quadp (p42, positive)

Percentage Accurate: 53.3% → 84.3%

Time: 14.9s

Alternatives: 10

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: 53.3% 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: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+108}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.16 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(\frac{{b}^{2}}{a} - 4 \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e+108)
   (/ b (- a))
   (if (<= b 1.16e-92)
     (/ (- (sqrt (* a (- (/ (pow b 2.0) a) (* 4.0 c)))) b) (* a 2.0))
     (/ -0.5 (+ (* 0.5 (/ b c)) (* -0.5 (/ a b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e+108) {
		tmp = b / -a;
	} else if (b <= 1.16e-92) {
		tmp = (sqrt((a * ((pow(b, 2.0) / a) - (4.0 * c)))) - b) / (a * 2.0);
	} else {
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3d+108)) then
        tmp = b / -a
    else if (b <= 1.16d-92) then
        tmp = (sqrt((a * (((b ** 2.0d0) / a) - (4.0d0 * c)))) - b) / (a * 2.0d0)
    else
        tmp = (-0.5d0) / ((0.5d0 * (b / c)) + ((-0.5d0) * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e+108) {
		tmp = b / -a;
	} else if (b <= 1.16e-92) {
		tmp = (Math.sqrt((a * ((Math.pow(b, 2.0) / a) - (4.0 * c)))) - b) / (a * 2.0);
	} else {
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e+108:
		tmp = b / -a
	elif b <= 1.16e-92:
		tmp = (math.sqrt((a * ((math.pow(b, 2.0) / a) - (4.0 * c)))) - b) / (a * 2.0)
	else:
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e+108)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.16e-92)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(Float64((b ^ 2.0) / a) - Float64(4.0 * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-0.5 / Float64(Float64(0.5 * Float64(b / c)) + Float64(-0.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e+108)
		tmp = b / -a;
	elseif (b <= 1.16e-92)
		tmp = (sqrt((a * (((b ^ 2.0) / a) - (4.0 * c)))) - b) / (a * 2.0);
	else
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e+108], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.16e-92], N[(N[(N[Sqrt[N[(a * N[(N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision] - N[(4.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(N[(0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.99999999999999984e108

   1. Initial program 59.7%

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

      1. *-commutative 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in b around -inf 95.9%

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

      1. associate-*r/ 95.9%

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

      2. mul-1-neg 95.9%

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

   7. Simplified 95.9%

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

   if -2.99999999999999984e108 < b < 1.1599999999999999e-92

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

   3. Simplified 87.1%

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

   5. Taylor expanded in a around inf 87.1%

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

   if 1.1599999999999999e-92 < b

   1. Initial program 14.6%

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

      1. *-commutative 14.6%

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

   3. Simplified 14.6%

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

      1. add-cube-cbrt 14.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}} \]

      2. pow3 14.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}\right)}^{3}} \]

   6. Applied egg-rr 14.6%

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

      1. rem-cube-cbrt 14.6%

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

      2. clear-num 14.6%

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

      3. *-commutative 14.6%

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

      4. *-un-lft-identity 14.6%

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

      5. times-frac 14.6%

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

      6. metadata-eval 14.6%

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

      7. fma-undefine 14.6%

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

      8. *-commutative 14.6%

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

      9. associate-*r* 14.6%

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

      10. add-sqr-sqrt 10.8%

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

      11. pow2 10.8%

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

      12. hypot-define 24.8%

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

   8. Applied egg-rr 24.8%

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

      1. associate-/r* 24.8%

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

      2. metadata-eval 24.8%

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

   10. Simplified 24.8%

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

   11. Taylor expanded in a around 0 0.0%

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

       1. associate-*r/ 0.0%

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

       2. *-commutative 0.0%

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

       3. times-frac 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 90.1%

          \[\leadsto \frac{-0.5}{\frac{-2}{\color{blue}{-4}} \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}} \]

       6. metadata-eval 90.1%

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

   13. Simplified 90.1%

       \[\leadsto \frac{-0.5}{\color{blue}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+108}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.16 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(\frac{{b}^{2}}{a} - 4 \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+154)
   (/ b (- a))
   (if (<= b 2.5e-78)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ -0.5 (+ (* 0.5 (/ b c)) (* -0.5 (/ a b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+154) {
		tmp = b / -a;
	} else if (b <= 2.5e-78) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d+154)) then
        tmp = b / -a
    else if (b <= 2.5d-78) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = (-0.5d0) / ((0.5d0 * (b / c)) + ((-0.5d0) * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+154) {
		tmp = b / -a;
	} else if (b <= 2.5e-78) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+154:
		tmp = b / -a
	elif b <= 2.5e-78:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+154)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 2.5e-78)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-0.5 / Float64(Float64(0.5 * Float64(b / c)) + Float64(-0.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+154)
		tmp = b / -a;
	elseif (b <= 2.5e-78)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+154], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 2.5e-78], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(N[(0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000004e154

   1. Initial program 39.5%

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

      1. *-commutative 39.5%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in b around -inf 93.8%

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

      1. associate-*r/ 93.8%

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

      2. mul-1-neg 93.8%

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

   7. Simplified 93.8%

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

   if -5.00000000000000004e154 < b < 2.4999999999999998e-78

   1. Initial program 88.7%

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

   if 2.4999999999999998e-78 < b

   1. Initial program 14.6%

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

      1. *-commutative 14.6%

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

   3. Simplified 14.6%

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

      1. add-cube-cbrt 14.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}} \]

      2. pow3 14.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}\right)}^{3}} \]

   6. Applied egg-rr 14.6%

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

      1. rem-cube-cbrt 14.6%

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

      2. clear-num 14.6%

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

      3. *-commutative 14.6%

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

      4. *-un-lft-identity 14.6%

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

      5. times-frac 14.6%

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

      6. metadata-eval 14.6%

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

      7. fma-undefine 14.6%

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

      8. *-commutative 14.6%

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

      9. associate-*r* 14.6%

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

      10. add-sqr-sqrt 10.8%

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

      11. pow2 10.8%

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

      12. hypot-define 24.8%

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

   8. Applied egg-rr 24.8%

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

      1. associate-/r* 24.8%

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

      2. metadata-eval 24.8%

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

   10. Simplified 24.8%

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

   11. Taylor expanded in a around 0 0.0%

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

       1. associate-*r/ 0.0%

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

       2. *-commutative 0.0%

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

       3. times-frac 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 90.1%

          \[\leadsto \frac{-0.5}{\frac{-2}{\color{blue}{-4}} \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}} \]

       6. metadata-eval 90.1%

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

   13. Simplified 90.1%

       \[\leadsto \frac{-0.5}{\color{blue}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-148}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{b - \sqrt{\left(a \cdot c\right) \cdot -4}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e-148)
   (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))
   (if (<= b 6.4e-83)
     (/ (- b (sqrt (* (* a c) -4.0))) (* a -2.0))
     (/ -0.5 (+ (* 0.5 (/ b c)) (* -0.5 (/ a b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-148) {
		tmp = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 6.4e-83) {
		tmp = (b - sqrt(((a * c) * -4.0))) / (a * -2.0);
	} else {
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.35d-148)) then
        tmp = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
    else if (b <= 6.4d-83) then
        tmp = (b - sqrt(((a * c) * (-4.0d0)))) / (a * (-2.0d0))
    else
        tmp = (-0.5d0) / ((0.5d0 * (b / c)) + ((-0.5d0) * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-148) {
		tmp = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 6.4e-83) {
		tmp = (b - Math.sqrt(((a * c) * -4.0))) / (a * -2.0);
	} else {
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.35e-148:
		tmp = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
	elif b <= 6.4e-83:
		tmp = (b - math.sqrt(((a * c) * -4.0))) / (a * -2.0)
	else:
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e-148)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	elseif (b <= 6.4e-83)
		tmp = Float64(Float64(b - sqrt(Float64(Float64(a * c) * -4.0))) / Float64(a * -2.0));
	else
		tmp = Float64(-0.5 / Float64(Float64(0.5 * Float64(b / c)) + Float64(-0.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.35e-148)
		tmp = b * ((c / (b ^ 2.0)) + (-1.0 / a));
	elseif (b <= 6.4e-83)
		tmp = (b - sqrt(((a * c) * -4.0))) / (a * -2.0);
	else
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e-148], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.4e-83], N[(N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(N[(0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.34999999999999994e-148

   1. Initial program 73.9%

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

      1. *-commutative 73.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in b around -inf 76.8%

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

      1. mul-1-neg 76.8%

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

      2. *-commutative 76.8%

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

      3. distribute-rgt-neg-in 76.8%

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

      4. +-commutative 76.8%

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

      5. mul-1-neg 76.8%

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

      6. unsub-neg 76.8%

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

   7. Simplified 76.8%

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

   if -1.34999999999999994e-148 < b < 6.4000000000000002e-83

   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. *-commutative 86.9%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in b around 0 86.8%

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

      1. associate-*r* 86.8%

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

      2. *-commutative 86.8%

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

      3. *-commutative 86.8%

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

   7. Simplified 86.8%

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

      1. add-cbrt-cube 49.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2} \cdot \frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\right) \cdot \frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}}} \]

      2. pow1/3 14.5%

         \[\leadsto \color{blue}{{\left(\left(\frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2} \cdot \frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\right) \cdot \frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\right)}^{0.3333333333333333}} \]

   9. Applied egg-rr 14.3%

      \[\leadsto \color{blue}{{\left({\left(0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\right)}^{3}\right)}^{0.3333333333333333}} \]

   11. Applied egg-rr 86.8%

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

       1. *-commutative 86.8%

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

       2. rem-square-sqrt 0.0%

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

       3. unpow2 0.0%

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

       4. associate-*r* 0.0%

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

       5. *-commutative 0.0%

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

       6. unpow2 0.0%

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

       7. rem-square-sqrt 86.8%

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

   13. Simplified 86.8%

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

   if 6.4000000000000002e-83 < b

   1. Initial program 14.6%

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

      1. *-commutative 14.6%

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

   3. Simplified 14.6%

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

      1. add-cube-cbrt 14.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}} \]

      2. pow3 14.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}\right)}^{3}} \]

   6. Applied egg-rr 14.6%

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

      1. rem-cube-cbrt 14.6%

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

      2. clear-num 14.6%

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

      3. *-commutative 14.6%

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

      4. *-un-lft-identity 14.6%

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

      5. times-frac 14.6%

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

      6. metadata-eval 14.6%

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

      7. fma-undefine 14.6%

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

      8. *-commutative 14.6%

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

      9. associate-*r* 14.6%

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

      10. add-sqr-sqrt 10.8%

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

      11. pow2 10.8%

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

      12. hypot-define 24.8%

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

   8. Applied egg-rr 24.8%

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

      1. associate-/r* 24.8%

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

      2. metadata-eval 24.8%

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

   10. Simplified 24.8%

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

   11. Taylor expanded in a around 0 0.0%

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

       1. associate-*r/ 0.0%

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

       2. *-commutative 0.0%

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

       3. times-frac 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 90.1%

          \[\leadsto \frac{-0.5}{\frac{-2}{\color{blue}{-4}} \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}} \]

       6. metadata-eval 90.1%

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

   13. Simplified 90.1%

       \[\leadsto \frac{-0.5}{\color{blue}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-148}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{b - \sqrt{\left(a \cdot c\right) \cdot -4}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-148}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 6.3 \cdot 10^{-84}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e-148)
   (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))
   (if (<= b 6.3e-84)
     (/ (+ b (sqrt (* c (* a -4.0)))) (* a 2.0))
     (/ -0.5 (+ (* 0.5 (/ b c)) (* -0.5 (/ a b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-148) {
		tmp = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 6.3e-84) {
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.35d-148)) then
        tmp = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
    else if (b <= 6.3d-84) then
        tmp = (b + sqrt((c * (a * (-4.0d0))))) / (a * 2.0d0)
    else
        tmp = (-0.5d0) / ((0.5d0 * (b / c)) + ((-0.5d0) * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-148) {
		tmp = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 6.3e-84) {
		tmp = (b + Math.sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.35e-148:
		tmp = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
	elif b <= 6.3e-84:
		tmp = (b + math.sqrt((c * (a * -4.0)))) / (a * 2.0)
	else:
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e-148)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	elseif (b <= 6.3e-84)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * 2.0));
	else
		tmp = Float64(-0.5 / Float64(Float64(0.5 * Float64(b / c)) + Float64(-0.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.35e-148)
		tmp = b * ((c / (b ^ 2.0)) + (-1.0 / a));
	elseif (b <= 6.3e-84)
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * 2.0);
	else
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e-148], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.3e-84], N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(N[(0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.34999999999999994e-148

   1. Initial program 73.9%

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

      1. *-commutative 73.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in b around -inf 76.8%

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

      1. mul-1-neg 76.8%

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

      2. *-commutative 76.8%

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

      3. distribute-rgt-neg-in 76.8%

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

      4. +-commutative 76.8%

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

      5. mul-1-neg 76.8%

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

      6. unsub-neg 76.8%

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

   7. Simplified 76.8%

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

   if -1.34999999999999994e-148 < b < 6.3000000000000004e-84

   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. *-commutative 86.9%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in b around 0 86.8%

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

      1. associate-*r* 86.8%

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

      2. *-commutative 86.8%

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

      3. *-commutative 86.8%

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

   7. Simplified 86.8%

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

      1. *-un-lft-identity 86.8%

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

      2. times-frac 86.7%

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

      3. add-sqr-sqrt 39.3%

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

      4. sqrt-unprod 86.7%

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

      5. sqr-neg 86.7%

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

      6. sqrt-prod 47.5%

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

      7. add-sqr-sqrt 86.6%

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

   9. Applied egg-rr 86.6%

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

       1. times-frac 86.7%

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

       2. *-lft-identity 86.7%

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

   11. Simplified 86.7%

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

   if 6.3000000000000004e-84 < b

   1. Initial program 14.6%

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

      1. *-commutative 14.6%

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

   3. Simplified 14.6%

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

      1. add-cube-cbrt 14.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}} \]

      2. pow3 14.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}\right)}^{3}} \]

   6. Applied egg-rr 14.6%

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

      1. rem-cube-cbrt 14.6%

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

      2. clear-num 14.6%

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

      3. *-commutative 14.6%

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

      4. *-un-lft-identity 14.6%

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

      5. times-frac 14.6%

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

      6. metadata-eval 14.6%

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

      7. fma-undefine 14.6%

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

      8. *-commutative 14.6%

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

      9. associate-*r* 14.6%

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

      10. add-sqr-sqrt 10.8%

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

      11. pow2 10.8%

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

      12. hypot-define 24.8%

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

   8. Applied egg-rr 24.8%

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

      1. associate-/r* 24.8%

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

      2. metadata-eval 24.8%

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

   10. Simplified 24.8%

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

   11. Taylor expanded in a around 0 0.0%

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

       1. associate-*r/ 0.0%

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

       2. *-commutative 0.0%

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

       3. times-frac 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 90.1%

          \[\leadsto \frac{-0.5}{\frac{-2}{\color{blue}{-4}} \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}} \]

       6. metadata-eval 90.1%

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

   13. Simplified 90.1%

       \[\leadsto \frac{-0.5}{\color{blue}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-148}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 6.3 \cdot 10^{-84}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-148}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{0.5}{\frac{a}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e-148)
   (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))
   (if (<= b 7.5e-88)
     (/ 0.5 (/ a (+ b (sqrt (* a (* c -4.0))))))
     (/ -0.5 (+ (* 0.5 (/ b c)) (* -0.5 (/ a b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-148) {
		tmp = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 7.5e-88) {
		tmp = 0.5 / (a / (b + sqrt((a * (c * -4.0)))));
	} else {
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.35d-148)) then
        tmp = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
    else if (b <= 7.5d-88) then
        tmp = 0.5d0 / (a / (b + sqrt((a * (c * (-4.0d0))))))
    else
        tmp = (-0.5d0) / ((0.5d0 * (b / c)) + ((-0.5d0) * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-148) {
		tmp = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 7.5e-88) {
		tmp = 0.5 / (a / (b + Math.sqrt((a * (c * -4.0)))));
	} else {
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.35e-148:
		tmp = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
	elif b <= 7.5e-88:
		tmp = 0.5 / (a / (b + math.sqrt((a * (c * -4.0)))))
	else:
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e-148)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	elseif (b <= 7.5e-88)
		tmp = Float64(0.5 / Float64(a / Float64(b + sqrt(Float64(a * Float64(c * -4.0))))));
	else
		tmp = Float64(-0.5 / Float64(Float64(0.5 * Float64(b / c)) + Float64(-0.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.35e-148)
		tmp = b * ((c / (b ^ 2.0)) + (-1.0 / a));
	elseif (b <= 7.5e-88)
		tmp = 0.5 / (a / (b + sqrt((a * (c * -4.0)))));
	else
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e-148], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.5e-88], N[(0.5 / N[(a / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(N[(0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.34999999999999994e-148

   1. Initial program 73.9%

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

      1. *-commutative 73.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in b around -inf 76.8%

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

      1. mul-1-neg 76.8%

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

      2. *-commutative 76.8%

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

      3. distribute-rgt-neg-in 76.8%

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

      4. +-commutative 76.8%

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

      5. mul-1-neg 76.8%

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

      6. unsub-neg 76.8%

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

   7. Simplified 76.8%

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

   if -1.34999999999999994e-148 < b < 7.50000000000000041e-88

   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. *-commutative 86.9%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in b around 0 86.8%

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

      1. associate-*r* 86.8%

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

      2. *-commutative 86.8%

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

      3. *-commutative 86.8%

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

   7. Simplified 86.8%

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

      1. add-cbrt-cube 49.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2} \cdot \frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\right) \cdot \frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}}} \]

      2. pow1/3 14.5%

         \[\leadsto \color{blue}{{\left(\left(\frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2} \cdot \frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\right) \cdot \frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\right)}^{0.3333333333333333}} \]

   9. Applied egg-rr 14.3%

      \[\leadsto \color{blue}{{\left({\left(0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\right)}^{3}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 49.0%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\right)}^{3}}} \]

       2. rem-cbrt-cube 86.7%

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

       3. clear-num 86.6%

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

       4. un-div-inv 86.6%

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

       5. add-sqr-sqrt 47.5%

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

       6. sqrt-unprod 86.6%

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

       7. sqr-neg 86.6%

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

       8. sqrt-unprod 39.2%

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

       9. add-sqr-sqrt 86.7%

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

       10. sqrt-prod 58.3%

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

       11. *-commutative 58.3%

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

       12. add-sqr-sqrt 28.4%

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

       13. sqrt-unprod 58.4%

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

       14. sqr-neg 58.4%

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

       15. sqrt-unprod 30.0%

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

       16. add-sqr-sqrt 58.4%

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

       17. sqrt-unprod 86.6%

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

       18. associate-*l* 86.6%

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

   11. Applied egg-rr 86.6%

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

   if 7.50000000000000041e-88 < b

   1. Initial program 14.6%

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

      1. *-commutative 14.6%

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

   3. Simplified 14.6%

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

      1. add-cube-cbrt 14.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}} \]

      2. pow3 14.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}\right)}^{3}} \]

   6. Applied egg-rr 14.6%

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

      1. rem-cube-cbrt 14.6%

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

      2. clear-num 14.6%

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

      3. *-commutative 14.6%

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

      4. *-un-lft-identity 14.6%

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

      5. times-frac 14.6%

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

      6. metadata-eval 14.6%

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

      7. fma-undefine 14.6%

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

      8. *-commutative 14.6%

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

      9. associate-*r* 14.6%

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

      10. add-sqr-sqrt 10.8%

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

      11. pow2 10.8%

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

      12. hypot-define 24.8%

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

   8. Applied egg-rr 24.8%

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

      1. associate-/r* 24.8%

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

      2. metadata-eval 24.8%

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

   10. Simplified 24.8%

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

   11. Taylor expanded in a around 0 0.0%

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

       1. associate-*r/ 0.0%

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

       2. *-commutative 0.0%

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

       3. times-frac 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 90.1%

          \[\leadsto \frac{-0.5}{\frac{-2}{\color{blue}{-4}} \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}} \]

       6. metadata-eval 90.1%

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

   13. Simplified 90.1%

       \[\leadsto \frac{-0.5}{\color{blue}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-148}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{0.5}{\frac{a}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ b (- a)) (/ -0.5 (+ (* 0.5 (/ b c)) (* -0.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b / -a
    else
        tmp = (-0.5d0) / ((0.5d0 * (b / c)) + ((-0.5d0) * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b / -a
	else:
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(-0.5 / Float64(Float64(0.5 * Float64(b / c)) + Float64(-0.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = b / -a;
	else
		tmp = -0.5 / ((0.5 * (b / c)) + (-0.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b / (-a)), $MachinePrecision], N[(-0.5 / N[(N[(0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   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. *-commutative 76.6%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in b around -inf 59.4%

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

      1. associate-*r/ 59.4%

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

      2. mul-1-neg 59.4%

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

   7. Simplified 59.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 33.2%

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

      1. *-commutative 33.2%

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

   3. Simplified 33.2%

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

      1. add-cube-cbrt 32.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}} \]

      2. pow3 32.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}}\right)}^{3}} \]

   6. Applied egg-rr 32.7%

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

      1. rem-cube-cbrt 33.2%

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

      2. clear-num 33.1%

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

      3. *-commutative 33.1%

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

      4. *-un-lft-identity 33.1%

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

      5. times-frac 33.1%

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

      6. metadata-eval 33.1%

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

      7. fma-undefine 33.1%

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

      8. *-commutative 33.1%

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

      9. associate-*r* 33.1%

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

      10. add-sqr-sqrt 30.3%

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

      11. pow2 30.3%

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

      12. hypot-define 40.8%

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

   8. Applied egg-rr 40.8%

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

      1. associate-/r* 40.8%

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

      2. metadata-eval 40.8%

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

   10. Simplified 40.8%

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

   11. Taylor expanded in a around 0 0.0%

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

       1. associate-*r/ 0.0%

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

       2. *-commutative 0.0%

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

       3. times-frac 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 69.9%

          \[\leadsto \frac{-0.5}{\frac{-2}{\color{blue}{-4}} \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}} \]

       6. metadata-eval 69.9%

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

   13. Simplified 69.9%

       \[\leadsto \frac{-0.5}{\color{blue}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.2% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   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. *-commutative 76.6%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in b around -inf 59.4%

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

      1. associate-*r/ 59.4%

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

      2. mul-1-neg 59.4%

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

   7. Simplified 59.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 33.2%

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

      1. *-commutative 33.2%

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

   3. Simplified 33.2%

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

   5. Taylor expanded in b around inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. neg-mul-1 69.9%

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

   7. Simplified 69.9%

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

4. Final simplification 64.9%

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

Alternative 8: 42.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-10}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 3.7e-10) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.7e-10) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.7d-10) then
        tmp = b / -a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.7e-10) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.7e-10:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.7e-10)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.7e-10)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.7e-10], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.70000000000000015e-10

   1. Initial program 75.8%

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

      1. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in b around -inf 44.3%

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

      1. associate-*r/ 44.3%

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

      2. mul-1-neg 44.3%

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

   7. Simplified 44.3%

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

   if 3.70000000000000015e-10 < b

   1. Initial program 13.7%

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

      1. *-commutative 13.7%

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

   3. Simplified 13.7%

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

   5. Taylor expanded in b around inf 93.0%

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

      1. associate-*r/ 93.0%

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

      2. neg-mul-1 93.0%

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

   7. Simplified 93.0%

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

      1. div-inv 92.8%

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

      2. add-sqr-sqrt 43.2%

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

      3. sqrt-unprod 45.0%

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

      4. sqr-neg 45.0%

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

      5. sqrt-unprod 14.4%

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

      6. add-sqr-sqrt 32.1%

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

   9. Applied egg-rr 32.1%

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

       1. associate-*r/ 32.1%

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

       2. *-rgt-identity 32.1%

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

   11. Simplified 32.1%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-10}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 11.0% 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 53.7%

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

   1. *-commutative 53.7%

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

3. Simplified 53.7%

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

5. Taylor expanded in b around inf 37.7%

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

   1. associate-*r/ 37.7%

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

   2. neg-mul-1 37.7%

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

7. Simplified 37.7%

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

   1. div-inv 37.7%

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

   2. add-sqr-sqrt 17.2%

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

   3. sqrt-unprod 19.4%

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

   4. sqr-neg 19.4%

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

   5. sqrt-unprod 6.3%

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

   6. add-sqr-sqrt 13.5%

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

9. Applied egg-rr 13.5%

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

    1. associate-*r/ 13.5%

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

    2. *-rgt-identity 13.5%

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

11. Simplified 13.5%

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

Alternative 10: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.7%

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

   1. *-commutative 53.7%

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

3. Simplified 53.7%

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

5. Taylor expanded in b around -inf 29.4%

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

   1. associate-*r/ 29.4%

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

   2. mul-1-neg 29.4%

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

7. Simplified 29.4%

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

   1. add-sqr-sqrt 28.0%

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

   2. sqrt-unprod 23.2%

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

   3. sqr-neg 23.2%

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

   4. sqrt-prod 1.9%

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

   5. add-sqr-sqrt 2.6%

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

   6. *-un-lft-identity 2.6%

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

   7. *-un-lft-identity 2.6%

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

   8. times-frac 2.6%

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

   9. metadata-eval 2.6%

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

9. Applied egg-rr 2.6%

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

    1. *-lft-identity 2.6%

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

11. Simplified 2.6%

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

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024182 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

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

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