Quadratic roots, full range

Percentage Accurate: 52.2% → 86.0%

Time: 14.6s

Alternatives: 8

Speedup: 12.9×

Specification

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.2% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\left|-2 \cdot \left(a \cdot \frac{c}{b} - b\right)\right|}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+154)
   (* 0.5 (/ (fabs (* -2.0 (- (* a (/ c b)) b))) a))
   (if (<= b 3.4e-49)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+154) {
		tmp = 0.5 * (fabs((-2.0 * ((a * (c / b)) - b))) / a);
	} else if (b <= 3.4e-49) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} 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+154)) then
        tmp = 0.5d0 * (abs(((-2.0d0) * ((a * (c / b)) - b))) / a)
    else if (b <= 3.4d-49) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    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+154) {
		tmp = 0.5 * (Math.abs((-2.0 * ((a * (c / b)) - b))) / a);
	} else if (b <= 3.4e-49) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+154:
		tmp = 0.5 * (math.fabs((-2.0 * ((a * (c / b)) - b))) / a)
	elif b <= 3.4e-49:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+154)
		tmp = Float64(0.5 * Float64(abs(Float64(-2.0 * Float64(Float64(a * Float64(c / b)) - b))) / a));
	elseif (b <= 3.4e-49)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+154)
		tmp = 0.5 * (abs((-2.0 * ((a * (c / b)) - b))) / a);
	elseif (b <= 3.4e-49)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+154], N[(0.5 * N[(N[Abs[N[(-2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.4e-49], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000004e154

   1. Initial program 53.4%

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

      1. *-commutative 53.4%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in a around 0 1.8%

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

      1. add-sqr-sqrt 1.7%

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

      2. sqrt-unprod 2.1%

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

      3. pow2 2.1%

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

      4. add-sqr-sqrt 21.9%

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

      5. sqrt-unprod 53.1%

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

      6. sqr-neg 53.1%

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

      7. sqrt-prod 0.0%

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

      8. add-sqr-sqrt 53.5%

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

      9. +-commutative 53.5%

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

      10. fma-define 53.5%

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

      11. associate-/l* 53.5%

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

   7. Applied egg-rr 53.5%

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

      1. unpow2 53.5%

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

      2. rem-sqrt-square 94.7%

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

   9. Simplified 94.7%

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

   10. Taylor expanded in b around 0 83.5%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\left|-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b\right|}{a}} \]
   11. Step-by-step derivation

       1. metadata-eval 83.5%

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

       2. cancel-sign-sub-inv 83.5%

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

       3. distribute-lft-out-- 83.5%

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

       4. associate-*r/ 94.7%

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

   12. Simplified 94.7%

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

   if -5.00000000000000004e154 < b < 3.40000000000000005e-49

   1. Initial program 81.6%

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

   if 3.40000000000000005e-49 < b

   1. Initial program 18.3%

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

      1. *-commutative 18.3%

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

   3. Simplified 18.3%

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

   5. Taylor expanded in b around inf 86.8%

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

      1. associate-*r/ 86.8%

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

      2. mul-1-neg 86.8%

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

   7. Simplified 86.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\left|-2 \cdot \left(a \cdot \frac{c}{b} - b\right)\right|}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \frac{\left|-2 \cdot \left(a \cdot \frac{c}{b} - b\right)\right|}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.1e-38)
   (* 0.5 (/ (fabs (* -2.0 (- (* a (/ c b)) b))) a))
   (if (<= b 3.4e-53)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.1e-38) {
		tmp = 0.5 * (fabs((-2.0 * ((a * (c / b)) - b))) / a);
	} else if (b <= 3.4e-53) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} 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 <= (-4.1d-38)) then
        tmp = 0.5d0 * (abs(((-2.0d0) * ((a * (c / b)) - b))) / a)
    else if (b <= 3.4d-53) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    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 <= -4.1e-38) {
		tmp = 0.5 * (Math.abs((-2.0 * ((a * (c / b)) - b))) / a);
	} else if (b <= 3.4e-53) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.1e-38:
		tmp = 0.5 * (math.fabs((-2.0 * ((a * (c / b)) - b))) / a)
	elif b <= 3.4e-53:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.1e-38)
		tmp = Float64(0.5 * Float64(abs(Float64(-2.0 * Float64(Float64(a * Float64(c / b)) - b))) / a));
	elseif (b <= 3.4e-53)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.1e-38)
		tmp = 0.5 * (abs((-2.0 * ((a * (c / b)) - b))) / a);
	elseif (b <= 3.4e-53)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.1e-38], N[(0.5 * N[(N[Abs[N[(-2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.4e-53], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.0999999999999998e-38

   1. Initial program 70.9%

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

      1. *-commutative 70.9%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in a around 0 1.9%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 2.5%

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

      3. pow2 2.5%

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

      4. add-sqr-sqrt 14.0%

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

      5. sqrt-unprod 29.0%

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

      6. sqr-neg 29.0%

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

      7. sqrt-prod 0.0%

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

      8. add-sqr-sqrt 66.3%

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

      9. +-commutative 66.3%

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

      10. fma-define 66.3%

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

      11. associate-/l* 66.3%

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

   7. Applied egg-rr 66.3%

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

      1. unpow2 66.3%

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

      2. rem-sqrt-square 87.7%

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

   9. Simplified 87.7%

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

   10. Taylor expanded in b around 0 81.9%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\left|-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b\right|}{a}} \]
   11. Step-by-step derivation

       1. metadata-eval 81.9%

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

       2. cancel-sign-sub-inv 81.9%

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

       3. distribute-lft-out-- 81.9%

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

       4. associate-*r/ 87.7%

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

   12. Simplified 87.7%

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

   if -4.0999999999999998e-38 < b < 3.4e-53

   1. Initial program 77.8%

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

      1. *-commutative 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in b around 0 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*r* 70.8%

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

   7. Simplified 70.8%

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

   if 3.4e-53 < b

   1. Initial program 18.3%

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

      1. *-commutative 18.3%

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

   3. Simplified 18.3%

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

   5. Taylor expanded in b around inf 86.8%

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

      1. associate-*r/ 86.8%

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

      2. mul-1-neg 86.8%

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

   7. Simplified 86.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \frac{\left|-2 \cdot \left(a \cdot \frac{c}{b} - b\right)\right|}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e-40)
   (/ (- b) a)
   (if (<= b 4.3e-52)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e-40) {
		tmp = -b / a;
	} else if (b <= 4.3e-52) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} 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 <= (-2.5d-40)) then
        tmp = -b / a
    else if (b <= 4.3d-52) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    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 <= -2.5e-40) {
		tmp = -b / a;
	} else if (b <= 4.3e-52) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.5e-40:
		tmp = -b / a
	elif b <= 4.3e-52:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e-40)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 4.3e-52)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.5e-40)
		tmp = -b / a;
	elseif (b <= 4.3e-52)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e-40], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 4.3e-52], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.49999999999999982e-40

   1. Initial program 70.9%

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

      1. *-commutative 70.9%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in b around -inf 87.5%

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

      1. mul-1-neg 87.5%

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

   7. Simplified 87.5%

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

   if -2.49999999999999982e-40 < b < 4.3000000000000003e-52

   1. Initial program 77.8%

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

      1. *-commutative 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in b around 0 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*r* 70.8%

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

   7. Simplified 70.8%

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

   if 4.3000000000000003e-52 < b

   1. Initial program 18.3%

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

      1. *-commutative 18.3%

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

   3. Simplified 18.3%

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

   5. Taylor expanded in b around inf 86.8%

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

      1. associate-*r/ 86.8%

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

      2. mul-1-neg 86.8%

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

   7. Simplified 86.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-55}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-63}:\\ \;\;\;\;-0.5 \cdot \left(-\sqrt{c \cdot \frac{-4}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-55)
   (/ (- b) a)
   (if (<= b 1.15e-63) (* -0.5 (- (sqrt (* c (/ -4.0 a))))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-55) {
		tmp = -b / a;
	} else if (b <= 1.15e-63) {
		tmp = -0.5 * -sqrt((c * (-4.0 / 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 <= (-3d-55)) then
        tmp = -b / a
    else if (b <= 1.15d-63) then
        tmp = (-0.5d0) * -sqrt((c * ((-4.0d0) / 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 <= -3e-55) {
		tmp = -b / a;
	} else if (b <= 1.15e-63) {
		tmp = -0.5 * -Math.sqrt((c * (-4.0 / a)));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e-55:
		tmp = -b / a
	elif b <= 1.15e-63:
		tmp = -0.5 * -math.sqrt((c * (-4.0 / a)))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-55)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.15e-63)
		tmp = Float64(-0.5 * Float64(-sqrt(Float64(c * Float64(-4.0 / a)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e-55)
		tmp = -b / a;
	elseif (b <= 1.15e-63)
		tmp = -0.5 * -sqrt((c * (-4.0 / a)));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-55], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.15e-63], N[(-0.5 * (-N[Sqrt[N[(c * N[(-4.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.00000000000000016e-55

   1. Initial program 72.0%

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

      1. *-commutative 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in b around -inf 85.7%

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

      1. mul-1-neg 85.7%

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

   7. Simplified 85.7%

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

   if -3.00000000000000016e-55 < b < 1.15e-63

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

   3. Simplified 77.4%

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

      1. *-un-lft-identity 77.4%

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

      2. add-cube-cbrt 76.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{1 \cdot \left(b \cdot b\right) - \color{blue}{\left(\sqrt[3]{\left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      3. prod-diff 76.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(1, b \cdot b, -\sqrt[3]{\left(4 \cdot a\right) \cdot c} \cdot \left(\sqrt[3]{\left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\left(4 \cdot a\right) \cdot c}, \sqrt[3]{\left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}, \sqrt[3]{\left(4 \cdot a\right) \cdot c} \cdot \left(\sqrt[3]{\left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}\right)\right)}}}{a \cdot 2} \]

   6. Applied egg-rr 76.4%

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

      1. +-commutative 76.4%

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

      2. fma-undefine 76.4%

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

      3. associate-+l+ 76.4%

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

      4. distribute-lft-neg-out 76.4%

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

      5. *-commutative 76.4%

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

      6. *-commutative 76.4%

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

      7. fma-define 76.4%

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

      8. associate-+r+ 76.4%

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

   8. Simplified 76.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-\sqrt[3]{a \cdot \left(c \cdot 4\right)} \cdot {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{2}\right) + \left(a \cdot \left(c \cdot 0\right) + {b}^{2}\right)}}}{a \cdot 2} \]

   9. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto -0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right)} \]

       2. unpow2 0.0%

          \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

       3. rem-square-sqrt 38.7%

          \[\leadsto -0.5 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

       4. rem-cube-cbrt 39.0%

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

       5. associate-/l* 39.1%

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

   11. Simplified 39.1%

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

   if 1.15e-63 < b

   1. Initial program 19.8%

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

      1. *-commutative 19.8%

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

   3. Simplified 19.8%

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

   5. Taylor expanded in b around inf 85.3%

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

      1. associate-*r/ 85.3%

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

      2. mul-1-neg 85.3%

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

   7. Simplified 85.3%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-55}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-63}:\\ \;\;\;\;-0.5 \cdot \left(-\sqrt{c \cdot \frac{-4}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.6% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-309) (/ (- b) a) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-309) {
		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 <= (-1d-309)) 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 <= -1e-309) {
		tmp = -b / a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-309:
		tmp = -b / a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-309)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-309)
		tmp = -b / a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-309], N[((-b) / a), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.000000000000002e-309

   1. Initial program 76.7%

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

      1. *-commutative 76.7%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in b around -inf 63.9%

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

      1. mul-1-neg 63.9%

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

   7. Simplified 63.9%

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

   if -1.000000000000002e-309 < b

   1. Initial program 32.5%

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

      1. *-commutative 32.5%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in b around inf 66.0%

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

      1. associate-*r/ 66.0%

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

      2. mul-1-neg 66.0%

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

   7. Simplified 66.0%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 5.4e+48) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.4e+48) {
		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 <= 5.4d+48) 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 <= 5.4e+48) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.4e+48:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.4e+48)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.4e+48)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.4e+48], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.40000000000000007e48

   1. Initial program 67.9%

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

      1. *-commutative 67.9%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in b around -inf 42.4%

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

      1. mul-1-neg 42.4%

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

   7. Simplified 42.4%

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

   if 5.40000000000000007e48 < b

   1. Initial program 13.9%

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

      1. *-commutative 13.9%

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

   3. Simplified 13.9%

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

   5. Taylor expanded in a around 0 39.8%

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

      1. frac-2neg 39.8%

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

      2. div-inv 39.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 38.7%

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

      5. sqr-neg 38.7%

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

      6. sqrt-prod 38.7%

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

      7. add-sqr-sqrt 38.7%

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

   7. Applied egg-rr 38.7%

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

      1. distribute-lft-neg-out 38.7%

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

      2. associate-*r/ 38.7%

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

      3. *-rgt-identity 38.7%

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

      4. times-frac 38.7%

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

      5. associate-*r/ 38.9%

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

      6. metadata-eval 38.9%

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

      7. *-commutative 38.9%

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

      8. *-lft-identity 38.9%

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

      9. associate-*r/ 38.7%

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

      10. mul-1-neg 38.7%

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

      11. metadata-eval 38.7%

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

      12. times-frac 38.7%

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

      13. neg-mul-1 38.7%

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

      14. distribute-rgt-neg-in 38.7%

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

      15. mul-1-neg 38.7%

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

      16. times-frac 38.9%

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

      17. /-rgt-identity 38.9%

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

      18. mul-1-neg 38.9%

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

   9. Simplified 38.9%

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

   10. Taylor expanded in b around 0 37.9%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 53.6%

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

   1. *-commutative 53.6%

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

3. Simplified 53.6%

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

5. Taylor expanded in a around 0 12.7%

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

   1. frac-2neg 12.7%

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

   2. div-inv 12.7%

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

   3. add-sqr-sqrt 0.8%

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

   4. sqrt-unprod 11.5%

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

   5. sqr-neg 11.5%

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

   6. sqrt-prod 10.9%

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

   7. add-sqr-sqrt 12.5%

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

7. Applied egg-rr 12.5%

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

   1. distribute-lft-neg-out 12.5%

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

   2. associate-*r/ 12.5%

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

   3. *-rgt-identity 12.5%

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

   4. times-frac 12.5%

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

   5. associate-*r/ 12.5%

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

   6. metadata-eval 12.5%

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

   7. *-commutative 12.5%

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

   8. *-lft-identity 12.5%

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

   9. associate-*r/ 12.5%

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

   10. mul-1-neg 12.5%

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

   11. metadata-eval 12.5%

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

   12. times-frac 12.5%

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

   13. neg-mul-1 12.5%

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

   14. distribute-rgt-neg-in 12.5%

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

   15. mul-1-neg 12.5%

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

   16. times-frac 12.5%

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

   17. /-rgt-identity 12.5%

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

   18. mul-1-neg 12.5%

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

9. Simplified 12.5%

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

10. Taylor expanded in b around 0 12.3%

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

Alternative 8: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

   1. *-commutative 53.6%

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

3. Simplified 53.6%

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

   1. *-un-lft-identity 53.6%

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

   2. *-un-lft-identity 53.6%

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

   3. prod-diff 53.6%

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

   4. *-commutative 53.6%

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

   5. *-un-lft-identity 53.6%

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

   6. fma-define 53.6%

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

   7. *-un-lft-identity 53.6%

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

   8. +-commutative 53.6%

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

   9. add-sqr-sqrt 36.6%

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

   10. sqrt-unprod 50.1%

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

   11. sqr-neg 50.1%

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

   12. sqrt-prod 13.8%

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

   13. add-sqr-sqrt 36.4%

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

   14. pow2 36.4%

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

   15. add-sqr-sqrt 22.8%

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

   16. sqrt-unprod 36.4%

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

   17. sqr-neg 36.4%

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

   18. sqrt-prod 13.8%

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

   19. add-sqr-sqrt 36.1%

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

   20. *-commutative 36.1%

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

   21. *-un-lft-identity 36.1%

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

6. Applied egg-rr 36.1%

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

   1. +-commutative 36.1%

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

   2. associate-+l+ 36.1%

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

   3. fma-undefine 36.1%

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

   4. *-rgt-identity 36.1%

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

8. Simplified 36.1%

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

9. Taylor expanded in b around -inf 2.6%

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

Reproduce

herbie shell --seed 2024106 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))