quadm (p42, negative)

Percentage Accurate: 52.7% → 85.3%

Time: 18.7s

Alternatives: 9

Speedup: 12.9×

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: 52.7% 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: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-63}:\\ \;\;\;\;\frac{0.5}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e-63)
   (/ 0.5 (+ (* -0.5 (/ b c)) (* 0.5 (/ a b))))
   (if (<= b 1.8e+130)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-63) {
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)));
	} else if (b <= 1.8e+130) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (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 <= (-1.35d-63)) then
        tmp = 0.5d0 / (((-0.5d0) * (b / c)) + (0.5d0 * (a / b)))
    else if (b <= 1.8d+130) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (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 <= -1.35e-63) {
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)));
	} else if (b <= 1.8e+130) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.35e-63:
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)))
	elif b <= 1.8e+130:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e-63)
		tmp = Float64(0.5 / Float64(Float64(-0.5 * Float64(b / c)) + Float64(0.5 * Float64(a / b))));
	elseif (b <= 1.8e+130)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / 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 <= -1.35e-63)
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)));
	elseif (b <= 1.8e+130)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e-63], N[(0.5 / N[(N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e+130], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $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 < -1.3500000000000001e-63

   1. Initial program 17.3%

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

   3. Applied egg-rr 17.3%

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

      1. *-commutative 17.3%

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

      2. associate-/r* 17.3%

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

      3. metadata-eval 17.3%

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

   5. Simplified 17.3%

      \[\leadsto \color{blue}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}} \]
   6. Step-by-step derivation

      1. frac-2neg 17.3%

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

      2. associate-*r/ 17.3%

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

      3. metadata-eval 17.3%

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

   7. Applied egg-rr 17.3%

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

      1. *-commutative 17.3%

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

      2. associate-/l* 17.3%

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

   9. Simplified 17.3%

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

   10. Taylor expanded in b around -inf 88.2%

       \[\leadsto \frac{0.5}{\color{blue}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}} \]

   if -1.3500000000000001e-63 < b < 1.8000000000000001e130

   1. Initial program 89.7%

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

   if 1.8000000000000001e130 < b

   1. Initial program 49.6%

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

   2. Taylor expanded in b around inf 97.1%

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

      1. +-commutative 97.1%

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

      2. mul-1-neg 97.1%

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

      3. unsub-neg 97.1%

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

   4. Simplified 97.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-63}:\\ \;\;\;\;\frac{0.5}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{0.5}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-66}:\\ \;\;\;\;b \cdot \frac{-0.5}{a} - 0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e-68)
   (/ 0.5 (+ (* -0.5 (/ b c)) (* 0.5 (/ a b))))
   (if (<= b 1.55e-66)
     (- (* b (/ -0.5 a)) (* 0.5 (/ (sqrt (* a (* c -4.0))) a)))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e-68) {
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)));
	} else if (b <= 1.55e-66) {
		tmp = (b * (-0.5 / a)) - (0.5 * (sqrt((a * (c * -4.0))) / 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 <= (-4.6d-68)) then
        tmp = 0.5d0 / (((-0.5d0) * (b / c)) + (0.5d0 * (a / b)))
    else if (b <= 1.55d-66) then
        tmp = (b * ((-0.5d0) / a)) - (0.5d0 * (sqrt((a * (c * (-4.0d0)))) / 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 <= -4.6e-68) {
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)));
	} else if (b <= 1.55e-66) {
		tmp = (b * (-0.5 / a)) - (0.5 * (Math.sqrt((a * (c * -4.0))) / a));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.6e-68:
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)))
	elif b <= 1.55e-66:
		tmp = (b * (-0.5 / a)) - (0.5 * (math.sqrt((a * (c * -4.0))) / a))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.6e-68)
		tmp = Float64(0.5 / Float64(Float64(-0.5 * Float64(b / c)) + Float64(0.5 * Float64(a / b))));
	elseif (b <= 1.55e-66)
		tmp = Float64(Float64(b * Float64(-0.5 / a)) - Float64(0.5 * Float64(sqrt(Float64(a * Float64(c * -4.0))) / 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 <= -4.6e-68)
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)));
	elseif (b <= 1.55e-66)
		tmp = (b * (-0.5 / a)) - (0.5 * (sqrt((a * (c * -4.0))) / a));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.6e-68], N[(0.5 / N[(N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.55e-66], N[(N[(b * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.59999999999999994e-68

   1. Initial program 17.3%

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

   3. Applied egg-rr 17.3%

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

      1. *-commutative 17.3%

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

      2. associate-/r* 17.3%

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

      3. metadata-eval 17.3%

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

   5. Simplified 17.3%

      \[\leadsto \color{blue}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}} \]
   6. Step-by-step derivation

      1. frac-2neg 17.3%

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

      2. associate-*r/ 17.3%

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

      3. metadata-eval 17.3%

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

   7. Applied egg-rr 17.3%

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

      1. *-commutative 17.3%

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

      2. associate-/l* 17.3%

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

   9. Simplified 17.3%

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

   10. Taylor expanded in b around -inf 88.2%

       \[\leadsto \frac{0.5}{\color{blue}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}} \]

   if -4.59999999999999994e-68 < b < 1.5499999999999999e-66

   1. Initial program 86.2%

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

   2. Taylor expanded in b around 0 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-*r* 81.4%

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

   4. Simplified 81.4%

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

      1. div-sub 81.4%

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

      2. sub-neg 81.4%

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

      3. div-inv 81.4%

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

      4. add-sqr-sqrt 40.4%

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

      5. sqrt-unprod 79.3%

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

      6. sqr-neg 79.3%

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

      7. unpow2 79.3%

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

      8. unpow2 79.3%

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

      9. sqrt-prod 39.1%

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

      10. add-sqr-sqrt 79.4%

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

      11. metadata-eval 79.4%

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

      12. associate-/r* 79.4%

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

      13. metadata-eval 79.4%

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

      14. metadata-eval 79.4%

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

      15. div-inv 79.2%

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

      16. metadata-eval 79.2%

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

      17. associate-/r* 79.2%

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

      18. metadata-eval 79.2%

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

      19. metadata-eval 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. sub-neg 79.2%

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

      2. distribute-rgt-out-- 79.2%

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

   8. Simplified 79.2%

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

   10. Applied egg-rr 81.4%

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

   if 1.5499999999999999e-66 < b

   1. Initial program 68.5%

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

   2. Taylor expanded in b around inf 88.4%

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

      1. +-commutative 88.4%

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

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

   4. Simplified 88.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{0.5}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-66}:\\ \;\;\;\;b \cdot \frac{-0.5}{a} - 0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.3e-65)
   (/ 0.5 (+ (* -0.5 (/ b c)) (* 0.5 (/ a b))))
   (if (<= b 5.4e-66)
     (/ (- (- b) (sqrt (* a (* c -4.0)))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-65) {
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)));
	} else if (b <= 5.4e-66) {
		tmp = (-b - sqrt((a * (c * -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 <= (-3.3d-65)) then
        tmp = 0.5d0 / (((-0.5d0) * (b / c)) + (0.5d0 * (a / b)))
    else if (b <= 5.4d-66) then
        tmp = (-b - sqrt((a * (c * (-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 <= -3.3e-65) {
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)));
	} else if (b <= 5.4e-66) {
		tmp = (-b - Math.sqrt((a * (c * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.3e-65:
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)))
	elif b <= 5.4e-66:
		tmp = (-b - math.sqrt((a * (c * -4.0)))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.3e-65)
		tmp = Float64(0.5 / Float64(Float64(-0.5 * Float64(b / c)) + Float64(0.5 * Float64(a / b))));
	elseif (b <= 5.4e-66)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(a * Float64(c * -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 <= -3.3e-65)
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)));
	elseif (b <= 5.4e-66)
		tmp = (-b - sqrt((a * (c * -4.0)))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.3e-65], N[(0.5 / N[(N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.4e-66], N[(N[((-b) - N[Sqrt[N[(a * N[(c * -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 < -3.3000000000000001e-65

   1. Initial program 17.3%

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

   3. Applied egg-rr 17.3%

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

      1. *-commutative 17.3%

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

      2. associate-/r* 17.3%

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

      3. metadata-eval 17.3%

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

   5. Simplified 17.3%

      \[\leadsto \color{blue}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}} \]
   6. Step-by-step derivation

      1. frac-2neg 17.3%

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

      2. associate-*r/ 17.3%

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

      3. metadata-eval 17.3%

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

   7. Applied egg-rr 17.3%

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

      1. *-commutative 17.3%

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

      2. associate-/l* 17.3%

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

   9. Simplified 17.3%

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

   10. Taylor expanded in b around -inf 88.2%

       \[\leadsto \frac{0.5}{\color{blue}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}} \]

   if -3.3000000000000001e-65 < b < 5.39999999999999992e-66

   1. Initial program 86.2%

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

   2. Taylor expanded in b around 0 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-*r* 81.4%

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

   4. Simplified 81.4%

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

   if 5.39999999999999992e-66 < b

   1. Initial program 68.5%

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

   2. Taylor expanded in b around inf 88.4%

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

      1. +-commutative 88.4%

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

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

   4. Simplified 88.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-65}:\\ \;\;\;\;\frac{0.5}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-67}:\\ \;\;\;\;\frac{0.5}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.25e-67)
   (/ 0.5 (+ (* -0.5 (/ b c)) (* 0.5 (/ a b))))
   (if (<= b 3.5e-69)
     (* (/ -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 <= -2.25e-67) {
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)));
	} else if (b <= 3.5e-69) {
		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 <= (-2.25d-67)) then
        tmp = 0.5d0 / (((-0.5d0) * (b / c)) + (0.5d0 * (a / b)))
    else if (b <= 3.5d-69) 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 <= -2.25e-67) {
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)));
	} else if (b <= 3.5e-69) {
		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 <= -2.25e-67:
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)))
	elif b <= 3.5e-69:
		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 <= -2.25e-67)
		tmp = Float64(0.5 / Float64(Float64(-0.5 * Float64(b / c)) + Float64(0.5 * Float64(a / b))));
	elseif (b <= 3.5e-69)
		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 <= -2.25e-67)
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)));
	elseif (b <= 3.5e-69)
		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, -2.25e-67], N[(0.5 / N[(N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.5e-69], 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 < -2.25000000000000008e-67

   1. Initial program 17.3%

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

   3. Applied egg-rr 17.3%

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

      1. *-commutative 17.3%

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

      2. associate-/r* 17.3%

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

      3. metadata-eval 17.3%

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

   5. Simplified 17.3%

      \[\leadsto \color{blue}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}} \]
   6. Step-by-step derivation

      1. frac-2neg 17.3%

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

      2. associate-*r/ 17.3%

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

      3. metadata-eval 17.3%

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

   7. Applied egg-rr 17.3%

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

      1. *-commutative 17.3%

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

      2. associate-/l* 17.3%

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

   9. Simplified 17.3%

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

   10. Taylor expanded in b around -inf 88.2%

       \[\leadsto \frac{0.5}{\color{blue}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}} \]

   if -2.25000000000000008e-67 < b < 3.5000000000000001e-69

   1. Initial program 86.2%

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

   2. Taylor expanded in b around 0 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-*r* 81.4%

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

   4. Simplified 81.4%

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

      1. div-sub 81.4%

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

      2. sub-neg 81.4%

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

      3. div-inv 81.4%

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

      4. add-sqr-sqrt 40.4%

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

      5. sqrt-unprod 79.3%

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

      6. sqr-neg 79.3%

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

      7. unpow2 79.3%

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

      8. unpow2 79.3%

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

      9. sqrt-prod 39.1%

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

      10. add-sqr-sqrt 79.4%

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

      11. metadata-eval 79.4%

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

      12. associate-/r* 79.4%

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

      13. metadata-eval 79.4%

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

      14. metadata-eval 79.4%

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

      15. div-inv 79.2%

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

      16. metadata-eval 79.2%

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

      17. associate-/r* 79.2%

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

      18. metadata-eval 79.2%

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

      19. metadata-eval 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. sub-neg 79.2%

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

      2. distribute-rgt-out-- 79.2%

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

   8. Simplified 79.2%

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

   10. Applied egg-rr 81.4%

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

       1. *-commutative 81.4%

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

       2. distribute-lft-neg-in 81.4%

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

       3. metadata-eval 81.4%

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

       4. associate-*r/ 81.4%

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

       5. associate-*l/ 81.3%

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

       6. distribute-lft-in 81.2%

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

   12. Simplified 81.2%

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

   if 3.5000000000000001e-69 < b

   1. Initial program 68.5%

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

   2. Taylor expanded in b around inf 88.4%

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

      1. +-commutative 88.4%

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

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

   4. Simplified 88.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-67}:\\ \;\;\;\;\frac{0.5}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5: 67.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{0.5}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310)
   (/ 0.5 (+ (* -0.5 (/ b c)) (* 0.5 (/ a b))))
   (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / 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 <= (-4d-310)) then
        tmp = 0.5d0 / (((-0.5d0) * (b / c)) + (0.5d0 * (a / 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 <= -4e-310) {
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(0.5 / Float64(Float64(-0.5 * Float64(b / c)) + Float64(0.5 * Float64(a / 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 <= -4e-310)
		tmp = 0.5 / ((-0.5 * (b / c)) + (0.5 * (a / b)));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(0.5 / N[(N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 36.7%

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

   3. Applied egg-rr 36.7%

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

      1. *-commutative 36.7%

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

      2. associate-/r* 36.7%

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

      3. metadata-eval 36.7%

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

   5. Simplified 36.7%

      \[\leadsto \color{blue}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) \cdot \frac{-0.5}{a}} \]
   6. Step-by-step derivation

      1. frac-2neg 36.7%

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

      2. associate-*r/ 36.7%

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

      3. metadata-eval 36.7%

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

   7. Applied egg-rr 36.7%

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

      1. *-commutative 36.7%

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

      2. associate-/l* 36.7%

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

   9. Simplified 36.7%

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

   10. Taylor expanded in b around -inf 63.7%

       \[\leadsto \frac{0.5}{\color{blue}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}} \]

   if -3.999999999999988e-310 < b

   1. Initial program 75.0%

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

   2. Taylor expanded in b around inf 70.1%

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

      1. +-commutative 70.1%

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

      2. mul-1-neg 70.1%

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

      3. unsub-neg 70.1%

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

   4. Simplified 70.1%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{0.5}{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6: 67.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ (- c) b) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = -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 <= (-4d-310)) then
        tmp = -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 <= -4e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = -c / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(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 <= -4e-310)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[((-c) / b), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 36.7%

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

   2. Taylor expanded in b around -inf 63.5%

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

      1. associate-*r/ 63.5%

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

      2. neg-mul-1 63.5%

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

   4. Simplified 63.5%

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

   if -3.999999999999988e-310 < b

   1. Initial program 75.0%

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

   2. Taylor expanded in b around inf 70.1%

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

      1. +-commutative 70.1%

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

      2. mul-1-neg 70.1%

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

      3. unsub-neg 70.1%

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

   4. Simplified 70.1%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 7: 42.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{+69}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -2.45e+69) (/ c b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e+69) {
		tmp = c / b;
	} else {
		tmp = -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 <= (-2.45d+69)) then
        tmp = c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e+69) {
		tmp = c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.45e+69:
		tmp = c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.45e+69)
		tmp = Float64(c / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.45e+69)
		tmp = c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.45e+69], N[(c / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.45e69

   1. Initial program 15.5%

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

   2. Taylor expanded in b around -inf 82.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{2 \cdot a} \]
   3. Step-by-step derivation

      1. associate-/l* 88.3%

         \[\leadsto \frac{-2 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{2 \cdot a} \]

   4. Simplified 88.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a}{\frac{b}{c}}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. div-inv 88.2%

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

      2. associate-*l* 88.2%

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

      3. associate-/r/ 76.6%

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

      4. *-commutative 76.6%

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

      5. metadata-eval 76.6%

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

      6. associate-/r* 76.6%

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

      7. metadata-eval 76.6%

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

      8. metadata-eval 76.6%

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

   6. Applied egg-rr 76.6%

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

      1. associate-*r* 76.6%

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

      2. associate-*r/ 82.0%

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

      3. *-commutative 82.0%

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

      4. *-commutative 82.0%

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

      5. *-commutative 82.0%

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

      6. associate-*r/ 76.6%

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

      7. associate-*r* 76.6%

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

      8. associate-*r/ 82.0%

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

      9. *-commutative 82.0%

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

      10. associate-*r/ 88.4%

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

      11. associate-*r/ 88.4%

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

      12. metadata-eval 88.4%

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

   8. Simplified 88.4%

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

      1. *-commutative 88.4%

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

      2. associate-*r/ 82.0%

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

      3. associate-*r/ 86.7%

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

   10. Applied egg-rr 86.7%

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

       1. *-commutative 86.7%

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

       2. associate-*r/ 86.8%

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

       3. *-commutative 86.8%

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

       4. mul-1-neg 86.8%

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

       5. distribute-rgt-neg-in 86.8%

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

   12. Simplified 86.8%

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

       1. expm1-log1p-u 68.9%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot \left(-c\right)}{a}\right)\right)}}{b} \]

       2. expm1-udef 52.8%

          \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot \left(-c\right)}{a}\right)} - 1}}{b} \]

       3. associate-/l* 43.0%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{\frac{a}{-c}}}\right)} - 1}{b} \]

       4. add-sqr-sqrt 26.3%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{a}{\frac{a}{\color{blue}{\sqrt{-c} \cdot \sqrt{-c}}}}\right)} - 1}{b} \]

       5. sqrt-unprod 41.8%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{a}{\frac{a}{\color{blue}{\sqrt{\left(-c\right) \cdot \left(-c\right)}}}}\right)} - 1}{b} \]

       6. sqr-neg 41.8%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{a}{\frac{a}{\sqrt{\color{blue}{c \cdot c}}}}\right)} - 1}{b} \]

       7. sqrt-unprod 17.2%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{a}{\frac{a}{\color{blue}{\sqrt{c} \cdot \sqrt{c}}}}\right)} - 1}{b} \]

       8. add-sqr-sqrt 32.7%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{a}{\frac{a}{\color{blue}{c}}}\right)} - 1}{b} \]

   14. Applied egg-rr 32.7%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{a}{\frac{a}{c}}\right)} - 1}}{b} \]
   15. Step-by-step derivation

       1. expm1-def 32.5%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\frac{a}{c}}\right)\right)}}{b} \]

       2. expm1-log1p 32.9%

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

       3. associate-/l* 32.8%

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

       4. *-commutative 32.8%

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

       5. associate-/l* 32.9%

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

       6. *-inverses 32.9%

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

       7. /-rgt-identity 32.9%

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

   16. Simplified 32.9%

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

   if -2.45e69 < b

   1. Initial program 70.9%

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

   2. Taylor expanded in b around inf 50.6%

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

      1. associate-*r/ 50.6%

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

      2. mul-1-neg 50.6%

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

   4. Simplified 50.6%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{+69}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8: 67.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-300}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.4e-300) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.4e-300) {
		tmp = -c / b;
	} else {
		tmp = -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 <= (-6.4d-300)) then
        tmp = -c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.4e-300) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.4e-300:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.4e-300)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.4e-300)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.4e-300], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.40000000000000043e-300

   1. Initial program 36.1%

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

   2. Taylor expanded in b around -inf 64.0%

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

      1. associate-*r/ 64.0%

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

      2. neg-mul-1 64.0%

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

   4. Simplified 64.0%

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

   if -6.40000000000000043e-300 < b

   1. Initial program 75.2%

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

   2. Taylor expanded in b around inf 68.8%

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

      1. associate-*r/ 68.8%

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

      2. mul-1-neg 68.8%

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

   4. Simplified 68.8%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-300}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 9: 10.9% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in b around -inf 24.3%

   \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{2 \cdot a} \]
3. Step-by-step derivation

   1. associate-/l* 26.7%

      \[\leadsto \frac{-2 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}{2 \cdot a} \]

4. Simplified 26.7%

   \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a}{\frac{b}{c}}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. div-inv 26.7%

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

   2. associate-*l* 26.7%

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

   3. associate-/r/ 23.8%

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

   4. *-commutative 23.8%

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

   5. metadata-eval 23.8%

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

   6. associate-/r* 23.8%

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

   7. metadata-eval 23.8%

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

   8. metadata-eval 23.8%

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

6. Applied egg-rr 23.8%

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

   1. associate-*r* 23.8%

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

   2. associate-*r/ 24.2%

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

   3. *-commutative 24.2%

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

   4. *-commutative 24.2%

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

   5. *-commutative 24.2%

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

   6. associate-*r/ 23.8%

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

   7. associate-*r* 23.8%

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

   8. associate-*r/ 24.2%

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

   9. *-commutative 24.2%

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

   10. associate-*r/ 26.7%

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

   11. associate-*r/ 26.7%

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

   12. metadata-eval 26.7%

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

8. Simplified 26.7%

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

   1. *-commutative 26.7%

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

   2. associate-*r/ 24.2%

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

   3. associate-*r/ 25.8%

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

10. Applied egg-rr 25.8%

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

    1. *-commutative 25.8%

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

    2. associate-*r/ 25.8%

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

    3. *-commutative 25.8%

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

    4. mul-1-neg 25.8%

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

    5. distribute-rgt-neg-in 25.8%

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

12. Simplified 25.8%

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

    1. expm1-log1p-u 21.0%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot \left(-c\right)}{a}\right)\right)}}{b} \]

    2. expm1-udef 15.7%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot \left(-c\right)}{a}\right)} - 1}}{b} \]

    3. associate-/l* 13.8%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{\frac{a}{-c}}}\right)} - 1}{b} \]

    4. add-sqr-sqrt 9.0%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{a}{\frac{a}{\color{blue}{\sqrt{-c} \cdot \sqrt{-c}}}}\right)} - 1}{b} \]

    5. sqrt-unprod 14.3%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{a}{\frac{a}{\color{blue}{\sqrt{\left(-c\right) \cdot \left(-c\right)}}}}\right)} - 1}{b} \]

    6. sqr-neg 14.3%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{a}{\frac{a}{\sqrt{\color{blue}{c \cdot c}}}}\right)} - 1}{b} \]

    7. sqrt-unprod 5.4%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{a}{\frac{a}{\color{blue}{\sqrt{c} \cdot \sqrt{c}}}}\right)} - 1}{b} \]

    8. add-sqr-sqrt 9.9%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{a}{\frac{a}{\color{blue}{c}}}\right)} - 1}{b} \]

14. Applied egg-rr 9.9%

    \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{a}{\frac{a}{c}}\right)} - 1}}{b} \]
15. Step-by-step derivation

    1. expm1-def 9.8%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\frac{a}{c}}\right)\right)}}{b} \]

    2. expm1-log1p 11.5%

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

    3. associate-/l* 10.3%

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

    4. *-commutative 10.3%

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

    5. associate-/l* 10.4%

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

    6. *-inverses 10.4%

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

    7. /-rgt-identity 10.4%

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

16. Simplified 10.4%

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

17. Final simplification 10.4%

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

Developer target: 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 2023336 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (if (< b 0.0) (/ c (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0))) (/ (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (- a)))

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