The quadratic formula (r2)

Percentage Accurate: 51.8% → 85.9%

Time: 12.1s

Alternatives: 8

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.8% 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: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-59}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e-59)
   (- 0.0 (/ c b))
   (if (<= b 1.85e+90)
     (/ (/ (+ b (sqrt (+ (* b b) (* c (* a -4.0))))) -2.0) a)
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-59) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.85e+90) {
		tmp = ((b + sqrt(((b * b) + (c * (a * -4.0))))) / -2.0) / a;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.2d-59)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.85d+90) then
        tmp = ((b + sqrt(((b * b) + (c * (a * (-4.0d0)))))) / (-2.0d0)) / a
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-59) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.85e+90) {
		tmp = ((b + Math.sqrt(((b * b) + (c * (a * -4.0))))) / -2.0) / a;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.2e-59:
		tmp = 0.0 - (c / b)
	elif b <= 1.85e+90:
		tmp = ((b + math.sqrt(((b * b) + (c * (a * -4.0))))) / -2.0) / a
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e-59)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.85e+90)
		tmp = Float64(Float64(Float64(b + sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))) / -2.0) / a);
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.2e-59)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.85e+90)
		tmp = ((b + sqrt(((b * b) + (c * (a * -4.0))))) / -2.0) / a;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e-59], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.85e+90], N[(N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.19999999999999993e-59

   1. Initial program 15.3%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 15.3%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 87.4%

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

   7. Simplified 87.4%

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

   if -4.19999999999999993e-59 < b < 1.85e90

   1. Initial program 86.9%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 86.9%

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

   if 1.85e90 < b

   1. Initial program 61.1%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 61.1%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 95.7%

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

   7. Simplified 95.7%

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

Alternative 2: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-65}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-65)
   (- 0.0 (/ c b))
   (if (<= b 6.8e+89)
     (* (/ -0.5 a) (+ b (sqrt (+ (* b b) (* -4.0 (* c a))))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-65) {
		tmp = 0.0 - (c / b);
	} else if (b <= 6.8e+89) {
		tmp = (-0.5 / a) * (b + sqrt(((b * b) + (-4.0 * (c * a)))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3d-65)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 6.8d+89) then
        tmp = ((-0.5d0) / a) * (b + sqrt(((b * b) + ((-4.0d0) * (c * a)))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-65) {
		tmp = 0.0 - (c / b);
	} else if (b <= 6.8e+89) {
		tmp = (-0.5 / a) * (b + Math.sqrt(((b * b) + (-4.0 * (c * a)))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e-65:
		tmp = 0.0 - (c / b)
	elif b <= 6.8e+89:
		tmp = (-0.5 / a) * (b + math.sqrt(((b * b) + (-4.0 * (c * a)))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-65)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 6.8e+89)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a))))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e-65)
		tmp = 0.0 - (c / b);
	elseif (b <= 6.8e+89)
		tmp = (-0.5 / a) * (b + sqrt(((b * b) + (-4.0 * (c * a)))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-65], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e+89], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.99999999999999998e-65

   1. Initial program 15.3%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 15.3%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 87.4%

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

   7. Simplified 87.4%

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

   if -2.99999999999999998e-65 < b < 6.8000000000000004e89

   1. Initial program 86.9%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 86.9%

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

      1. div-inv N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. metadata-eval N/A

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

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

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

      8. rem-square-sqrt N/A

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

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

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

      10. rem-square-sqrt N/A

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

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

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

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

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-lowering-*.f64 86.7%

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

   6. Applied egg-rr 86.7%

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

   if 6.8000000000000004e89 < b

   1. Initial program 61.1%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 61.1%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 95.7%

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

   7. Simplified 95.7%

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

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.75 \cdot 10^{-65}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-122}:\\ \;\;\;\;\frac{\frac{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.75e-65)
   (- 0.0 (/ c b))
   (if (<= b 1.9e-122)
     (/ (/ (+ b (sqrt (* -4.0 (* c a)))) -2.0) a)
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.75e-65) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.9e-122) {
		tmp = ((b + sqrt((-4.0 * (c * a)))) / -2.0) / a;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.75d-65)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.9d-122) then
        tmp = ((b + sqrt(((-4.0d0) * (c * a)))) / (-2.0d0)) / a
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.75e-65) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.9e-122) {
		tmp = ((b + Math.sqrt((-4.0 * (c * a)))) / -2.0) / a;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.75e-65:
		tmp = 0.0 - (c / b)
	elif b <= 1.9e-122:
		tmp = ((b + math.sqrt((-4.0 * (c * a)))) / -2.0) / a
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.75e-65)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.9e-122)
		tmp = Float64(Float64(Float64(b + sqrt(Float64(-4.0 * Float64(c * a)))) / -2.0) / a);
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.75e-65)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.9e-122)
		tmp = ((b + sqrt((-4.0 * (c * a)))) / -2.0) / a;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.75e-65], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.9e-122], N[(N[(N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7500000000000001e-65

   1. Initial program 15.3%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 15.3%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 87.4%

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

   7. Simplified 87.4%

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

   if -3.7500000000000001e-65 < b < 1.9e-122

   1. Initial program 78.4%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 78.4%

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

   5. Taylor expanded in b around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 78.3%

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

   7. Simplified 78.3%

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

   if 1.9e-122 < b

   1. Initial program 77.1%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 77.1%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 84.1%

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

   7. Simplified 84.1%

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

Alternative 4: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{-65}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-122}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.8e-65)
   (- 0.0 (/ c b))
   (if (<= b 1.95e-122)
     (* (/ -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 <= -9.8e-65) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.95e-122) {
		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 <= (-9.8d-65)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.95d-122) 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 <= -9.8e-65) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.95e-122) {
		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 <= -9.8e-65:
		tmp = 0.0 - (c / b)
	elif b <= 1.95e-122:
		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 <= -9.8e-65)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.95e-122)
		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 <= -9.8e-65)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.95e-122)
		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, -9.8e-65], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-122], 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 < -9.79999999999999929e-65

   1. Initial program 15.3%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 15.3%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 87.4%

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

   7. Simplified 87.4%

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

   if -9.79999999999999929e-65 < b < 1.94999999999999995e-122

   1. Initial program 78.4%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 78.4%

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

      1. div-inv N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. metadata-eval N/A

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

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

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

      8. rem-square-sqrt N/A

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

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

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

      10. rem-square-sqrt N/A

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

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

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

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

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-lowering-*.f64 78.3%

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

   6. Applied egg-rr 78.3%

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

   7. Taylor expanded in b around 0

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

      1. *-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 78.2%

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

   9. Simplified 78.2%

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

   if 1.94999999999999995e-122 < b

   1. Initial program 77.1%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 77.1%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 84.1%

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

   7. Simplified 84.1%

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

Alternative 5: 69.0% 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 32.4%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 32.4%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 67.4%

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

   7. Simplified 67.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 77.9%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 77.9%

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

   5. Taylor expanded in 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 75.3%

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

   7. Simplified 75.3%

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

Alternative 6: 68.8% 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 32.4%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 32.4%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 67.4%

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

   7. Simplified 67.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 77.9%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 77.9%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 74.8%

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

   7. Simplified 74.8%

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

4. Final simplification 71.5%

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

Alternative 7: 44.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-53}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -2.05e-53) 0.0 (- 0.0 (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-53) {
		tmp = 0.0;
	} 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 <= (-2.05d-53)) then
        tmp = 0.0d0
    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 <= -2.05e-53) {
		tmp = 0.0;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.05e-53:
		tmp = 0.0
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.05e-53)
		tmp = 0.0;
	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 <= -2.05e-53)
		tmp = 0.0;
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.05e-53], 0.0, N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.05e-53

   1. Initial program 15.6%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 15.6%

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

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. associate-/l* N/A

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

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

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

      20. associate-/l* N/A

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

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

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

      22. /-lowering-/.f64 28.0%

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

   7. Simplified 28.0%

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

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b + -1 \cdot b}{a}} \]
   9. 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 27.5%

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

   10. Simplified 27.5%

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

       1. div0 27.5%

          \[\leadsto 0 \]

   12. Applied egg-rr 27.5%

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

   if -2.05e-53 < b

   1. Initial program 76.6%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 76.6%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 61.1%

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

   7. Simplified 61.1%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-53}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 11.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 57.8%

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

   1. associate-/r* N/A

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

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

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

3. Simplified 57.8%

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

5. Taylor expanded in b around -inf

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

   1. associate-*r* N/A

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

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

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

   3. mul-1-neg N/A

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

   4. neg-sub0 N/A

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

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

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

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

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

   7. associate-*r/ N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. associate-*r/ N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

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

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

   16. *-commutative N/A

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

   17. associate-*r* N/A

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

   18. associate-/l* N/A

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

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

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

   20. associate-/l* N/A

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

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

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

   22. /-lowering-/.f64 10.3%

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

7. Simplified 10.3%

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

8. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b + -1 \cdot b}{a}} \]
9. 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 10.4%

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

10. Simplified 10.4%

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

    1. div0 10.4%

       \[\leadsto 0 \]

12. Applied egg-rr 10.4%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Developer Target 1: 70.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t\_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 a))))
     (/ (- (- b) t_0) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp = 0
	if b < 0.0:
		tmp = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a))));
	else
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	tmp = 0.0;
	if (b < 0.0)
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	else
		tmp = (-b - t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024144 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (sqrt (- (* b b) (* 4 (* a c)))))) (let ((r1 (/ (+ (- b) d) (* 2 a)))) (let ((r2 (/ (- (- b) d) (* 2 a)))) (if (< b 0) (/ c (* a r1)) r2)))))

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