quadp (p42, positive)

Percentage Accurate: 52.8% → 85.1%

Time: 15.2s

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: 52.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e+75)
   (- (/ c b) (/ b a))
   (if (<= b 2.1e-31)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (/ (- (- c) (* a (pow (/ c b) 2.0))) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e+75) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.1e-31) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = (-c - (a * pow((c / b), 2.0))) / 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 <= (-3.5d+75)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.1d-31) then
        tmp = (sqrt(((b * b) - (4.0d0 * (c * a)))) - b) / (a * 2.0d0)
    else
        tmp = (-c - (a * ((c / b) ** 2.0d0))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e+75) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.1e-31) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = (-c - (a * Math.pow((c / b), 2.0))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.5e+75:
		tmp = (c / b) - (b / a)
	elif b <= 2.1e-31:
		tmp = (math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0)
	else:
		tmp = (-c - (a * math.pow((c / b), 2.0))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e+75)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.1e-31)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / b) ^ 2.0))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.5e+75)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.1e-31)
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	else
		tmp = (-c - (a * ((c / b) ^ 2.0))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.5e+75], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.1e-31], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.4999999999999998e75

   1. Initial program 57.3%

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

      1. *-commutative 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in b around -inf 95.5%

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

      1. mul-1-neg 95.5%

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

      2. *-commutative 95.5%

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

      3. distribute-rgt-neg-in 95.5%

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

      4. +-commutative 95.5%

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

      5. mul-1-neg 95.5%

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

      6. unsub-neg 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in a around inf 95.8%

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

      1. +-commutative 95.8%

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

      2. neg-mul-1 95.8%

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

      3. unsub-neg 95.8%

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

   10. Simplified 95.8%

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

   if -3.4999999999999998e75 < b < 2.09999999999999991e-31

   1. Initial program 76.4%

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

   if 2.09999999999999991e-31 < b

   1. Initial program 17.6%

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

      1. *-commutative 17.6%

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

   3. Simplified 17.6%

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

      1. frac-2neg 17.6%

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

      2. div-inv 17.6%

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

   6. Applied egg-rr 17.6%

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

      1. *-commutative 17.6%

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

   8. Simplified 17.6%

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

      1. fma-undefine 17.6%

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

   10. Applied egg-rr 17.6%

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

       1. unpow2 17.6%

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

   12. Applied egg-rr 17.6%

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

   13. Taylor expanded in b around inf 63.3%

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

       1. mul-1-neg 63.3%

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

       2. unsub-neg 63.3%

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

       3. mul-1-neg 63.3%

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

       4. associate-/l* 65.9%

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

       5. unpow2 65.9%

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

       6. unpow2 65.9%

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

       7. times-frac 82.2%

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

       8. unpow2 82.2%

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

   15. Simplified 82.2%

       \[\leadsto \color{blue}{\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{+73}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.25e+73)
   (- (/ c b) (/ b a))
   (if (<= b 4.5e-33)
     (* (- b (sqrt (+ (* b b) (* a (* c -4.0))))) (/ -0.5 a))
     (/ (- (- c) (* a (pow (/ c b) 2.0))) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e+73) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.5e-33) {
		tmp = (b - sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a);
	} else {
		tmp = (-c - (a * pow((c / b), 2.0))) / 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 <= (-1.25d+73)) then
        tmp = (c / b) - (b / a)
    else if (b <= 4.5d-33) then
        tmp = (b - sqrt(((b * b) + (a * (c * (-4.0d0)))))) * ((-0.5d0) / a)
    else
        tmp = (-c - (a * ((c / b) ** 2.0d0))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e+73) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.5e-33) {
		tmp = (b - Math.sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a);
	} else {
		tmp = (-c - (a * Math.pow((c / b), 2.0))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.25e+73:
		tmp = (c / b) - (b / a)
	elif b <= 4.5e-33:
		tmp = (b - math.sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a)
	else:
		tmp = (-c - (a * math.pow((c / b), 2.0))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.25e+73)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4.5e-33)
		tmp = Float64(Float64(b - sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0))))) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / b) ^ 2.0))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.25e+73)
		tmp = (c / b) - (b / a);
	elseif (b <= 4.5e-33)
		tmp = (b - sqrt(((b * b) + (a * (c * -4.0))))) * (-0.5 / a);
	else
		tmp = (-c - (a * ((c / b) ^ 2.0))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.25e+73], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.5e-33], N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.24999999999999994e73

   1. Initial program 57.3%

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

      1. *-commutative 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in b around -inf 95.5%

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

      1. mul-1-neg 95.5%

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

      2. *-commutative 95.5%

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

      3. distribute-rgt-neg-in 95.5%

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

      4. +-commutative 95.5%

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

      5. mul-1-neg 95.5%

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

      6. unsub-neg 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in a around inf 95.8%

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

      1. +-commutative 95.8%

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

      2. neg-mul-1 95.8%

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

      3. unsub-neg 95.8%

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

   10. Simplified 95.8%

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

   if -1.24999999999999994e73 < b < 4.49999999999999991e-33

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

   3. Simplified 76.4%

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

      1. frac-2neg 76.4%

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

      2. div-inv 76.3%

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

   6. Applied egg-rr 76.3%

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

      1. *-commutative 76.3%

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

   8. Simplified 76.3%

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

      1. fma-undefine 76.3%

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

   10. Applied egg-rr 76.3%

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

       1. unpow2 76.3%

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

   12. Applied egg-rr 76.3%

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

   13. Taylor expanded in a around 0 76.3%

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

   if 4.49999999999999991e-33 < b

   1. Initial program 17.6%

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

      1. *-commutative 17.6%

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

   3. Simplified 17.6%

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

      1. frac-2neg 17.6%

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

      2. div-inv 17.6%

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

   6. Applied egg-rr 17.6%

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

      1. *-commutative 17.6%

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

   8. Simplified 17.6%

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

      1. fma-undefine 17.6%

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

   10. Applied egg-rr 17.6%

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

       1. unpow2 17.6%

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

   12. Applied egg-rr 17.6%

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

   13. Taylor expanded in b around inf 63.3%

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

       1. mul-1-neg 63.3%

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

       2. unsub-neg 63.3%

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

       3. mul-1-neg 63.3%

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

       4. associate-/l* 65.9%

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

       5. unpow2 65.9%

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

       6. unpow2 65.9%

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

       7. times-frac 82.2%

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

       8. unpow2 82.2%

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

   15. Simplified 82.2%

       \[\leadsto \color{blue}{\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{+73}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.16 \cdot 10^{-57}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.16e-57)
   (- (/ c b) (/ b a))
   (if (<= b 3.6e-33)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ (- (- c) (* a (pow (/ c b) 2.0))) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.16e-57) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.6e-33) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (-c - (a * pow((c / b), 2.0))) / 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 <= (-1.16d-57)) then
        tmp = (c / b) - (b / a)
    else if (b <= 3.6d-33) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (-c - (a * ((c / b) ** 2.0d0))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.16e-57) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.6e-33) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (-c - (a * Math.pow((c / b), 2.0))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.16e-57:
		tmp = (c / b) - (b / a)
	elif b <= 3.6e-33:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = (-c - (a * math.pow((c / b), 2.0))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.16e-57)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 3.6e-33)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / b) ^ 2.0))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.16e-57)
		tmp = (c / b) - (b / a);
	elseif (b <= 3.6e-33)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = (-c - (a * ((c / b) ^ 2.0))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.16e-57], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.6e-33], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.15999999999999996e-57

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in b around -inf 89.8%

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

      1. mul-1-neg 89.8%

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

      2. *-commutative 89.8%

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

      3. distribute-rgt-neg-in 89.8%

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

      4. +-commutative 89.8%

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

      5. mul-1-neg 89.8%

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

      6. unsub-neg 89.8%

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

   7. Simplified 89.8%

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

   8. Taylor expanded in a around inf 90.0%

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

      1. +-commutative 90.0%

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

      2. neg-mul-1 90.0%

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

      3. unsub-neg 90.0%

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

   10. Simplified 90.0%

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

   if -1.15999999999999996e-57 < b < 3.60000000000000034e-33

   1. Initial program 67.2%

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

      1. *-commutative 67.2%

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

   3. Simplified 67.2%

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

      1. add-sqr-sqrt 66.9%

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

      2. pow2 66.9%

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

      3. pow1/2 66.9%

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

      4. sqrt-pow1 67.0%

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

      5. sub-neg 67.0%

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

      6. +-commutative 67.0%

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

      7. distribute-lft-neg-in 67.0%

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

      8. *-commutative 67.0%

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

      9. associate-*r* 67.0%

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

      10. fma-define 67.0%

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

      11. metadata-eval 67.0%

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

      12. pow2 67.0%

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

      13. metadata-eval 67.0%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(-4 \cdot c, a, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a \cdot 2} \]

   6. Applied egg-rr 67.0%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(-4 \cdot c, a, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}{a \cdot 2} \]

   7. Taylor expanded in b around 0 61.2%

      \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(-4 \cdot \left(a \cdot c\right)\right)}^{0.25}\right)}}^{2}}{a \cdot 2} \]
   8. Step-by-step derivation

      1. *-commutative 61.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\left(a \cdot c\right) \cdot -4\right)}}^{0.25}\right)}^{2}}{a \cdot 2} \]

   9. Simplified 61.2%

      \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(\left(a \cdot c\right) \cdot -4\right)}^{0.25}\right)}}^{2}}{a \cdot 2} \]
   10. Step-by-step derivation

       1. +-commutative 61.2%

          \[\leadsto \frac{\color{blue}{{\left({\left(\left(a \cdot c\right) \cdot -4\right)}^{0.25}\right)}^{2} + \left(-b\right)}}{a \cdot 2} \]

       2. *-un-lft-identity 61.2%

          \[\leadsto \frac{\color{blue}{1 \cdot {\left({\left(\left(a \cdot c\right) \cdot -4\right)}^{0.25}\right)}^{2}} + \left(-b\right)}{a \cdot 2} \]

       3. fma-define 61.2%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1, {\left({\left(\left(a \cdot c\right) \cdot -4\right)}^{0.25}\right)}^{2}, -b\right)}}{a \cdot 2} \]

       4. pow-pow 61.4%

          \[\leadsto \frac{\mathsf{fma}\left(1, \color{blue}{{\left(\left(a \cdot c\right) \cdot -4\right)}^{\left(0.25 \cdot 2\right)}}, -b\right)}{a \cdot 2} \]

       5. associate-*l* 61.4%

          \[\leadsto \frac{\mathsf{fma}\left(1, {\color{blue}{\left(a \cdot \left(c \cdot -4\right)\right)}}^{\left(0.25 \cdot 2\right)}, -b\right)}{a \cdot 2} \]

       6. *-commutative 61.4%

          \[\leadsto \frac{\mathsf{fma}\left(1, {\color{blue}{\left(\left(c \cdot -4\right) \cdot a\right)}}^{\left(0.25 \cdot 2\right)}, -b\right)}{a \cdot 2} \]

       7. metadata-eval 61.4%

          \[\leadsto \frac{\mathsf{fma}\left(1, {\left(\left(c \cdot -4\right) \cdot a\right)}^{\color{blue}{0.5}}, -b\right)}{a \cdot 2} \]

       8. pow1/2 61.4%

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

       9. associate-*l* 61.4%

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

   11. Applied egg-rr 61.4%

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

       1. fma-undefine 61.4%

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

       2. *-lft-identity 61.4%

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

       3. unsub-neg 61.4%

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

       4. *-commutative 61.4%

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

   13. Simplified 61.4%

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

   if 3.60000000000000034e-33 < b

   1. Initial program 17.6%

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

      1. *-commutative 17.6%

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

   3. Simplified 17.6%

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

      1. frac-2neg 17.6%

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

      2. div-inv 17.6%

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

   6. Applied egg-rr 17.6%

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

      1. *-commutative 17.6%

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

   8. Simplified 17.6%

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

      1. fma-undefine 17.6%

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

   10. Applied egg-rr 17.6%

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

       1. unpow2 17.6%

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

   12. Applied egg-rr 17.6%

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

   13. Taylor expanded in b around inf 63.3%

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

       1. mul-1-neg 63.3%

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

       2. unsub-neg 63.3%

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

       3. mul-1-neg 63.3%

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

       4. associate-/l* 65.9%

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

       5. unpow2 65.9%

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

       6. unpow2 65.9%

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

       7. times-frac 82.2%

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

       8. unpow2 82.2%

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

   15. Simplified 82.2%

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

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-55}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\left(c \cdot a\right) \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.4e-55)
   (- (/ c b) (/ b a))
   (if (<= b 3.2e-32)
     (* (/ -0.5 a) (- b (sqrt (* (* c a) -4.0))))
     (/ (- (- c) (* a (pow (/ c b) 2.0))) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.4e-55) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.2e-32) {
		tmp = (-0.5 / a) * (b - sqrt(((c * a) * -4.0)));
	} else {
		tmp = (-c - (a * pow((c / b), 2.0))) / 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 <= (-6.4d-55)) then
        tmp = (c / b) - (b / a)
    else if (b <= 3.2d-32) then
        tmp = ((-0.5d0) / a) * (b - sqrt(((c * a) * (-4.0d0))))
    else
        tmp = (-c - (a * ((c / b) ** 2.0d0))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.4e-55) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.2e-32) {
		tmp = (-0.5 / a) * (b - Math.sqrt(((c * a) * -4.0)));
	} else {
		tmp = (-c - (a * Math.pow((c / b), 2.0))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.4e-55:
		tmp = (c / b) - (b / a)
	elif b <= 3.2e-32:
		tmp = (-0.5 / a) * (b - math.sqrt(((c * a) * -4.0)))
	else:
		tmp = (-c - (a * math.pow((c / b), 2.0))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.4e-55)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 3.2e-32)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(Float64(c * a) * -4.0))));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(c / b) ^ 2.0))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.4e-55)
		tmp = (c / b) - (b / a);
	elseif (b <= 3.2e-32)
		tmp = (-0.5 / a) * (b - sqrt(((c * a) * -4.0)));
	else
		tmp = (-c - (a * ((c / b) ^ 2.0))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.4e-55], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.2e-32], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.4000000000000003e-55

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in b around -inf 89.8%

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

      1. mul-1-neg 89.8%

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

      2. *-commutative 89.8%

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

      3. distribute-rgt-neg-in 89.8%

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

      4. +-commutative 89.8%

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

      5. mul-1-neg 89.8%

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

      6. unsub-neg 89.8%

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

   7. Simplified 89.8%

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

   8. Taylor expanded in a around inf 90.0%

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

      1. +-commutative 90.0%

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

      2. neg-mul-1 90.0%

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

      3. unsub-neg 90.0%

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

   10. Simplified 90.0%

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

   if -6.4000000000000003e-55 < b < 3.2000000000000002e-32

   1. Initial program 67.2%

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

      1. *-commutative 67.2%

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

   3. Simplified 67.2%

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

      1. frac-2neg 67.2%

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

      2. div-inv 67.1%

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

   6. Applied egg-rr 67.1%

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

      1. *-commutative 67.1%

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

   8. Simplified 67.1%

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

   9. Taylor expanded in c around inf 61.3%

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

       1. *-commutative 61.3%

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

   11. Simplified 61.3%

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

   12. Taylor expanded in a around 0 61.3%

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

   if 3.2000000000000002e-32 < b

   1. Initial program 17.6%

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

      1. *-commutative 17.6%

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

   3. Simplified 17.6%

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

      1. frac-2neg 17.6%

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

      2. div-inv 17.6%

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

   6. Applied egg-rr 17.6%

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

      1. *-commutative 17.6%

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

   8. Simplified 17.6%

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

      1. fma-undefine 17.6%

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

   10. Applied egg-rr 17.6%

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

       1. unpow2 17.6%

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

   12. Applied egg-rr 17.6%

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

   13. Taylor expanded in b around inf 63.3%

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

       1. mul-1-neg 63.3%

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

       2. unsub-neg 63.3%

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

       3. mul-1-neg 63.3%

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

       4. associate-/l* 65.9%

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

       5. unpow2 65.9%

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

       6. unpow2 65.9%

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

       7. times-frac 82.2%

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

       8. unpow2 82.2%

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

   15. Simplified 82.2%

       \[\leadsto \color{blue}{\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-55}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\left(c \cdot a\right) \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-57}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-33}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\left(c \cdot a\right) \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e-57)
   (- (/ c b) (/ b a))
   (if (<= b 4e-33) (* (/ -0.5 a) (- b (sqrt (* (* c a) -4.0)))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e-57) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4e-33) {
		tmp = (-0.5 / a) * (b - sqrt(((c * a) * -4.0)));
	} 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 <= (-7d-57)) then
        tmp = (c / b) - (b / a)
    else if (b <= 4d-33) then
        tmp = ((-0.5d0) / a) * (b - sqrt(((c * a) * (-4.0d0))))
    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 <= -7e-57) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4e-33) {
		tmp = (-0.5 / a) * (b - Math.sqrt(((c * a) * -4.0)));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7e-57:
		tmp = (c / b) - (b / a)
	elif b <= 4e-33:
		tmp = (-0.5 / a) * (b - math.sqrt(((c * a) * -4.0)))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7e-57)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4e-33)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(Float64(c * a) * -4.0))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7e-57)
		tmp = (c / b) - (b / a);
	elseif (b <= 4e-33)
		tmp = (-0.5 / a) * (b - sqrt(((c * a) * -4.0)));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7e-57], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e-33], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.99999999999999983e-57

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in b around -inf 89.8%

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

      1. mul-1-neg 89.8%

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

      2. *-commutative 89.8%

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

      3. distribute-rgt-neg-in 89.8%

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

      4. +-commutative 89.8%

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

      5. mul-1-neg 89.8%

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

      6. unsub-neg 89.8%

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

   7. Simplified 89.8%

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

   8. Taylor expanded in a around inf 90.0%

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

      1. +-commutative 90.0%

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

      2. neg-mul-1 90.0%

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

      3. unsub-neg 90.0%

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

   10. Simplified 90.0%

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

   if -6.99999999999999983e-57 < b < 4.0000000000000002e-33

   1. Initial program 67.2%

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

      1. *-commutative 67.2%

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

   3. Simplified 67.2%

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

      1. frac-2neg 67.2%

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

      2. div-inv 67.1%

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

   6. Applied egg-rr 67.1%

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

      1. *-commutative 67.1%

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

   8. Simplified 67.1%

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

   9. Taylor expanded in c around inf 61.3%

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

       1. *-commutative 61.3%

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

   11. Simplified 61.3%

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

   12. Taylor expanded in a around 0 61.3%

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

   if 4.0000000000000002e-33 < b

   1. Initial program 17.6%

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

      1. *-commutative 17.6%

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

   3. Simplified 17.6%

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

   5. Taylor expanded in b around inf 82.1%

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

      1. associate-*r/ 82.1%

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

      2. neg-mul-1 82.1%

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

   7. Simplified 82.1%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-57}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-33}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\left(c \cdot a\right) \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 71.3%

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

      1. *-commutative 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in b around -inf 74.2%

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

      1. mul-1-neg 74.2%

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

      2. *-commutative 74.2%

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

      3. distribute-rgt-neg-in 74.2%

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

      4. +-commutative 74.2%

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

      5. mul-1-neg 74.2%

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

      6. unsub-neg 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in a around inf 74.6%

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

      1. +-commutative 74.6%

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

      2. neg-mul-1 74.6%

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

      3. unsub-neg 74.6%

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

   10. Simplified 74.6%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.4%

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

      1. *-commutative 31.4%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in b around inf 62.8%

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

      1. associate-*r/ 62.8%

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

      2. neg-mul-1 62.8%

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

   7. Simplified 62.8%

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

4. Final simplification 68.8%

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

Alternative 7: 67.2% 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(c / Float64(-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 71.3%

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

      1. *-commutative 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in b around -inf 74.2%

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

      1. mul-1-neg 74.2%

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

      2. distribute-neg-frac2 74.2%

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

   7. Simplified 74.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.4%

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

      1. *-commutative 31.4%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in b around inf 62.8%

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

      1. associate-*r/ 62.8%

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

      2. neg-mul-1 62.8%

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

   7. Simplified 62.8%

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

4. Final simplification 68.6%

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

Alternative 8: 42.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+76}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1.3e+76) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.3e+76) {
		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 <= 1.3d+76) 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 <= 1.3e+76) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.3e+76:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.3e+76)
		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 <= 1.3e+76)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.3e+76], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3e76

   1. Initial program 63.2%

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

      1. *-commutative 63.2%

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

   3. Simplified 63.2%

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

   5. Taylor expanded in b around -inf 50.3%

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

      1. mul-1-neg 50.3%

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

      2. distribute-neg-frac2 50.3%

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

   7. Simplified 50.3%

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

   if 1.3e76 < b

   1. Initial program 13.8%

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

      1. *-commutative 13.8%

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

   3. Simplified 13.8%

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

   5. Taylor expanded in b around -inf 2.7%

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

      1. mul-1-neg 2.7%

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

      2. *-commutative 2.7%

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

      3. distribute-rgt-neg-in 2.7%

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

      4. +-commutative 2.7%

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

      5. mul-1-neg 2.7%

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

      6. unsub-neg 2.7%

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

   7. Simplified 2.7%

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

   8. Taylor expanded in a around inf 25.3%

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

Alternative 9: 10.2% 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 51.7%

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

   1. *-commutative 51.7%

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

3. Simplified 51.7%

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

5. Taylor expanded in b around -inf 38.9%

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

   1. mul-1-neg 38.9%

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

   2. *-commutative 38.9%

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

   3. distribute-rgt-neg-in 38.9%

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

   4. +-commutative 38.9%

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

   5. mul-1-neg 38.9%

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

   6. unsub-neg 38.9%

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

7. Simplified 38.9%

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

8. Taylor expanded in a around inf 8.3%

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

Alternative 10: 2.6% 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 51.7%

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

   1. *-commutative 51.7%

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

3. Simplified 51.7%

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

   1. clear-num 51.5%

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

   2. associate-/r/ 51.6%

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

   3. *-commutative 51.6%

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

   4. associate-/r* 51.6%

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

   5. metadata-eval 51.6%

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

   6. add-sqr-sqrt 36.1%

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

   7. sqrt-unprod 48.9%

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

   8. sqr-neg 48.9%

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

   9. sqrt-prod 12.8%

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

   10. add-sqr-sqrt 28.6%

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

   11. sub-neg 28.6%

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

   12. +-commutative 28.6%

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

   13. distribute-lft-neg-in 28.6%

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

   14. *-commutative 28.6%

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

   15. associate-*r* 28.6%

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

   16. fma-define 28.6%

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

   17. metadata-eval 28.6%

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

   18. pow2 28.6%

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

6. Applied egg-rr 28.6%

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

7. Taylor expanded in a around 0 2.6%

   \[\leadsto \color{blue}{\frac{b}{a}} \]
8. 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 2024112 
(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)))