Cubic critical

Percentage Accurate: 52.5% → 86.0%

Time: 11.8s

Alternatives: 9

Speedup: 11.6×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e+119)
   (/ (/ b -1.5) a)
   (if (<= b 2.7e-90)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e+119) {
		tmp = (b / -1.5) / a;
	} else if (b <= 2.7e-90) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-6d+119)) then
        tmp = (b / (-1.5d0)) / a
    else if (b <= 2.7d-90) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e+119) {
		tmp = (b / -1.5) / a;
	} else if (b <= 2.7e-90) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e+119:
		tmp = (b / -1.5) / a
	elif b <= 2.7e-90:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e+119)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 2.7e-90)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6e+119)
		tmp = (b / -1.5) / a;
	elseif (b <= 2.7e-90)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e+119], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.7e-90], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.00000000000000002e119

   1. Initial program 39.3%

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

      1. sqr-neg 39.3%

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

      2. sqr-neg 39.3%

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

      3. associate-*l* 39.3%

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

   3. Simplified 39.3%

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

      1. clear-num 39.3%

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

      2. associate-/r/ 39.3%

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

      3. associate-/r* 39.3%

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

      4. metadata-eval 39.3%

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

      5. add-sqr-sqrt 39.2%

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

      6. sqrt-unprod 39.3%

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

      7. sqr-neg 39.3%

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

      8. sqrt-prod 0.0%

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

      9. add-sqr-sqrt 30.2%

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

      10. fma-neg 30.2%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}\right) \]

      11. distribute-lft-neg-in 30.2%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right) \cdot \left(a \cdot c\right)}\right)}\right) \]

      12. *-commutative 30.2%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(-3\right) \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      13. associate-*r* 30.2%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(-3\right) \cdot c\right) \cdot a}\right)}\right) \]

      14. metadata-eval 30.2%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{-3} \cdot c\right) \cdot a\right)}\right) \]

   6. Applied egg-rr 30.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right)} \]

   7. Taylor expanded in a around 0 0.6%

      \[\leadsto \color{blue}{0.6666666666666666 \cdot \frac{b}{a}} \]
   8. Step-by-step derivation

      1. *-commutative 0.6%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot 0.6666666666666666} \]

   9. Simplified 0.6%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot 0.6666666666666666} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.3%

          \[\leadsto \color{blue}{\sqrt{\frac{b}{a} \cdot 0.6666666666666666} \cdot \sqrt{\frac{b}{a} \cdot 0.6666666666666666}} \]

       2. sqrt-unprod 29.6%

          \[\leadsto \color{blue}{\sqrt{\left(\frac{b}{a} \cdot 0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot 0.6666666666666666\right)}} \]

       3. swap-sqr 29.6%

          \[\leadsto \sqrt{\color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}} \]

       4. metadata-eval 29.6%

          \[\leadsto \sqrt{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \color{blue}{0.4444444444444444}} \]

       5. metadata-eval 29.6%

          \[\leadsto \sqrt{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \color{blue}{\left(-0.6666666666666666 \cdot -0.6666666666666666\right)}} \]

       6. swap-sqr 29.6%

          \[\leadsto \sqrt{\color{blue}{\left(\frac{b}{a} \cdot -0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)}} \]

       7. sqrt-unprod 45.2%

          \[\leadsto \color{blue}{\sqrt{\frac{b}{a} \cdot -0.6666666666666666} \cdot \sqrt{\frac{b}{a} \cdot -0.6666666666666666}} \]

       8. add-sqr-sqrt 95.8%

          \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

       9. metadata-eval 95.8%

          \[\leadsto \frac{b}{a} \cdot \color{blue}{\frac{1}{-1.5}} \]

       10. div-inv 95.9%

           \[\leadsto \color{blue}{\frac{\frac{b}{a}}{-1.5}} \]

       11. associate-/l/ 96.0%

           \[\leadsto \color{blue}{\frac{b}{-1.5 \cdot a}} \]

       12. associate-/r* 96.2%

           \[\leadsto \color{blue}{\frac{\frac{b}{-1.5}}{a}} \]

   11. Applied egg-rr 96.2%

       \[\leadsto \color{blue}{\frac{\frac{b}{-1.5}}{a}} \]

   if -6.00000000000000002e119 < b < 2.69999999999999996e-90

   1. Initial program 85.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   if 2.69999999999999996e-90 < b

   1. Initial program 17.2%

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

      1. sqr-neg 17.2%

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

      2. sqr-neg 17.2%

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

      3. associate-*l* 17.2%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in b around inf 87.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 87.7%

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

   7. Simplified 87.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.8e+119)
   (/ (/ b -1.5) a)
   (if (<= b 4.5e-90)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.8e+119) {
		tmp = (b / -1.5) / a;
	} else if (b <= 4.5e-90) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-9.8d+119)) then
        tmp = (b / (-1.5d0)) / a
    else if (b <= 4.5d-90) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.8e+119) {
		tmp = (b / -1.5) / a;
	} else if (b <= 4.5e-90) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.8e+119:
		tmp = (b / -1.5) / a
	elif b <= 4.5e-90:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.8e+119)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 4.5e-90)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.8e+119)
		tmp = (b / -1.5) / a;
	elseif (b <= 4.5e-90)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.8e+119], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.5e-90], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.79999999999999992e119

   1. Initial program 39.3%

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

      1. sqr-neg 39.3%

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

      2. sqr-neg 39.3%

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

      3. associate-*l* 39.3%

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

   3. Simplified 39.3%

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

      1. clear-num 39.3%

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

      2. associate-/r/ 39.3%

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

      3. associate-/r* 39.3%

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

      4. metadata-eval 39.3%

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

      5. add-sqr-sqrt 39.2%

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

      6. sqrt-unprod 39.3%

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

      7. sqr-neg 39.3%

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

      8. sqrt-prod 0.0%

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

      9. add-sqr-sqrt 30.2%

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

      10. fma-neg 30.2%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}\right) \]

      11. distribute-lft-neg-in 30.2%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right) \cdot \left(a \cdot c\right)}\right)}\right) \]

      12. *-commutative 30.2%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(-3\right) \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      13. associate-*r* 30.2%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(-3\right) \cdot c\right) \cdot a}\right)}\right) \]

      14. metadata-eval 30.2%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{-3} \cdot c\right) \cdot a\right)}\right) \]

   6. Applied egg-rr 30.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right)} \]

   7. Taylor expanded in a around 0 0.6%

      \[\leadsto \color{blue}{0.6666666666666666 \cdot \frac{b}{a}} \]
   8. Step-by-step derivation

      1. *-commutative 0.6%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot 0.6666666666666666} \]

   9. Simplified 0.6%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot 0.6666666666666666} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.3%

          \[\leadsto \color{blue}{\sqrt{\frac{b}{a} \cdot 0.6666666666666666} \cdot \sqrt{\frac{b}{a} \cdot 0.6666666666666666}} \]

       2. sqrt-unprod 29.6%

          \[\leadsto \color{blue}{\sqrt{\left(\frac{b}{a} \cdot 0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot 0.6666666666666666\right)}} \]

       3. swap-sqr 29.6%

          \[\leadsto \sqrt{\color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}} \]

       4. metadata-eval 29.6%

          \[\leadsto \sqrt{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \color{blue}{0.4444444444444444}} \]

       5. metadata-eval 29.6%

          \[\leadsto \sqrt{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \color{blue}{\left(-0.6666666666666666 \cdot -0.6666666666666666\right)}} \]

       6. swap-sqr 29.6%

          \[\leadsto \sqrt{\color{blue}{\left(\frac{b}{a} \cdot -0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)}} \]

       7. sqrt-unprod 45.2%

          \[\leadsto \color{blue}{\sqrt{\frac{b}{a} \cdot -0.6666666666666666} \cdot \sqrt{\frac{b}{a} \cdot -0.6666666666666666}} \]

       8. add-sqr-sqrt 95.8%

          \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

       9. metadata-eval 95.8%

          \[\leadsto \frac{b}{a} \cdot \color{blue}{\frac{1}{-1.5}} \]

       10. div-inv 95.9%

           \[\leadsto \color{blue}{\frac{\frac{b}{a}}{-1.5}} \]

       11. associate-/l/ 96.0%

           \[\leadsto \color{blue}{\frac{b}{-1.5 \cdot a}} \]

       12. associate-/r* 96.2%

           \[\leadsto \color{blue}{\frac{\frac{b}{-1.5}}{a}} \]

   11. Applied egg-rr 96.2%

       \[\leadsto \color{blue}{\frac{\frac{b}{-1.5}}{a}} \]

   if -9.79999999999999992e119 < b < 4.50000000000000009e-90

   1. Initial program 85.5%

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

      1. sqr-neg 85.5%

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

      2. sqr-neg 85.5%

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

      3. associate-*l* 85.4%

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

   3. Simplified 85.4%

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

   if 4.50000000000000009e-90 < b

   1. Initial program 17.2%

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

      1. sqr-neg 17.2%

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

      2. sqr-neg 17.2%

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

      3. associate-*l* 17.2%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in b around inf 87.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 87.7%

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

   7. Simplified 87.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-81)
   (/ (/ b -1.5) a)
   (if (<= b 9e-91) (/ (sqrt (* a (* c -3.0))) (* a 3.0)) (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-81) {
		tmp = (b / -1.5) / a;
	} else if (b <= 9e-91) {
		tmp = sqrt((a * (c * -3.0))) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-8.5d-81)) then
        tmp = (b / (-1.5d0)) / a
    else if (b <= 9d-91) then
        tmp = sqrt((a * (c * (-3.0d0)))) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-81) {
		tmp = (b / -1.5) / a;
	} else if (b <= 9e-91) {
		tmp = Math.sqrt((a * (c * -3.0))) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e-81:
		tmp = (b / -1.5) / a
	elif b <= 9e-91:
		tmp = math.sqrt((a * (c * -3.0))) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-81)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 9e-91)
		tmp = Float64(sqrt(Float64(a * Float64(c * -3.0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e-81)
		tmp = (b / -1.5) / a;
	elseif (b <= 9e-91)
		tmp = sqrt((a * (c * -3.0))) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e-81], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 9e-91], N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.5000000000000001e-81

   1. Initial program 60.4%

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

      1. sqr-neg 60.4%

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

      2. sqr-neg 60.4%

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

      3. associate-*l* 60.4%

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

   3. Simplified 60.4%

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

      1. clear-num 60.3%

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

      2. associate-/r/ 60.4%

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

      3. associate-/r* 60.3%

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

      4. metadata-eval 60.3%

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

      5. add-sqr-sqrt 60.1%

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

      6. sqrt-unprod 60.3%

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

      7. sqr-neg 60.3%

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

      8. sqrt-prod 0.0%

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

      9. add-sqr-sqrt 26.9%

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

      10. fma-neg 26.9%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}\right) \]

      11. distribute-lft-neg-in 26.9%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right) \cdot \left(a \cdot c\right)}\right)}\right) \]

      12. *-commutative 26.9%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(-3\right) \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      13. associate-*r* 26.9%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(-3\right) \cdot c\right) \cdot a}\right)}\right) \]

      14. metadata-eval 26.9%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{-3} \cdot c\right) \cdot a\right)}\right) \]

   6. Applied egg-rr 26.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right)} \]

   7. Taylor expanded in a around 0 0.8%

      \[\leadsto \color{blue}{0.6666666666666666 \cdot \frac{b}{a}} \]
   8. Step-by-step derivation

      1. *-commutative 0.8%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot 0.6666666666666666} \]

   9. Simplified 0.8%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot 0.6666666666666666} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.5%

          \[\leadsto \color{blue}{\sqrt{\frac{b}{a} \cdot 0.6666666666666666} \cdot \sqrt{\frac{b}{a} \cdot 0.6666666666666666}} \]

       2. sqrt-unprod 25.7%

          \[\leadsto \color{blue}{\sqrt{\left(\frac{b}{a} \cdot 0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot 0.6666666666666666\right)}} \]

       3. swap-sqr 25.7%

          \[\leadsto \sqrt{\color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}} \]

       4. metadata-eval 25.7%

          \[\leadsto \sqrt{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \color{blue}{0.4444444444444444}} \]

       5. metadata-eval 25.7%

          \[\leadsto \sqrt{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \color{blue}{\left(-0.6666666666666666 \cdot -0.6666666666666666\right)}} \]

       6. swap-sqr 25.7%

          \[\leadsto \sqrt{\color{blue}{\left(\frac{b}{a} \cdot -0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)}} \]

       7. sqrt-unprod 42.9%

          \[\leadsto \color{blue}{\sqrt{\frac{b}{a} \cdot -0.6666666666666666} \cdot \sqrt{\frac{b}{a} \cdot -0.6666666666666666}} \]

       8. add-sqr-sqrt 89.1%

          \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

       9. metadata-eval 89.1%

          \[\leadsto \frac{b}{a} \cdot \color{blue}{\frac{1}{-1.5}} \]

       10. div-inv 89.3%

           \[\leadsto \color{blue}{\frac{\frac{b}{a}}{-1.5}} \]

       11. associate-/l/ 89.3%

           \[\leadsto \color{blue}{\frac{b}{-1.5 \cdot a}} \]

       12. associate-/r* 89.4%

           \[\leadsto \color{blue}{\frac{\frac{b}{-1.5}}{a}} \]

   11. Applied egg-rr 89.4%

       \[\leadsto \color{blue}{\frac{\frac{b}{-1.5}}{a}} \]

   if -8.5000000000000001e-81 < b < 8.99999999999999952e-91

   1. Initial program 79.5%

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

      1. sqr-neg 79.5%

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

      2. sqr-neg 79.5%

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

      3. associate-*l* 79.4%

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

   3. Simplified 79.4%

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

      1. add-cube-cbrt 78.9%

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

      2. pow3 78.8%

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

      3. associate-*r* 78.7%

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

      4. *-commutative 78.7%

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

      5. associate-*l* 78.7%

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

   6. Applied egg-rr 78.7%

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

   7. Taylor expanded in a around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 73.6%

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

      5. mul-1-neg 73.6%

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

      6. rem-cube-cbrt 73.6%

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

   9. Simplified 73.6%

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

   if 8.99999999999999952e-91 < b

   1. Initial program 17.2%

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

      1. sqr-neg 17.2%

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

      2. sqr-neg 17.2%

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

      3. associate-*l* 17.2%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in b around inf 87.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 87.7%

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

   7. Simplified 87.7%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{c \cdot \frac{-3}{a}} \cdot \left(--0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e-155)
   (/ (/ b -1.5) a)
   (if (<= b 4.8e-108)
     (* (sqrt (* c (/ -3.0 a))) (- -0.3333333333333333))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e-155) {
		tmp = (b / -1.5) / a;
	} else if (b <= 4.8e-108) {
		tmp = sqrt((c * (-3.0 / a))) * -(-0.3333333333333333);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-4.4d-155)) then
        tmp = (b / (-1.5d0)) / a
    else if (b <= 4.8d-108) then
        tmp = sqrt((c * ((-3.0d0) / a))) * -(-0.3333333333333333d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e-155) {
		tmp = (b / -1.5) / a;
	} else if (b <= 4.8e-108) {
		tmp = Math.sqrt((c * (-3.0 / a))) * -(-0.3333333333333333);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.4e-155:
		tmp = (b / -1.5) / a
	elif b <= 4.8e-108:
		tmp = math.sqrt((c * (-3.0 / a))) * -(-0.3333333333333333)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.4e-155)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 4.8e-108)
		tmp = Float64(sqrt(Float64(c * Float64(-3.0 / a))) * Float64(-(-0.3333333333333333)));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.4e-155)
		tmp = (b / -1.5) / a;
	elseif (b <= 4.8e-108)
		tmp = sqrt((c * (-3.0 / a))) * -(-0.3333333333333333);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.4e-155], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.8e-108], N[(N[Sqrt[N[(c * N[(-3.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (--0.3333333333333333)), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.3999999999999998e-155

   1. Initial program 64.7%

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

      1. sqr-neg 64.7%

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

      2. sqr-neg 64.7%

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

      3. associate-*l* 64.6%

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

   3. Simplified 64.6%

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

      1. clear-num 64.6%

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

      2. associate-/r/ 64.6%

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

      3. associate-/r* 64.5%

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

      4. metadata-eval 64.5%

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

      5. add-sqr-sqrt 64.4%

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

      6. sqrt-unprod 64.5%

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

      7. sqr-neg 64.5%

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

      8. sqrt-prod 0.0%

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

      9. add-sqr-sqrt 32.4%

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

      10. fma-neg 32.4%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}\right) \]

      11. distribute-lft-neg-in 32.4%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right) \cdot \left(a \cdot c\right)}\right)}\right) \]

      12. *-commutative 32.4%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(-3\right) \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      13. associate-*r* 32.4%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(-3\right) \cdot c\right) \cdot a}\right)}\right) \]

      14. metadata-eval 32.4%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{-3} \cdot c\right) \cdot a\right)}\right) \]

   6. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right)} \]

   7. Taylor expanded in a around 0 1.0%

      \[\leadsto \color{blue}{0.6666666666666666 \cdot \frac{b}{a}} \]
   8. Step-by-step derivation

      1. *-commutative 1.0%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot 0.6666666666666666} \]

   9. Simplified 1.0%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot 0.6666666666666666} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.6%

          \[\leadsto \color{blue}{\sqrt{\frac{b}{a} \cdot 0.6666666666666666} \cdot \sqrt{\frac{b}{a} \cdot 0.6666666666666666}} \]

       2. sqrt-unprod 23.6%

          \[\leadsto \color{blue}{\sqrt{\left(\frac{b}{a} \cdot 0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot 0.6666666666666666\right)}} \]

       3. swap-sqr 23.6%

          \[\leadsto \sqrt{\color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}} \]

       4. metadata-eval 23.6%

          \[\leadsto \sqrt{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \color{blue}{0.4444444444444444}} \]

       5. metadata-eval 23.6%

          \[\leadsto \sqrt{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \color{blue}{\left(-0.6666666666666666 \cdot -0.6666666666666666\right)}} \]

       6. swap-sqr 23.6%

          \[\leadsto \sqrt{\color{blue}{\left(\frac{b}{a} \cdot -0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)}} \]

       7. sqrt-unprod 38.4%

          \[\leadsto \color{blue}{\sqrt{\frac{b}{a} \cdot -0.6666666666666666} \cdot \sqrt{\frac{b}{a} \cdot -0.6666666666666666}} \]

       8. add-sqr-sqrt 80.7%

          \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

       9. metadata-eval 80.7%

          \[\leadsto \frac{b}{a} \cdot \color{blue}{\frac{1}{-1.5}} \]

       10. div-inv 80.9%

           \[\leadsto \color{blue}{\frac{\frac{b}{a}}{-1.5}} \]

       11. associate-/l/ 80.9%

           \[\leadsto \color{blue}{\frac{b}{-1.5 \cdot a}} \]

       12. associate-/r* 81.0%

           \[\leadsto \color{blue}{\frac{\frac{b}{-1.5}}{a}} \]

   11. Applied egg-rr 81.0%

       \[\leadsto \color{blue}{\frac{\frac{b}{-1.5}}{a}} \]

   if -4.3999999999999998e-155 < b < 4.80000000000000034e-108

   1. Initial program 76.7%

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

      1. sqr-neg 76.7%

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

      2. sqr-neg 76.7%

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

      3. associate-*l* 76.6%

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

   3. Simplified 76.6%

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

      1. add-cube-cbrt 76.1%

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

      2. pow3 76.1%

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

      3. associate-*r* 76.0%

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

      4. *-commutative 76.0%

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

      5. associate-*l* 75.9%

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

   6. Applied egg-rr 75.9%

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

   7. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}\right) \]

      3. rem-square-sqrt 29.4%

         \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}\right) \]

      4. rem-cube-cbrt 29.7%

         \[\leadsto -0.3333333333333333 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot \color{blue}{-3}}{a}}\right) \]

      5. associate-/l* 29.6%

         \[\leadsto -0.3333333333333333 \cdot \left(-1 \cdot \sqrt{\color{blue}{c \cdot \frac{-3}{a}}}\right) \]

   9. Simplified 29.6%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-3}{a}}\right)} \]

   if 4.80000000000000034e-108 < b

   1. Initial program 18.0%

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

      1. sqr-neg 18.0%

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

      2. sqr-neg 18.0%

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

      3. associate-*l* 18.0%

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

   3. Simplified 18.0%

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

   5. Taylor expanded in b around inf 85.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{c \cdot \frac{-3}{a}} \cdot \left(--0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-291}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.7e-291) (/ (/ b -1.5) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.7e-291) {
		tmp = (b / -1.5) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.7d-291) then
        tmp = (b / (-1.5d0)) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.7e-291) {
		tmp = (b / -1.5) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.7e-291:
		tmp = (b / -1.5) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.7e-291)
		tmp = Float64(Float64(b / -1.5) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.7e-291)
		tmp = (b / -1.5) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.7e-291], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7000000000000001e-291

   1. Initial program 65.9%

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

      1. sqr-neg 65.9%

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

      2. sqr-neg 65.9%

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

      3. associate-*l* 65.9%

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

   3. Simplified 65.9%

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

      1. clear-num 65.8%

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

      2. associate-/r/ 65.8%

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

      3. associate-/r* 65.8%

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

      4. metadata-eval 65.8%

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

      5. add-sqr-sqrt 65.0%

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

      6. sqrt-unprod 65.7%

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

      7. sqr-neg 65.7%

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

      8. sqrt-prod 0.7%

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

      9. add-sqr-sqrt 39.1%

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

      10. fma-neg 39.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}\right) \]

      11. distribute-lft-neg-in 39.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right) \cdot \left(a \cdot c\right)}\right)}\right) \]

      12. *-commutative 39.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(-3\right) \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      13. associate-*r* 39.0%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(-3\right) \cdot c\right) \cdot a}\right)}\right) \]

      14. metadata-eval 39.0%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{-3} \cdot c\right) \cdot a\right)}\right) \]

   6. Applied egg-rr 39.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right)} \]

   7. Taylor expanded in a around 0 1.4%

      \[\leadsto \color{blue}{0.6666666666666666 \cdot \frac{b}{a}} \]
   8. Step-by-step derivation

      1. *-commutative 1.4%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot 0.6666666666666666} \]

   9. Simplified 1.4%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot 0.6666666666666666} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 1.0%

          \[\leadsto \color{blue}{\sqrt{\frac{b}{a} \cdot 0.6666666666666666} \cdot \sqrt{\frac{b}{a} \cdot 0.6666666666666666}} \]

       2. sqrt-unprod 20.1%

          \[\leadsto \color{blue}{\sqrt{\left(\frac{b}{a} \cdot 0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot 0.6666666666666666\right)}} \]

       3. swap-sqr 20.1%

          \[\leadsto \sqrt{\color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}} \]

       4. metadata-eval 20.1%

          \[\leadsto \sqrt{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \color{blue}{0.4444444444444444}} \]

       5. metadata-eval 20.1%

          \[\leadsto \sqrt{\left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \color{blue}{\left(-0.6666666666666666 \cdot -0.6666666666666666\right)}} \]

       6. swap-sqr 20.1%

          \[\leadsto \sqrt{\color{blue}{\left(\frac{b}{a} \cdot -0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)}} \]

       7. sqrt-unprod 32.2%

          \[\leadsto \color{blue}{\sqrt{\frac{b}{a} \cdot -0.6666666666666666} \cdot \sqrt{\frac{b}{a} \cdot -0.6666666666666666}} \]

       8. add-sqr-sqrt 68.1%

          \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

       9. metadata-eval 68.1%

          \[\leadsto \frac{b}{a} \cdot \color{blue}{\frac{1}{-1.5}} \]

       10. div-inv 68.2%

           \[\leadsto \color{blue}{\frac{\frac{b}{a}}{-1.5}} \]

       11. associate-/l/ 68.2%

           \[\leadsto \color{blue}{\frac{b}{-1.5 \cdot a}} \]

       12. associate-/r* 68.3%

           \[\leadsto \color{blue}{\frac{\frac{b}{-1.5}}{a}} \]

   11. Applied egg-rr 68.3%

       \[\leadsto \color{blue}{\frac{\frac{b}{-1.5}}{a}} \]

   if 3.7000000000000001e-291 < b

   1. Initial program 34.2%

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

      1. sqr-neg 34.2%

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

      2. sqr-neg 34.2%

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

      3. associate-*l* 34.1%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in b around inf 66.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 66.6%

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

   7. Simplified 66.6%

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

Alternative 6: 68.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-291}:\\ \;\;\;\;\frac{b}{-1.5 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.7e-291) (/ b (* -1.5 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.7e-291) {
		tmp = b / (-1.5 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.7d-291) then
        tmp = b / ((-1.5d0) * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.7e-291) {
		tmp = b / (-1.5 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.7e-291:
		tmp = b / (-1.5 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.7e-291)
		tmp = Float64(b / Float64(-1.5 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.7e-291)
		tmp = b / (-1.5 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.7e-291], N[(b / N[(-1.5 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7000000000000001e-291

   1. Initial program 65.9%

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

      1. sqr-neg 65.9%

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

      2. sqr-neg 65.9%

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

      3. associate-*l* 65.9%

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

   3. Simplified 65.9%

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

      1. frac-2neg 65.9%

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

      2. div-inv 65.8%

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

   6. Applied egg-rr 65.8%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
   7. Step-by-step derivation

      1. *-commutative 65.8%

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

      2. *-commutative 65.8%

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

      3. associate-/r* 65.8%

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

   8. Simplified 65.8%

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

   9. Taylor expanded in b around -inf 68.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   10. Step-by-step derivation

       1. *-commutative 68.1%

          \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

       2. associate-*l/ 68.2%

          \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

       3. associate-/l* 68.1%

          \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   11. Simplified 68.1%

       \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]
   12. Step-by-step derivation

       1. clear-num 68.1%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 68.2%

          \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

       3. div-inv 68.2%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 68.2%

          \[\leadsto \frac{b}{a \cdot \color{blue}{-1.5}} \]

   13. Applied egg-rr 68.2%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if 3.7000000000000001e-291 < b

   1. Initial program 34.2%

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

      1. sqr-neg 34.2%

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

      2. sqr-neg 34.2%

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

      3. associate-*l* 34.1%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in b around inf 66.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 66.6%

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

   7. Simplified 66.6%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-291}:\\ \;\;\;\;\frac{b}{-1.5 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-291}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.7e-291) (* (/ b a) -0.6666666666666666) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.7e-291) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.7d-291) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.7e-291) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.7e-291:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.7e-291)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.7e-291)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.7e-291], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7000000000000001e-291

   1. Initial program 65.9%

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

      1. sqr-neg 65.9%

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

      2. sqr-neg 65.9%

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

      3. associate-*l* 65.9%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in b around -inf 68.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 68.1%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   7. Simplified 68.1%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   if 3.7000000000000001e-291 < b

   1. Initial program 34.2%

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

      1. sqr-neg 34.2%

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

      2. sqr-neg 34.2%

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

      3. associate-*l* 34.1%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in b around inf 66.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. *-commutative 66.6%

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

   7. Simplified 66.6%

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

Alternative 8: 35.5% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \cdot -0.6666666666666666 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ b a) -0.6666666666666666))

C

double code(double a, double b, double c) {
	return (b / a) * -0.6666666666666666;
}

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) * (-0.6666666666666666d0)
end function

Java

public static double code(double a, double b, double c) {
	return (b / a) * -0.6666666666666666;
}

Python

def code(a, b, c):
	return (b / a) * -0.6666666666666666

Julia

function code(a, b, c)
	return Float64(Float64(b / a) * -0.6666666666666666)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (b / a) * -0.6666666666666666;
end

Wolfram

code[a_, b_, c_] := N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 51.5%

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

   1. sqr-neg 51.5%

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

   2. sqr-neg 51.5%

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

   3. associate-*l* 51.5%

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

3. Simplified 51.5%

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

5. Taylor expanded in b around -inf 38.4%

   \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
6. Step-by-step derivation

   1. *-commutative 38.4%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

7. Simplified 38.4%

   \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
8. Add Preprocessing

Alternative 9: 35.5% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ b \cdot \frac{-0.6666666666666666}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* b (/ -0.6666666666666666 a)))

C

double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / 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 * ((-0.6666666666666666d0) / a)
end function

Java

public static double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / a);
}

Python

def code(a, b, c):
	return b * (-0.6666666666666666 / a)

Julia

function code(a, b, c)
	return Float64(b * Float64(-0.6666666666666666 / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = b * (-0.6666666666666666 / a);
end

Wolfram

code[a_, b_, c_] := N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

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

   1. sqr-neg 51.5%

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

   2. sqr-neg 51.5%

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

   3. associate-*l* 51.5%

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

3. Simplified 51.5%

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

   1. frac-2neg 51.5%

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

   2. div-inv 51.5%

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

6. Applied egg-rr 51.4%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}\right) \cdot \frac{1}{a \cdot -3}} \]
7. Step-by-step derivation

   1. *-commutative 51.4%

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

   2. *-commutative 51.4%

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

   3. associate-/r* 51.5%

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

8. Simplified 51.5%

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

9. Taylor expanded in b around -inf 38.4%

   \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
10. Step-by-step derivation

    1. *-commutative 38.4%

       \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

    2. associate-*l/ 38.4%

       \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

    3. associate-/l* 38.4%

       \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

11. Simplified 38.4%

    \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024132 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))