.0 a))
   (if (<= b 5.8e-107)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+125) {
		tmp = b / (0.0 - a);
	} else if (b <= 5.8e-107) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d+125)) then
        tmp = b / (0.0d0 - a)
    else if (b <= 5.8d-107) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = 0.0d0 - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+125) {
		tmp = b / (0.0 - a);
	} else if (b <= 5.8e-107) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+125:
		tmp = b / (0.0 - a)
	elif b <= 5.8e-107:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+125)
		tmp = Float64(b / Float64(0.0 - a));
	elseif (b <= 5.8e-107)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+125)
		tmp = b / (0.0 - a);
	elseif (b <= 5.8e-107)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+125], N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e-107], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.99999999999999962e125

   1. Initial program 51.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(-1 \cdot a\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \left(0 - \color{blue}{a}\right)\right) \]

      7. --lowering--.f64 97.2%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{\_.f64}\left(0, \color{blue}{a}\right)\right) \]

   5. Simplified 97.2%

      \[\leadsto \color{blue}{\frac{b}{0 - a}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

      2. neg-lowering-neg.f64 97.2%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{neg.f64}\left(a\right)\right) \]

   7. Applied egg-rr 97.2%

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

   if -4.99999999999999962e125 < b < 5.7999999999999996e-107

   1. Initial program 82.1%

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

   if 5.7999999999999996e-107 < b

   1. Initial program 11.9%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 93.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   5. Simplified 93.9%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

      3. /-lowering-/.f64 93.9%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(c, b\right)\right) \]

   7. Applied egg-rr 93.9%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+125}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+128}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-102}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e+128)
   (/ b (- 0.0 a))
   (if (<= b 2.6e-102)
     (* (/ -0.5 a) (- b (sqrt (+ (* b b) (* a (* c -4.0))))))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e+128) {
		tmp = b / (0.0 - a);
	} else if (b <= 2.6e-102) {
		tmp = (-0.5 / a) * (b - sqrt(((b * b) + (a * (c * -4.0)))));
	} else {
		tmp = 0.0 - (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+128)) then
        tmp = b / (0.0d0 - a)
    else if (b <= 2.6d-102) then
        tmp = ((-0.5d0) / a) * (b - sqrt(((b * b) + (a * (c * (-4.0d0))))))
    else
        tmp = 0.0d0 - (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+128) {
		tmp = b / (0.0 - a);
	} else if (b <= 2.6e-102) {
		tmp = (-0.5 / a) * (b - Math.sqrt(((b * b) + (a * (c * -4.0)))));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e+128:
		tmp = b / (0.0 - a)
	elif b <= 2.6e-102:
		tmp = (-0.5 / a) * (b - math.sqrt(((b * b) + (a * (c * -4.0)))))
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e+128)
		tmp = Float64(b / Float64(0.0 - a));
	elseif (b <= 2.6e-102)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0))))));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.4e+128)
		tmp = b / (0.0 - a);
	elseif (b <= 2.6e-102)
		tmp = (-0.5 / a) * (b - sqrt(((b * b) + (a * (c * -4.0)))));
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e+128], N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.6e-102], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4000000000000002e128

   1. Initial program 51.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(-1 \cdot a\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \left(0 - \color{blue}{a}\right)\right) \]

      7. --lowering--.f64 97.2%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{\_.f64}\left(0, \color{blue}{a}\right)\right) \]

   5. Simplified 97.2%

      \[\leadsto \color{blue}{\frac{b}{0 - a}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

      2. neg-lowering-neg.f64 97.2%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{neg.f64}\left(a\right)\right) \]

   7. Applied egg-rr 97.2%

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

   if -2.4000000000000002e128 < b < 2.59999999999999986e-102

   1. Initial program 82.1%

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

   4. Applied egg-rr 81.9%

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

   if 2.59999999999999986e-102 < b

   1. Initial program 11.9%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 93.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   5. Simplified 93.9%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

      3. /-lowering-/.f64 93.9%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(c, b\right)\right) \]

   7. Applied egg-rr 93.9%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+128}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-102}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-82}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-82)
   (- (/ c b) (/ b a))
   (if (<= b 1.6e-111)
     (/ (/ (- (sqrt (* a (* c -4.0))) b) a) 2.0)
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-82) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.6e-111) {
		tmp = ((sqrt((a * (c * -4.0))) - b) / a) / 2.0;
	} else {
		tmp = 0.0 - (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 <= (-2.3d-82)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.6d-111) then
        tmp = ((sqrt((a * (c * (-4.0d0)))) - b) / a) / 2.0d0
    else
        tmp = 0.0d0 - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-82) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.6e-111) {
		tmp = ((Math.sqrt((a * (c * -4.0))) - b) / a) / 2.0;
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.3e-82:
		tmp = (c / b) - (b / a)
	elif b <= 1.6e-111:
		tmp = ((math.sqrt((a * (c * -4.0))) - b) / a) / 2.0
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-82)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.6e-111)
		tmp = Float64(Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / a) / 2.0);
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.3e-82)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.6e-111)
		tmp = ((sqrt((a * (c * -4.0))) - b) / a) / 2.0;
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-82], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e-111], N[(N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.29999999999999997e-82

   1. Initial program 65.3%

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

   4. Applied egg-rr 65.2%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right) \cdot b\right) \]

      3. distribute-rgt-neg-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right), \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a} + -1 \cdot \frac{c}{{b}^{2}}\right), \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a} + \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a} - \frac{c}{{b}^{2}}\right), \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{a}\right), \left(\frac{c}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \left(\frac{c}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right), \left(0 - \color{blue}{b}\right)\right) \]

      14. --lowering--.f64 86.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{b}\right)\right) \]

   7. Simplified 86.2%

      \[\leadsto \color{blue}{\left(\frac{1}{a} - \frac{c}{b \cdot b}\right) \cdot \left(0 - b\right)} \]

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{\frac{a \cdot c}{b} + \left(\mathsf{neg}\left(b\right)\right)}{a} \]

      3. unsub-neg N/A

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{\color{blue}{b}}{a}\right)\right) \]

      12. /-lowering-/.f64 86.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   10. Simplified 86.5%

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

   if -2.29999999999999997e-82 < b < 1.5999999999999999e-111

   1. Initial program 78.5%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\left(a \cdot \left(c \cdot -4\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\left(a \cdot \left(-4 \cdot c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(-4 \cdot c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

      6. *-lowering-*.f64 76.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right) \]

   5. Simplified 76.1%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{a \cdot \left(c \cdot -4\right)}}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(2 \cdot a\right), \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot 2\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 2\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 2\right), \left(\sqrt{a \cdot \left(c \cdot -4\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 2\right), \left(\sqrt{a \cdot \left(c \cdot -4\right)} - \color{blue}{b}\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 2\right), \mathsf{\_.f64}\left(\left(\sqrt{a \cdot \left(c \cdot -4\right)}\right), \color{blue}{b}\right)\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 2\right), \mathsf{\_.f64}\left(\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{\frac{1}{2}}\right), b\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 2\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \left(c \cdot -4\right)\right), \frac{1}{2}\right), b\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 2\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right), \frac{1}{2}\right), b\right)\right)\right) \]

      12. *-lowering-*.f64 76.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, 2\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right), \frac{1}{2}\right), b\right)\right)\right) \]

   7. Applied egg-rr 76.0%

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

      1. clear-num N/A

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

      2. remove-double-div N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\left(a \cdot \left(c \cdot -4\right)\right)}^{\frac{1}{2}} - b}{a}\right), \color{blue}{2}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{\frac{1}{2}} - b\right), a\right), 2\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{\frac{1}{2}}\right), b\right), a\right), 2\right) \]

      7. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{a \cdot \left(c \cdot -4\right)}\right), b\right), a\right), 2\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(a \cdot \left(c \cdot -4\right)\right)\right), b\right), a\right), 2\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(a \cdot \left(\sqrt{c \cdot -4} \cdot \sqrt{c \cdot -4}\right)\right)\right), b\right), a\right), 2\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(a \cdot \left({\left(c \cdot -4\right)}^{\frac{1}{2}} \cdot \sqrt{c \cdot -4}\right)\right)\right), b\right), a\right), 2\right) \]

      11. unpow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(a \cdot \left({\left(c \cdot -4\right)}^{\frac{1}{2}} \cdot {\left(c \cdot -4\right)}^{\frac{1}{2}}\right)\right)\right), b\right), a\right), 2\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left({\left(c \cdot -4\right)}^{\frac{1}{2}} \cdot {\left(c \cdot -4\right)}^{\frac{1}{2}}\right)\right)\right), b\right), a\right), 2\right) \]

      13. unpow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(\sqrt{c \cdot -4} \cdot {\left(c \cdot -4\right)}^{\frac{1}{2}}\right)\right)\right), b\right), a\right), 2\right) \]

      14. unpow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(\sqrt{c \cdot -4} \cdot \sqrt{c \cdot -4}\right)\right)\right), b\right), a\right), 2\right) \]

      15. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right), b\right), a\right), 2\right) \]

      16. *-lowering-*.f64 76.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right), b\right), a\right), 2\right) \]

   9. Applied egg-rr 76.1%

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

   if 1.5999999999999999e-111 < b

   1. Initial program 11.9%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 93.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   5. Simplified 93.9%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

      3. /-lowering-/.f64 93.9%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(c, b\right)\right) \]

   7. Applied egg-rr 93.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-82}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-103}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e-76)
   (- (/ c b) (/ b a))
   (if (<= b 7e-103)
     (* (/ -0.5 a) (- b (sqrt (* c (* a -4.0)))))
     (- 0.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-76) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7e-103) {
		tmp = (-0.5 / a) * (b - sqrt((c * (a * -4.0))));
	} else {
		tmp = 0.0 - (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 <= (-1.6d-76)) then
        tmp = (c / b) - (b / a)
    else if (b <= 7d-103) then
        tmp = ((-0.5d0) / a) * (b - sqrt((c * (a * (-4.0d0)))))
    else
        tmp = 0.0d0 - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-76) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7e-103) {
		tmp = (-0.5 / a) * (b - Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.6e-76:
		tmp = (c / b) - (b / a)
	elif b <= 7e-103:
		tmp = (-0.5 / a) * (b - math.sqrt((c * (a * -4.0))))
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e-76)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 7e-103)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.6e-76)
		tmp = (c / b) - (b / a);
	elseif (b <= 7e-103)
		tmp = (-0.5 / a) * (b - sqrt((c * (a * -4.0))));
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e-76], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e-103], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5999999999999999e-76

   1. Initial program 65.3%

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

   4. Applied egg-rr 65.2%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right) \cdot b\right) \]

      3. distribute-rgt-neg-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right), \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a} + -1 \cdot \frac{c}{{b}^{2}}\right), \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a} + \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a} - \frac{c}{{b}^{2}}\right), \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{a}\right), \left(\frac{c}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \left(\frac{c}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right), \left(0 - \color{blue}{b}\right)\right) \]

      14. --lowering--.f64 86.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{b}\right)\right) \]

   7. Simplified 86.2%

      \[\leadsto \color{blue}{\left(\frac{1}{a} - \frac{c}{b \cdot b}\right) \cdot \left(0 - b\right)} \]

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{\frac{a \cdot c}{b} + \left(\mathsf{neg}\left(b\right)\right)}{a} \]

      3. unsub-neg N/A

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{\color{blue}{b}}{a}\right)\right) \]

      12. /-lowering-/.f64 86.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   10. Simplified 86.5%

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

   if -1.5999999999999999e-76 < b < 7.00000000000000032e-103

   1. Initial program 78.5%

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

   4. Applied egg-rr 78.3%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(-4 \cdot a\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-4 \cdot a\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 75.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right) \]

   7. Simplified 75.9%

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

   if 7.00000000000000032e-103 < b

   1. Initial program 11.9%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 93.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   5. Simplified 93.9%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

      3. /-lowering-/.f64 93.9%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(c, b\right)\right) \]

   7. Applied egg-rr 93.9%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-103}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (- (/ c b) (/ b a)) (- 0.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = 0.0 - (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 <= (-2d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = 0.0d0 - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 70.4%

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

   4. Applied egg-rr 70.3%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right) \cdot b\right) \]

      3. distribute-rgt-neg-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right), \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a} + -1 \cdot \frac{c}{{b}^{2}}\right), \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a} + \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a} - \frac{c}{{b}^{2}}\right), \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{a}\right), \left(\frac{c}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \left(\frac{c}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right), \left(0 - \color{blue}{b}\right)\right) \]

      14. --lowering--.f64 62.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{b}\right)\right) \]

   7. Simplified 62.9%

      \[\leadsto \color{blue}{\left(\frac{1}{a} - \frac{c}{b \cdot b}\right) \cdot \left(0 - b\right)} \]

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{\frac{a \cdot c}{b} + \left(\mathsf{neg}\left(b\right)\right)}{a} \]

      3. unsub-neg N/A

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{\color{blue}{b}}{a}\right)\right) \]

      12. /-lowering-/.f64 64.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   10. Simplified 64.6%

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

   if -1.999999999999994e-310 < b

   1. Initial program 21.4%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 80.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   5. Simplified 80.3%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

      3. /-lowering-/.f64 80.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(c, b\right)\right) \]

   7. Applied egg-rr 80.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ b (- 0.0 a)) (- 0.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = b / (0.0 - a);
	} else {
		tmp = 0.0 - (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 <= (-2d-310)) then
        tmp = b / (0.0d0 - a)
    else
        tmp = 0.0d0 - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = b / (0.0 - a);
	} else {
		tmp = 0.0 - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = b / (0.0 - a)
	else:
		tmp = 0.0 - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(b / Float64(0.0 - a));
	else
		tmp = Float64(0.0 - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = b / (0.0 - a);
	else
		tmp = 0.0 - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 70.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(-1 \cdot a\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \left(0 - \color{blue}{a}\right)\right) \]

      7. --lowering--.f64 64.5%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{\_.f64}\left(0, \color{blue}{a}\right)\right) \]

   5. Simplified 64.5%

      \[\leadsto \color{blue}{\frac{b}{0 - a}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

      2. neg-lowering-neg.f64 64.5%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{neg.f64}\left(a\right)\right) \]

   7. Applied egg-rr 64.5%

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

   if -1.999999999999994e-310 < b

   1. Initial program 21.4%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 80.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   5. Simplified 80.3%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

      3. /-lowering-/.f64 80.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(c, b\right)\right) \]

   7. Applied egg-rr 80.3%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{0 - a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.5% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{c}{b} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

   4. /-lowering-/.f64 34.2%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

5. Simplified 34.2%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

   3. /-lowering-/.f64 34.2%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(c, b\right)\right) \]

7. Applied egg-rr 34.2%

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

8. Final simplification 34.2%

   \[\leadsto 0 - \frac{c}{b} \]
9. Add Preprocessing

Alternative 8: 11.3% 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 50.3%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

   4. /-lowering-/.f64 34.2%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

5. Simplified 34.2%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
6. Step-by-step derivation

   1. flip3-- N/A

      \[\leadsto \frac{{0}^{3} - {\left(\frac{c}{b}\right)}^{3}}{\color{blue}{0 \cdot 0 + \left(\frac{c}{b} \cdot \frac{c}{b} + 0 \cdot \frac{c}{b}\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{0 - {\left(\frac{c}{b}\right)}^{3}}{\color{blue}{0} \cdot 0 + \left(\frac{c}{b} \cdot \frac{c}{b} + 0 \cdot \frac{c}{b}\right)} \]

   3. metadata-eval N/A

      \[\leadsto \frac{0 - {\left(\frac{c}{b}\right)}^{3}}{0 + \left(\color{blue}{\frac{c}{b} \cdot \frac{c}{b}} + 0 \cdot \frac{c}{b}\right)} \]

   4. +-lft-identity N/A

      \[\leadsto \frac{0 - {\left(\frac{c}{b}\right)}^{3}}{\frac{c}{b} \cdot \frac{c}{b} + \color{blue}{0 \cdot \frac{c}{b}}} \]

   5. distribute-rgt-out N/A

      \[\leadsto \frac{0 - {\left(\frac{c}{b}\right)}^{3}}{\frac{c}{b} \cdot \color{blue}{\left(\frac{c}{b} + 0\right)}} \]

   6. +-commutative N/A

      \[\leadsto \frac{0 - {\left(\frac{c}{b}\right)}^{3}}{\frac{c}{b} \cdot \left(0 + \color{blue}{\frac{c}{b}}\right)} \]

   7. +-lft-identity N/A

      \[\leadsto \frac{0 - {\left(\frac{c}{b}\right)}^{3}}{\frac{c}{b} \cdot \frac{c}{\color{blue}{b}}} \]

7. Applied egg-rr 1.3%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. associate-/l/ N/A

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

   5. clear-num N/A

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

   6. clear-num N/A

      \[\leadsto \frac{c}{b} \cdot \left(\frac{1}{c} \cdot \frac{1}{\frac{\frac{c}{b \cdot b} \cdot \frac{1}{\frac{\frac{c}{b}}{b}}}{c}}\right) \]

   7. associate-/r* N/A

      \[\leadsto \frac{c}{b} \cdot \left(\frac{1}{c} \cdot \frac{1}{\frac{\frac{c}{b \cdot b} \cdot \frac{1}{\frac{c}{b \cdot b}}}{c}}\right) \]

   8. rgt-mult-inverse N/A

      \[\leadsto \frac{c}{b} \cdot \left(\frac{1}{c} \cdot \frac{1}{\frac{1}{c}}\right) \]

   9. rgt-mult-inverse N/A

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

   10. *-rgt-identity N/A

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

   11. /-lowering-/.f64 10.9%

       \[\leadsto \mathsf{/.f64}\left(c, \color{blue}{b}\right) \]

9. Applied egg-rr 10.9%

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

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024191 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))