quadm (p42, negative)

Percentage Accurate: 52.6% → 86.5%

Time: 13.7s

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: 52.6% 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: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-56}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.8e-56)
   (- 0.0 (/ c b))
   (if (<= b 2.5e+87)
     (/ (/ (+ b (sqrt (+ (* b b) (* c (* a -4.0))))) -2.0) a)
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.8e-56) {
		tmp = 0.0 - (c / b);
	} else if (b <= 2.5e+87) {
		tmp = ((b + sqrt(((b * b) + (c * (a * -4.0))))) / -2.0) / a;
	} else {
		tmp = (c / b) - (b / a);
	}
	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-56)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 2.5d+87) then
        tmp = ((b + sqrt(((b * b) + (c * (a * (-4.0d0)))))) / (-2.0d0)) / a
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.8e-56) {
		tmp = 0.0 - (c / b);
	} else if (b <= 2.5e+87) {
		tmp = ((b + Math.sqrt(((b * b) + (c * (a * -4.0))))) / -2.0) / a;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.8e-56:
		tmp = 0.0 - (c / b)
	elif b <= 2.5e+87:
		tmp = ((b + math.sqrt(((b * b) + (c * (a * -4.0))))) / -2.0) / a
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.8e-56)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 2.5e+87)
		tmp = Float64(Float64(Float64(b + sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))) / -2.0) / a);
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.8e-56)
		tmp = 0.0 - (c / b);
	elseif (b <= 2.5e+87)
		tmp = ((b + sqrt(((b * b) + (c * (a * -4.0))))) / -2.0) / a;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.8e-56], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5e+87], N[(N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.79999999999999982e-56

   1. Initial program 16.3%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 16.3%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. 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 86.2%

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

   7. Simplified 86.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   8. 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 86.2%

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

   9. Applied egg-rr 86.2%

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

   if -5.79999999999999982e-56 < b < 2.4999999999999999e87

   1. Initial program 79.9%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 79.9%

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

   if 2.4999999999999999e87 < b

   1. Initial program 44.5%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 44.5%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 96.0%

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

   7. Simplified 96.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-56}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.46 \cdot 10^{-53}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+87}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.46e-53)
   (- 0.0 (/ c b))
   (if (<= b 1.3e+87)
     (* (/ -0.5 a) (+ b (sqrt (+ (* b b) (* -4.0 (* c a))))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.46e-53) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.3e+87) {
		tmp = (-0.5 / a) * (b + sqrt(((b * b) + (-4.0 * (c * a)))));
	} else {
		tmp = (c / b) - (b / a);
	}
	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 <= (-1.46d-53)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.3d+87) then
        tmp = ((-0.5d0) / a) * (b + sqrt(((b * b) + ((-4.0d0) * (c * a)))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.46e-53) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.3e+87) {
		tmp = (-0.5 / a) * (b + Math.sqrt(((b * b) + (-4.0 * (c * a)))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.46e-53:
		tmp = 0.0 - (c / b)
	elif b <= 1.3e+87:
		tmp = (-0.5 / a) * (b + math.sqrt(((b * b) + (-4.0 * (c * a)))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.46e-53)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.3e+87)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a))))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.46e-53)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.3e+87)
		tmp = (-0.5 / a) * (b + sqrt(((b * b) + (-4.0 * (c * a)))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.46e-53], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.3e+87], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.45999999999999989e-53

   1. Initial program 16.3%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 16.3%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. 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 86.2%

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

   7. Simplified 86.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   8. 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 86.2%

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

   9. Applied egg-rr 86.2%

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

   if -1.45999999999999989e-53 < b < 1.29999999999999999e87

   1. Initial program 79.9%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 79.9%

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

      1. div-inv N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. rem-square-sqrt N/A

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

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

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

      10. rem-square-sqrt N/A

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

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

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

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

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-lowering-*.f64 79.8%

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

   6. Applied egg-rr 79.8%

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

   if 1.29999999999999999e87 < b

   1. Initial program 44.5%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 44.5%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 96.0%

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

   7. Simplified 96.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.46 \cdot 10^{-53}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+87}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{-53}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.4e-53)
   (- 0.0 (/ c b))
   (if (<= b 1.5e-66)
     (/ (/ (+ b (sqrt (* -4.0 (* c a)))) -2.0) a)
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e-53) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.5e-66) {
		tmp = ((b + sqrt((-4.0 * (c * a)))) / -2.0) / a;
	} else {
		tmp = (c / b) - (b / a);
	}
	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 <= (-8.4d-53)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.5d-66) then
        tmp = ((b + sqrt(((-4.0d0) * (c * a)))) / (-2.0d0)) / a
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e-53) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.5e-66) {
		tmp = ((b + Math.sqrt((-4.0 * (c * a)))) / -2.0) / a;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.4e-53:
		tmp = 0.0 - (c / b)
	elif b <= 1.5e-66:
		tmp = ((b + math.sqrt((-4.0 * (c * a)))) / -2.0) / a
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.4e-53)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.5e-66)
		tmp = Float64(Float64(Float64(b + sqrt(Float64(-4.0 * Float64(c * a)))) / -2.0) / a);
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.4e-53)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.5e-66)
		tmp = ((b + sqrt((-4.0 * (c * a)))) / -2.0) / a;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.4e-53], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.5e-66], N[(N[(N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.3999999999999991e-53

   1. Initial program 16.3%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 16.3%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. 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 86.2%

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

   7. Simplified 86.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   8. 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 86.2%

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

   9. Applied egg-rr 86.2%

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

   if -8.3999999999999991e-53 < b < 1.5000000000000001e-66

   1. Initial program 80.2%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 80.2%

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

   5. Taylor expanded in b around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 77.6%

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

   7. Simplified 77.6%

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

   if 1.5000000000000001e-66 < b

   1. Initial program 56.2%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 56.2%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 85.7%

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

   7. Simplified 85.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{-53}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-64}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{-68}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-64)
   (- 0.0 (/ c b))
   (if (<= b 1.32e-68)
     (* (/ -0.5 a) (+ b (sqrt (* c (* a -4.0)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-64) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.32e-68) {
		tmp = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	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 <= (-6d-64)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.32d-68) then
        tmp = ((-0.5d0) / a) * (b + sqrt((c * (a * (-4.0d0)))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-64) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.32e-68) {
		tmp = (-0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e-64:
		tmp = 0.0 - (c / b)
	elif b <= 1.32e-68:
		tmp = (-0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e-64)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.32e-68)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6e-64)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.32e-68)
		tmp = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e-64], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.32e-68], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.0000000000000001e-64

   1. Initial program 16.3%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 16.3%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. 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 86.2%

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

   7. Simplified 86.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   8. 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 86.2%

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

   9. Applied egg-rr 86.2%

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

   if -6.0000000000000001e-64 < b < 1.32e-68

   1. Initial program 80.2%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 80.2%

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

      1. div-inv N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. rem-square-sqrt N/A

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

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

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

      10. rem-square-sqrt N/A

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

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

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

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

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-lowering-*.f64 80.1%

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

   6. Applied egg-rr 80.1%

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

   7. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 77.6%

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

   9. Simplified 77.6%

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

   if 1.32e-68 < b

   1. Initial program 56.2%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 56.2%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 85.7%

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

   7. Simplified 85.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-64}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{-68}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 35.4%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 35.4%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. 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 64.2%

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

   7. Simplified 64.2%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   8. 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 64.2%

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

   9. Applied egg-rr 64.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 60.7%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 60.7%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 71.6%

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

   7. Simplified 71.6%

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

4. Final simplification 67.7%

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

Alternative 6: 68.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-296}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-296) (- 0.0 (/ c b)) (- 0.0 (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-296) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-296)) then
        tmp = 0.0d0 - (c / b)
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-296) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-296:
		tmp = 0.0 - (c / b)
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-296)
		tmp = Float64(0.0 - Float64(c / b));
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-296)
		tmp = 0.0 - (c / b);
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-296], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1e-296

   1. Initial program 34.9%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 34.9%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. 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 64.7%

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

   7. Simplified 64.7%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   8. 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 64.7%

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

   9. Applied egg-rr 64.7%

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

   if -1e-296 < b

   1. Initial program 61.0%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 61.0%

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

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. 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-lowering-neg.f64 70.8%

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

   7. Simplified 70.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-296}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.7% 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 47.5%

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

   1. associate-/r* N/A

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

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

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

3. Simplified 47.5%

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

5. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
6. 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 34.4%

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

7. Simplified 34.4%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
8. 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 34.4%

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

9. Applied egg-rr 34.4%

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

10. Final simplification 34.4%

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

Alternative 8: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.5%

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

   1. associate-/r* N/A

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

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

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

3. Simplified 47.5%

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

   1. flip-+ N/A

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

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

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

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

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

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

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

   5. rem-square-sqrt N/A

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

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

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

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

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

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

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

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

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

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

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

   13. rem-square-sqrt N/A

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

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

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

6. Applied egg-rr 27.3%

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

7. Taylor expanded in b around -inf

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

   1. associate-*r* N/A

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

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

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

   3. mul-1-neg N/A

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

   4. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right), \mathsf{\_.f64}\left(b, \mathsf{*.f64}\left(\left(0 - b\right), \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), -2\right), a\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right), \mathsf{\_.f64}\left(b, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), -2\right), a\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right), \mathsf{\_.f64}\left(b, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{+.f64}\left(1, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right)\right), -2\right), a\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right), \mathsf{\_.f64}\left(b, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{+.f64}\left(1, \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{2}}\right)\right)\right)\right)\right), -2\right), a\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right), \mathsf{\_.f64}\left(b, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right)\right)\right)\right), -2\right), a\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right), \mathsf{\_.f64}\left(b, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(a \cdot c\right) \cdot -2\right), \left({b}^{2}\right)\right)\right)\right)\right)\right), -2\right), a\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right), \mathsf{\_.f64}\left(b, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(a \cdot c\right), -2\right), \left({b}^{2}\right)\right)\right)\right)\right)\right), -2\right), a\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right), \mathsf{\_.f64}\left(b, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -2\right), \left({b}^{2}\right)\right)\right)\right)\right)\right), -2\right), a\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right), \mathsf{\_.f64}\left(b, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -2\right), \left(b \cdot b\right)\right)\right)\right)\right)\right), -2\right), a\right) \]

   13. *-lowering-*.f64 4.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right), \mathsf{\_.f64}\left(b, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), -2\right), a\right) \]

9. Simplified 4.8%

   \[\leadsto \frac{\frac{\frac{b \cdot b - \left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}{b - \color{blue}{\left(0 - b\right) \cdot \left(1 + \frac{\left(a \cdot c\right) \cdot -2}{b \cdot b}\right)}}}{-2}}{a} \]

10. Taylor expanded in b around 0

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

    1. /-lowering-/.f64 2.7%

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

12. Simplified 2.7%

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

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

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) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

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 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	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 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	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 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	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(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	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 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	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[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024139 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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) (/ c (- sqtD (/ b 2))) (/ (+ (/ b 2) sqtD) (- a)))))

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