Cubic critical

Percentage Accurate: 51.3% → 85.2%

Time: 15.8s

Alternatives: 11

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: 51.3% 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: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{+89}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.65e+89)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 1.22e-33)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.65e+89) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.22e-33) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / 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 <= (-1.65d+89)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 1.22d-33) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.65e+89) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.22e-33) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.65e+89:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 1.22e-33:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.65e+89)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.22e-33)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.65e+89)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 1.22e-33)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.65e+89], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.22e-33], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.64999999999999987e89

   1. Initial program 51.9%

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

   3. Taylor expanded in b around -inf 95.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -1.64999999999999987e89 < b < 1.22e-33

   1. Initial program 80.9%

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

   if 1.22e-33 < b

   1. Initial program 8.2%

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

   3. Taylor expanded in b around inf 91.2%

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

      1. associate-*r/ 91.2%

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

   5. Simplified 91.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{+89}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.15 \cdot 10^{-100}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.15e-100)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 8.2e-34)
     (* (/ 1.0 (* a -3.0)) (- b (sqrt (* c (* a -3.0)))))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.15e-100) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 8.2e-34) {
		tmp = (1.0 / (a * -3.0)) * (b - sqrt((c * (a * -3.0))));
	} else {
		tmp = (c * -0.5) / 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 <= (-3.15d-100)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 8.2d-34) then
        tmp = (1.0d0 / (a * (-3.0d0))) * (b - sqrt((c * (a * (-3.0d0)))))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.15e-100) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 8.2e-34) {
		tmp = (1.0 / (a * -3.0)) * (b - Math.sqrt((c * (a * -3.0))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.15e-100:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 8.2e-34:
		tmp = (1.0 / (a * -3.0)) * (b - math.sqrt((c * (a * -3.0))))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.15e-100)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 8.2e-34)
		tmp = Float64(Float64(1.0 / Float64(a * -3.0)) * Float64(b - sqrt(Float64(c * Float64(a * -3.0)))));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.15e-100)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 8.2e-34)
		tmp = (1.0 / (a * -3.0)) * (b - sqrt((c * (a * -3.0))));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.15e-100], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.2e-34], N[(N[(1.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision] * N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.1499999999999998e-100

   1. Initial program 70.2%

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

   3. Taylor expanded in b around -inf 85.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -3.1499999999999998e-100 < b < 8.2000000000000007e-34

   1. Initial program 72.1%

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

   3. Taylor expanded in b around 0 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*l* 69.5%

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

   5. Simplified 69.5%

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

      1. frac-2neg 69.5%

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

      2. div-inv 69.5%

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

      3. distribute-neg-in 69.5%

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

      4. add-sqr-sqrt 30.4%

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

      5. sqrt-unprod 69.0%

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

      6. sqr-neg 69.0%

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

      7. sqrt-unprod 39.4%

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

      8. add-sqr-sqrt 68.8%

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

      9. sub-neg 68.8%

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

      10. add-sqr-sqrt 29.4%

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

      11. sqrt-unprod 68.7%

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

      12. sqr-neg 68.7%

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

      13. sqrt-unprod 39.2%

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

      14. add-sqr-sqrt 69.5%

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

      15. distribute-lft-neg-in 69.5%

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

      16. metadata-eval 69.5%

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

      17. associate-/l/ 69.3%

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

      18. div-inv 69.3%

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

      19. metadata-eval 69.3%

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

   7. Applied egg-rr 69.3%

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

      1. *-commutative 69.3%

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

      2. associate-*l/ 69.4%

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

      3. metadata-eval 69.4%

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

      4. associate-*r* 70.6%

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

      5. *-commutative 70.6%

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

      6. associate-*l* 70.7%

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

   9. Simplified 70.7%

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

       1. clear-num 70.7%

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

       2. inv-pow 70.7%

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

       3. div-inv 70.9%

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

       4. metadata-eval 70.9%

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

   11. Applied egg-rr 70.9%

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

       1. unpow-1 70.9%

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

   13. Simplified 70.9%

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

   if 8.2000000000000007e-34 < b

   1. Initial program 8.2%

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

   3. Taylor expanded in b around inf 91.2%

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

      1. associate-*r/ 91.2%

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

   5. Simplified 91.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.15 \cdot 10^{-100}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{a \cdot -3} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-124}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-33}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e-124)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 1.02e-33)
     (* 0.3333333333333333 (/ (+ b (sqrt (* a (* c -3.0)))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-124) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.02e-33) {
		tmp = 0.3333333333333333 * ((b + sqrt((a * (c * -3.0)))) / a);
	} else {
		tmp = (c * -0.5) / 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 <= (-1.6d-124)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 1.02d-33) then
        tmp = 0.3333333333333333d0 * ((b + sqrt((a * (c * (-3.0d0))))) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-124) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.02e-33) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((a * (c * -3.0)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.6e-124:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 1.02e-33:
		tmp = 0.3333333333333333 * ((b + math.sqrt((a * (c * -3.0)))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e-124)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.02e-33)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(a * Float64(c * -3.0)))) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.6e-124)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 1.02e-33)
		tmp = 0.3333333333333333 * ((b + sqrt((a * (c * -3.0)))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e-124], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.02e-33], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.60000000000000002e-124

   1. Initial program 70.2%

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

   3. Taylor expanded in b around -inf 83.8%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -1.60000000000000002e-124 < b < 1.02e-33

   1. Initial program 72.3%

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

   3. Taylor expanded in b around 0 72.0%

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

      1. *-commutative 72.0%

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

      2. associate-*l* 70.7%

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

   5. Simplified 70.7%

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

      1. *-un-lft-identity 70.7%

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

      2. times-frac 70.7%

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

      3. metadata-eval 70.7%

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

      4. +-commutative 70.7%

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

      5. add-sqr-sqrt 29.8%

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

      6. sqrt-unprod 70.1%

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

      7. sqr-neg 70.1%

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

      8. sqrt-unprod 41.1%

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

      9. add-sqr-sqrt 70.2%

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

   7. Applied egg-rr 70.2%

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

   if 1.02e-33 < b

   1. Initial program 8.2%

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

   3. Taylor expanded in b around inf 91.2%

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

      1. associate-*r/ 91.2%

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

   5. Simplified 91.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-124}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-33}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-100}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-100)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 4.5e-33)
     (* (- b (sqrt (* c (* a -3.0)))) (/ -0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-100) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 4.5e-33) {
		tmp = (b - sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3d-100)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 4.5d-33) then
        tmp = (b - sqrt((c * (a * (-3.0d0))))) * ((-0.3333333333333333d0) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-100) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 4.5e-33) {
		tmp = (b - Math.sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e-100:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 4.5e-33:
		tmp = (b - math.sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-100)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 4.5e-33)
		tmp = Float64(Float64(b - sqrt(Float64(c * Float64(a * -3.0)))) * Float64(-0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e-100)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 4.5e-33)
		tmp = (b - sqrt((c * (a * -3.0)))) * (-0.3333333333333333 / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-100], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.5e-33], N[(N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.0000000000000001e-100

   1. Initial program 70.2%

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

   3. Taylor expanded in b around -inf 85.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -3.0000000000000001e-100 < b < 4.49999999999999991e-33

   1. Initial program 72.1%

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

   3. Taylor expanded in b around 0 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*l* 69.5%

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

   5. Simplified 69.5%

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

      1. frac-2neg 69.5%

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

      2. div-inv 69.5%

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

      3. distribute-neg-in 69.5%

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

      4. add-sqr-sqrt 30.4%

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

      5. sqrt-unprod 69.0%

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

      6. sqr-neg 69.0%

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

      7. sqrt-unprod 39.4%

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

      8. add-sqr-sqrt 68.8%

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

      9. sub-neg 68.8%

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

      10. add-sqr-sqrt 29.4%

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

      11. sqrt-unprod 68.7%

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

      12. sqr-neg 68.7%

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

      13. sqrt-unprod 39.2%

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

      14. add-sqr-sqrt 69.5%

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

      15. distribute-lft-neg-in 69.5%

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

      16. metadata-eval 69.5%

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

      17. associate-/l/ 69.3%

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

      18. div-inv 69.3%

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

      19. metadata-eval 69.3%

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

   7. Applied egg-rr 69.3%

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

      1. *-commutative 69.3%

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

      2. associate-*l/ 69.4%

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

      3. metadata-eval 69.4%

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

      4. associate-*r* 70.6%

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

      5. *-commutative 70.6%

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

      6. associate-*l* 70.7%

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

   9. Simplified 70.7%

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

   if 4.49999999999999991e-33 < b

   1. Initial program 8.2%

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

   3. Taylor expanded in b around inf 91.2%

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

      1. associate-*r/ 91.2%

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

   5. Simplified 91.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-100}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.7 \cdot 10^{-102}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.7e-102)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 8.2e-34)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.7e-102) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 8.2e-34) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / 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 <= (-6.7d-102)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 8.2d-34) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.7e-102) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 8.2e-34) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.7e-102:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 8.2e-34:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.7e-102)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 8.2e-34)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.7e-102)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 8.2e-34)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.7e-102], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.2e-34], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.7e-102

   1. Initial program 70.2%

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

   3. Taylor expanded in b around -inf 85.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -6.7e-102 < b < 8.2000000000000007e-34

   1. Initial program 72.1%

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

   3. Taylor expanded in b around 0 70.7%

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

      1. associate-*r* 70.8%

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

      2. *-commutative 70.8%

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

      3. *-commutative 70.8%

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

   5. Simplified 70.8%

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

   if 8.2000000000000007e-34 < b

   1. Initial program 8.2%

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

   3. Taylor expanded in b around inf 91.2%

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

      1. associate-*r/ 91.2%

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

   5. Simplified 91.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.7 \cdot 10^{-102}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = (c * -0.5) / 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 = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else
        tmp = (c * (-0.5d0)) / 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 = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 69.9%

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

   3. Taylor expanded in b around -inf 67.9%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 30.1%

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

   3. Taylor expanded in b around inf 66.8%

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

      1. associate-*r/ 66.8%

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

   5. Simplified 66.8%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-287}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.5e-287) (/ -0.6666666666666666 (/ a b)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.5e-287) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 / (b / c);
	}
	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 <= 1.5d-287) then
        tmp = (-0.6666666666666666d0) / (a / b)
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.5e-287) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.5e-287:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.5e-287)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.5e-287)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.5e-287], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.49999999999999996e-287

   1. Initial program 70.1%

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

   3. Taylor expanded in b around -inf 67.1%

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

      1. *-commutative 67.1%

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

   5. Simplified 67.1%

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

      1. *-commutative 67.1%

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

      2. add-cbrt-cube 37.8%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(-0.6666666666666666 \cdot \frac{b}{a}\right) \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)\right) \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)}} \]

      3. pow1/3 19.3%

         \[\leadsto \color{blue}{{\left(\left(\left(-0.6666666666666666 \cdot \frac{b}{a}\right) \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)\right) \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)\right)}^{0.3333333333333333}} \]

      4. pow3 19.3%

         \[\leadsto {\color{blue}{\left({\left(-0.6666666666666666 \cdot \frac{b}{a}\right)}^{3}\right)}}^{0.3333333333333333} \]

      5. *-commutative 19.3%

         \[\leadsto {\left({\color{blue}{\left(\frac{b}{a} \cdot -0.6666666666666666\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. associate-*l/ 19.3%

         \[\leadsto {\left({\color{blue}{\left(\frac{b \cdot -0.6666666666666666}{a}\right)}}^{3}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 19.3%

      \[\leadsto \color{blue}{{\left({\left(\frac{b \cdot -0.6666666666666666}{a}\right)}^{3}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 37.7%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{b \cdot -0.6666666666666666}{a}\right)}^{3}}} \]

      2. rem-cbrt-cube 67.2%

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

      3. *-commutative 67.2%

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

      4. associate-/l* 67.1%

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

   9. Applied egg-rr 67.1%

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

   if 1.49999999999999996e-287 < b

   1. Initial program 29.5%

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

   3. Taylor expanded in b around inf 67.3%

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

      1. associate-*r/ 67.3%

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

      2. associate-/l* 66.5%

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

   5. Simplified 66.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-287}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-287}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.5e-287) (/ b (* a -1.5)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.5e-287) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 / (b / c);
	}
	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 <= 1.5d-287) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.5e-287) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.5e-287:
		tmp = b / (a * -1.5)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.5e-287)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.5e-287)
		tmp = b / (a * -1.5);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.5e-287], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.49999999999999996e-287

   1. Initial program 70.1%

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

   4. Applied egg-rr 66.3%

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

      1. associate-/r* 66.3%

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

   6. Simplified 66.3%

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

   7. Taylor expanded in b around -inf 67.2%

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

      1. *-commutative 67.2%

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

   9. Simplified 67.2%

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

   10. Taylor expanded in b around 0 67.1%

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

       1. *-commutative 67.1%

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

       2. metadata-eval 67.1%

          \[\leadsto \frac{b}{a} \cdot \color{blue}{\frac{2}{-3}} \]

       3. times-frac 66.6%

          \[\leadsto \color{blue}{\frac{b \cdot 2}{a \cdot -3}} \]

       4. associate-/l* 66.6%

          \[\leadsto \color{blue}{\frac{b}{\frac{a \cdot -3}{2}}} \]

       5. associate-/l* 67.2%

          \[\leadsto \frac{b}{\color{blue}{\frac{a}{\frac{2}{-3}}}} \]

       6. metadata-eval 67.2%

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

   12. Simplified 67.2%

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

   13. Taylor expanded in a around 0 67.3%

       \[\leadsto \frac{b}{\color{blue}{-1.5 \cdot a}} \]
   14. Step-by-step derivation

       1. *-commutative 67.3%

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

   15. Simplified 67.3%

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

   if 1.49999999999999996e-287 < b

   1. Initial program 29.5%

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

   3. Taylor expanded in b around inf 67.3%

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

      1. associate-*r/ 67.3%

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

      2. associate-/l* 66.5%

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

   5. Simplified 66.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-287}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-287}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.8e-287) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.8e-287) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.8d-287) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.8e-287) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.8e-287:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.8e-287)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.8e-287)
		tmp = b / (a * -1.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.8e-287], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8000000000000002e-287

   1. Initial program 70.1%

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

   4. Applied egg-rr 66.3%

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

      1. associate-/r* 66.3%

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

   6. Simplified 66.3%

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

   7. Taylor expanded in b around -inf 67.2%

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

      1. *-commutative 67.2%

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

   9. Simplified 67.2%

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

   10. Taylor expanded in b around 0 67.1%

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

       1. *-commutative 67.1%

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

       2. metadata-eval 67.1%

          \[\leadsto \frac{b}{a} \cdot \color{blue}{\frac{2}{-3}} \]

       3. times-frac 66.6%

          \[\leadsto \color{blue}{\frac{b \cdot 2}{a \cdot -3}} \]

       4. associate-/l* 66.6%

          \[\leadsto \color{blue}{\frac{b}{\frac{a \cdot -3}{2}}} \]

       5. associate-/l* 67.2%

          \[\leadsto \frac{b}{\color{blue}{\frac{a}{\frac{2}{-3}}}} \]

       6. metadata-eval 67.2%

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

   12. Simplified 67.2%

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

   13. Taylor expanded in a around 0 67.3%

       \[\leadsto \frac{b}{\color{blue}{-1.5 \cdot a}} \]
   14. Step-by-step derivation

       1. *-commutative 67.3%

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

   15. Simplified 67.3%

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

   if 2.8000000000000002e-287 < b

   1. Initial program 29.5%

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

   3. Taylor expanded in b around inf 67.3%

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

      1. associate-*r/ 67.3%

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

   5. Simplified 67.3%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-287}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.8% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in b around -inf 37.4%

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

   1. *-commutative 37.4%

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

5. Simplified 37.4%

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

6. Final simplification 37.4%

   \[\leadsto -0.6666666666666666 \cdot \frac{b}{a} \]
7. Add Preprocessing

Alternative 11: 34.8% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in b around -inf 37.4%

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

   1. *-commutative 37.4%

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

5. Simplified 37.4%

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

   1. *-commutative 37.4%

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

   2. add-cbrt-cube 21.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(-0.6666666666666666 \cdot \frac{b}{a}\right) \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)\right) \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)}} \]

   3. pow1/3 11.4%

      \[\leadsto \color{blue}{{\left(\left(\left(-0.6666666666666666 \cdot \frac{b}{a}\right) \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)\right) \cdot \left(-0.6666666666666666 \cdot \frac{b}{a}\right)\right)}^{0.3333333333333333}} \]

   4. pow3 11.4%

      \[\leadsto {\color{blue}{\left({\left(-0.6666666666666666 \cdot \frac{b}{a}\right)}^{3}\right)}}^{0.3333333333333333} \]

   5. *-commutative 11.4%

      \[\leadsto {\left({\color{blue}{\left(\frac{b}{a} \cdot -0.6666666666666666\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. associate-*l/ 11.4%

      \[\leadsto {\left({\color{blue}{\left(\frac{b \cdot -0.6666666666666666}{a}\right)}}^{3}\right)}^{0.3333333333333333} \]

7. Applied egg-rr 11.4%

   \[\leadsto \color{blue}{{\left({\left(\frac{b \cdot -0.6666666666666666}{a}\right)}^{3}\right)}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. unpow1/3 21.5%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{b \cdot -0.6666666666666666}{a}\right)}^{3}}} \]

   2. rem-cbrt-cube 37.4%

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

   3. *-commutative 37.4%

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

   4. associate-/l* 37.4%

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

9. Applied egg-rr 37.4%

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

10. Final simplification 37.4%

    \[\leadsto \frac{-0.6666666666666666}{\frac{a}{b}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))