quadm (p42, negative)

Percentage Accurate: 52.5% → 85.7%

Time: 14.4s

Alternatives: 10

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: 52.5% 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: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-30}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+112}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e-30)
   (- 0.0 (/ c b))
   (if (<= b 5e+112)
     (/ (+ b (sqrt (+ (* b b) (* a (* c -4.0))))) (* a -2.0))
     (- 0.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-30) {
		tmp = 0.0 - (c / b);
	} else if (b <= 5e+112) {
		tmp = (b + sqrt(((b * b) + (a * (c * -4.0))))) / (a * -2.0);
	} 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 <= (-9.5d-30)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 5d+112) then
        tmp = (b + sqrt(((b * b) + (a * (c * (-4.0d0)))))) / (a * (-2.0d0))
    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 <= -9.5e-30) {
		tmp = 0.0 - (c / b);
	} else if (b <= 5e+112) {
		tmp = (b + Math.sqrt(((b * b) + (a * (c * -4.0))))) / (a * -2.0);
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.5e-30:
		tmp = 0.0 - (c / b)
	elif b <= 5e+112:
		tmp = (b + math.sqrt(((b * b) + (a * (c * -4.0))))) / (a * -2.0)
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.5e-30)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 5e+112)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0))))) / Float64(a * -2.0));
	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 <= -9.5e-30)
		tmp = 0.0 - (c / b);
	elseif (b <= 5e+112)
		tmp = (b + sqrt(((b * b) + (a * (c * -4.0))))) / (a * -2.0);
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.5e-30], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+112], N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.49999999999999939e-30

   1. Initial program 7.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 7.5%

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}} \]
   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 96.8%

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

   7. Simplified 96.8%

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

   if -9.49999999999999939e-30 < b < 5e112

   1. Initial program 83.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 83.8%

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

   if 5e112 < b

   1. Initial program 42.3%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 42.3%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. rem-square-sqrt N/A

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

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

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

      11. rem-square-sqrt N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 42.2%

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

   6. Applied egg-rr 42.2%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   8. 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 94.7%

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

   9. Simplified 94.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-30}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+112}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-29}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+113}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e-29)
   (- 0.0 (/ c b))
   (if (<= b 1.1e+113)
     (/ -0.5 (/ a (+ b (sqrt (+ (* b b) (* a (* c -4.0)))))))
     (- 0.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-29) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.1e+113) {
		tmp = -0.5 / (a / (b + sqrt(((b * b) + (a * (c * -4.0))))));
	} 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 <= (-1.5d-29)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.1d+113) then
        tmp = (-0.5d0) / (a / (b + sqrt(((b * b) + (a * (c * (-4.0d0)))))))
    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 <= -1.5e-29) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.1e+113) {
		tmp = -0.5 / (a / (b + Math.sqrt(((b * b) + (a * (c * -4.0))))));
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.5e-29:
		tmp = 0.0 - (c / b)
	elif b <= 1.1e+113:
		tmp = -0.5 / (a / (b + math.sqrt(((b * b) + (a * (c * -4.0))))))
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e-29)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.1e+113)
		tmp = Float64(-0.5 / Float64(a / Float64(b + sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))))));
	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 <= -1.5e-29)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.1e+113)
		tmp = -0.5 / (a / (b + sqrt(((b * b) + (a * (c * -4.0))))));
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e-29], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e+113], N[(-0.5 / N[(a / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5000000000000001e-29

   1. Initial program 7.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 7.5%

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}} \]
   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 96.8%

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

   7. Simplified 96.8%

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

   if -1.5000000000000001e-29 < b < 1.10000000000000005e113

   1. Initial program 83.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 83.8%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. rem-square-sqrt N/A

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

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

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

      11. rem-square-sqrt N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 83.6%

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

   6. Applied egg-rr 83.6%

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

   if 1.10000000000000005e113 < b

   1. Initial program 42.3%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 42.3%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. rem-square-sqrt N/A

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

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

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

      11. rem-square-sqrt N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 42.2%

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

   6. Applied egg-rr 42.2%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   8. 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 94.7%

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

   9. Simplified 94.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-29}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+113}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-29}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+114}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-29)
   (- 0.0 (/ c b))
   (if (<= b 4e+114)
     (* (+ b (sqrt (+ (* b b) (* a (* c -4.0))))) (/ -0.5 a))
     (- 0.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-29) {
		tmp = 0.0 - (c / b);
	} else if (b <= 4e+114) {
		tmp = (b + sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a);
	} 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 <= (-2.9d-29)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 4d+114) then
        tmp = (b + sqrt(((b * b) + (a * (c * (-4.0d0)))))) * ((-0.5d0) / a)
    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 <= -2.9e-29) {
		tmp = 0.0 - (c / b);
	} else if (b <= 4e+114) {
		tmp = (b + Math.sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a);
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.9e-29:
		tmp = 0.0 - (c / b)
	elif b <= 4e+114:
		tmp = (b + math.sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a)
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e-29)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 4e+114)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0))))) * Float64(-0.5 / a));
	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 <= -2.9e-29)
		tmp = 0.0 - (c / b);
	elseif (b <= 4e+114)
		tmp = (b + sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a);
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e-29], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e+114], N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.90000000000000024e-29

   1. Initial program 7.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 7.5%

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}} \]
   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 96.8%

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

   7. Simplified 96.8%

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

   if -2.90000000000000024e-29 < b < 4e114

   1. Initial program 83.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 83.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. rem-square-sqrt N/A

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

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

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

      11. rem-square-sqrt N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 83.6%

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

   6. Applied egg-rr 83.6%

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

   if 4e114 < b

   1. Initial program 42.3%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 42.3%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. rem-square-sqrt N/A

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

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

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

      11. rem-square-sqrt N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 42.2%

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

   6. Applied egg-rr 42.2%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   8. 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 94.7%

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

   9. Simplified 94.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-29}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+114}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-29}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-29)
   (- 0.0 (/ c b))
   (if (<= b 2.4e-45)
     (/ (+ b (sqrt (* c (* a -4.0)))) (* a -2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-29) {
		tmp = 0.0 - (c / b);
	} else if (b <= 2.4e-45) {
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * -2.0);
	} 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 <= (-1d-29)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 2.4d-45) then
        tmp = (b + sqrt((c * (a * (-4.0d0))))) / (a * (-2.0d0))
    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 <= -1e-29) {
		tmp = 0.0 - (c / b);
	} else if (b <= 2.4e-45) {
		tmp = (b + Math.sqrt((c * (a * -4.0)))) / (a * -2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-29:
		tmp = 0.0 - (c / b)
	elif b <= 2.4e-45:
		tmp = (b + math.sqrt((c * (a * -4.0)))) / (a * -2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-29)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 2.4e-45)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * -2.0));
	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 <= -1e-29)
		tmp = 0.0 - (c / b);
	elseif (b <= 2.4e-45)
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * -2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-29], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-45], N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.99999999999999943e-30

   1. Initial program 7.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 7.5%

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}} \]
   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 96.8%

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

   7. Simplified 96.8%

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

   if -9.99999999999999943e-30 < b < 2.3999999999999999e-45

   1. Initial program 81.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 81.8%

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

   5. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 74.2%

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

   7. Simplified 74.2%

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

   if 2.3999999999999999e-45 < b

   1. Initial program 60.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}} \]
   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 84.9%

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

   7. Simplified 84.9%

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

Alternative 5: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{-30}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-47}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.8e-30)
   (- 0.0 (/ c b))
   (if (<= b 1.02e-47)
     (/ -0.5 (/ a (+ b (sqrt (* a (* c -4.0))))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.8e-30) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.02e-47) {
		tmp = -0.5 / (a / (b + sqrt((a * (c * -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 <= (-8.8d-30)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.02d-47) then
        tmp = (-0.5d0) / (a / (b + sqrt((a * (c * (-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 <= -8.8e-30) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.02e-47) {
		tmp = -0.5 / (a / (b + Math.sqrt((a * (c * -4.0)))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.8e-30:
		tmp = 0.0 - (c / b)
	elif b <= 1.02e-47:
		tmp = -0.5 / (a / (b + math.sqrt((a * (c * -4.0)))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.8e-30)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.02e-47)
		tmp = Float64(-0.5 / Float64(a / Float64(b + sqrt(Float64(a * Float64(c * -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 <= -8.8e-30)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.02e-47)
		tmp = -0.5 / (a / (b + sqrt((a * (c * -4.0)))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.8e-30], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.02e-47], N[(-0.5 / N[(a / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $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 < -8.79999999999999933e-30

   1. Initial program 7.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 7.5%

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}} \]
   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 96.8%

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

   7. Simplified 96.8%

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

   if -8.79999999999999933e-30 < b < 1.02000000000000002e-47

   1. Initial program 81.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 81.8%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. rem-square-sqrt N/A

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

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

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

      11. rem-square-sqrt N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 81.7%

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

   6. Applied egg-rr 81.7%

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

   7. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. rem-square-sqrt N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

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

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. *-lowering-*.f64 74.1%

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

   9. Simplified 74.1%

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

   if 1.02000000000000002e-47 < b

   1. Initial program 60.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}} \]
   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 84.9%

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

   7. Simplified 84.9%

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

Alternative 6: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-29}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-50}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.7e-29)
   (- 0.0 (/ c b))
   (if (<= b 1.95e-50)
     (* (/ -0.5 a) (+ b (sqrt (* a (* c -4.0)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.7e-29) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.95e-50) {
		tmp = (-0.5 / a) * (b + sqrt((a * (c * -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 <= (-1.7d-29)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.95d-50) then
        tmp = ((-0.5d0) / a) * (b + sqrt((a * (c * (-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 <= -1.7e-29) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.95e-50) {
		tmp = (-0.5 / a) * (b + Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.7e-29:
		tmp = 0.0 - (c / b)
	elif b <= 1.95e-50:
		tmp = (-0.5 / a) * (b + math.sqrt((a * (c * -4.0))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.7e-29)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.95e-50)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(a * Float64(c * -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 <= -1.7e-29)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.95e-50)
		tmp = (-0.5 / a) * (b + sqrt((a * (c * -4.0))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.7e-29], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-50], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(a * N[(c * -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 < -1.69999999999999986e-29

   1. Initial program 7.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 7.5%

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}} \]
   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 96.8%

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

   7. Simplified 96.8%

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

   if -1.69999999999999986e-29 < b < 1.9500000000000001e-50

   1. Initial program 81.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 81.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. rem-square-sqrt N/A

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

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

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

      11. rem-square-sqrt N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 81.6%

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

   6. Applied egg-rr 81.6%

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

   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. *-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. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 74.0%

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

   9. Simplified 74.0%

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

   if 1.9500000000000001e-50 < b

   1. Initial program 60.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}} \]
   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 84.9%

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

   7. Simplified 84.9%

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

Alternative 7: 67.7% 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 32.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 32.6%

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}} \]
   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 67.5%

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

   7. Simplified 67.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 70.4%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 70.4%

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}} \]
   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 64.6%

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

   7. Simplified 64.6%

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

Alternative 8: 67.5% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = 0.0 - (c / b)
	else:
		tmp = 0.0 - (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(0.0 - 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 = 0.0 - (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[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 32.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 32.6%

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}} \]
   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 67.5%

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

   7. Simplified 67.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 70.4%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 70.4%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. rem-square-sqrt N/A

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

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

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

      11. rem-square-sqrt N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 70.3%

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

   6. Applied egg-rr 70.3%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   8. 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 63.9%

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

   9. Simplified 63.9%

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

4. Final simplification 65.7%

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

Alternative 9: 43.4% accurate, 11.6× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = 0.0 / a;
	} 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 <= (-5d-310)) then
        tmp = 0.0d0 / a
    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 <= -5e-310) {
		tmp = 0.0 / a;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(0.0 / a);
	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 <= -5e-310)
		tmp = 0.0 / a;
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 32.5%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 32.6%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

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

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r* N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. /-lowering-/.f64 N/A

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

   7. Simplified 25.5%

      \[\leadsto \frac{b + \color{blue}{\left(0 - b \cdot \left(1 + \frac{\frac{a \cdot \left(c \cdot -2\right)}{b}}{b}\right)\right)}}{a \cdot -2} \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. +-lft-identity N/A

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

      3. flip-+ N/A

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

      4. associate-*r/ N/A

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

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

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

   9. Applied egg-rr 24.4%

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

   10. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b + -1 \cdot b}{a}} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

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

       2. distribute-rgt1-in N/A

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

       3. metadata-eval N/A

          \[\leadsto \frac{\frac{-1}{2} \cdot \left(0 \cdot b\right)}{a} \]

       4. mul0-lft N/A

          \[\leadsto \frac{\frac{-1}{2} \cdot 0}{a} \]

       5. metadata-eval N/A

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

       6. /-lowering-/.f64 24.0%

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

   12. Simplified 24.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 70.4%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 70.4%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

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

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

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

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

      9. rem-square-sqrt N/A

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

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

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

      11. rem-square-sqrt N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 70.3%

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

   6. Applied egg-rr 70.3%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   8. 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 63.9%

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

   9. Simplified 63.9%

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

4. Final simplification 43.8%

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

Alternative 10: 11.1% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.3%

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

   1. sub-neg N/A

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

   2. distribute-neg-out N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-neg-frac2 N/A

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

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

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

3. Simplified 51.4%

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

5. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

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

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

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

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

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

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

   6. associate-*r/ N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r* N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. /-lowering-/.f64 N/A

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

7. Simplified 13.9%

   \[\leadsto \frac{b + \color{blue}{\left(0 - b \cdot \left(1 + \frac{\frac{a \cdot \left(c \cdot -2\right)}{b}}{b}\right)\right)}}{a \cdot -2} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

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

   2. +-lft-identity N/A

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

   3. flip-+ N/A

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

   4. associate-*r/ N/A

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

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

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

9. Applied egg-rr 13.2%

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

10. Taylor expanded in a around 0

    \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b + -1 \cdot b}{a}} \]
11. Step-by-step derivation

    1. associate-*r/ N/A

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

    2. distribute-rgt1-in N/A

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

    3. metadata-eval N/A

       \[\leadsto \frac{\frac{-1}{2} \cdot \left(0 \cdot b\right)}{a} \]

    4. mul0-lft N/A

       \[\leadsto \frac{\frac{-1}{2} \cdot 0}{a} \]

    5. metadata-eval N/A

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

    6. /-lowering-/.f64 13.5%

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

12. Simplified 13.5%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
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 2024158 
(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)))