Cubic critical

Percentage Accurate: 51.8% → 84.7%

Time: 13.1s

Alternatives: 8

Speedup: 16.4×

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: 51.8% 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: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{+61}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-109}:\\ \;\;\;\;\left(\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b\right) \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.95e+61)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 2e-109)
     (* (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (/ 1.0 (* a 3.0)))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e+61) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 2e-109) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) * (1.0 / (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 <= (-1.95d+61)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 2d-109) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) * (1.0d0 / (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 <= -1.95e+61) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 2e-109) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) * (1.0 / (a * 3.0));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.95e+61:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 2e-109:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) * (1.0 / (a * 3.0))
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.95e+61)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 2e-109)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) * Float64(1.0 / 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 <= -1.95e+61)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 2e-109)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) * (1.0 / (a * 3.0));
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.95e+61], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-109], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(1.0 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.94999999999999994e61

   1. Initial program 65.3%

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

      1. sqr-neg 65.3%

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

      2. sqr-neg 65.3%

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

      3. associate-*l* 65.3%

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

   3. Simplified 65.3%

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

   4. Taylor expanded in b around -inf 88.0%

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

   if -1.94999999999999994e61 < b < 2e-109

   1. Initial program 87.9%

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

      1. neg-sub0 87.9%

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

      2. sqr-neg 87.9%

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

      3. associate-+l- 87.9%

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

      4. sub0-neg 87.9%

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

      5. neg-mul-1 87.9%

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

   3. Simplified 87.6%

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

      1. associate-*r* 87.7%

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

      2. metadata-eval 87.7%

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

      3. distribute-rgt-neg-in 87.7%

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

      4. *-commutative 87.7%

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

      5. fma-neg 87.7%

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

      6. associate-*r* 87.6%

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

      7. *-commutative 87.6%

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

      8. associate-*l* 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. associate-*r* 87.6%

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

   7. Simplified 87.6%

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

      1. div-inv 87.7%

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

      2. associate-*l* 87.6%

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

      3. clear-num 87.9%

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

   9. Applied egg-rr 87.9%

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

       1. clear-num 87.6%

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

       2. inv-pow 87.6%

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

       3. div-inv 87.9%

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

       4. metadata-eval 87.9%

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

   11. Applied egg-rr 87.9%

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

       1. unpow-1 87.9%

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

   13. Applied egg-rr 87.9%

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

   if 2e-109 < b

   1. Initial program 16.4%

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

      1. sqr-neg 16.4%

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

      2. sqr-neg 16.4%

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

      3. associate-*l* 16.4%

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

   3. Simplified 16.4%

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

   4. Taylor expanded in b around inf 91.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{+61}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-109}:\\ \;\;\;\;\left(\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b\right) \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 2: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{+61}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-102}:\\ \;\;\;\;\left(\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.95e+61)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 7.2e-102)
     (* (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (/ 0.3333333333333333 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e+61) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 7.2e-102) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) * (0.3333333333333333 / 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 <= (-1.95d+61)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 7.2d-102) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) * (0.3333333333333333d0 / 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 <= -1.95e+61) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 7.2e-102) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.95e+61:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 7.2e-102:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) * (0.3333333333333333 / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.95e+61)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 7.2e-102)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) * Float64(0.3333333333333333 / 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 <= -1.95e+61)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 7.2e-102)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) * (0.3333333333333333 / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.95e+61], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e-102], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.94999999999999994e61

   1. Initial program 65.3%

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

      1. sqr-neg 65.3%

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

      2. sqr-neg 65.3%

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

      3. associate-*l* 65.3%

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

   3. Simplified 65.3%

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

   4. Taylor expanded in b around -inf 88.0%

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

   if -1.94999999999999994e61 < b < 7.2e-102

   1. Initial program 87.9%

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

      1. neg-sub0 87.9%

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

      2. sqr-neg 87.9%

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

      3. associate-+l- 87.9%

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

      4. sub0-neg 87.9%

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

      5. neg-mul-1 87.9%

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

   3. Simplified 87.6%

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

      1. associate-*r* 87.7%

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

      2. metadata-eval 87.7%

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

      3. distribute-rgt-neg-in 87.7%

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

      4. *-commutative 87.7%

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

      5. fma-neg 87.7%

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

      6. associate-*r* 87.6%

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

      7. *-commutative 87.6%

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

      8. associate-*l* 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. associate-*r* 87.6%

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

   7. Simplified 87.6%

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

      1. div-inv 87.7%

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

      2. associate-*l* 87.6%

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

      3. clear-num 87.9%

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

   9. Applied egg-rr 87.9%

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

   if 7.2e-102 < b

   1. Initial program 16.4%

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

      1. sqr-neg 16.4%

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

      2. sqr-neg 16.4%

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

      3. associate-*l* 16.4%

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

   3. Simplified 16.4%

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

   4. Taylor expanded in b around inf 91.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{+61}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-102}:\\ \;\;\;\;\left(\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 3: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-108}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-108}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-108)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 5.8e-108)
     (* (/ 0.3333333333333333 a) (- (sqrt (* c (* a -3.0))) b))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-108) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 5.8e-108) {
		tmp = (0.3333333333333333 / a) * (sqrt((c * (a * -3.0))) - b);
	} 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 <= (-3.8d-108)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 5.8d-108) then
        tmp = (0.3333333333333333d0 / a) * (sqrt((c * (a * (-3.0d0)))) - b)
    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 <= -3.8e-108) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 5.8e-108) {
		tmp = (0.3333333333333333 / a) * (Math.sqrt((c * (a * -3.0))) - b);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.8e-108:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 5.8e-108:
		tmp = (0.3333333333333333 / a) * (math.sqrt((c * (a * -3.0))) - b)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-108)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 5.8e-108)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(c * Float64(a * -3.0))) - b));
	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 <= -3.8e-108)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 5.8e-108)
		tmp = (0.3333333333333333 / a) * (sqrt((c * (a * -3.0))) - b);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-108], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e-108], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.79999999999999973e-108

   1. Initial program 73.9%

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

      1. sqr-neg 73.9%

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

      2. sqr-neg 73.9%

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

      3. associate-*l* 73.9%

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

   3. Simplified 73.9%

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

   4. Taylor expanded in b around -inf 82.8%

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

   if -3.79999999999999973e-108 < b < 5.8000000000000002e-108

   1. Initial program 83.7%

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

      1. sqr-neg 83.7%

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

      2. sqr-neg 83.7%

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

      3. associate-*l* 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in b around 0 80.5%

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

      1. *-commutative 80.5%

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

      2. *-commutative 80.5%

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

      3. *-commutative 80.5%

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

      4. associate-*l* 80.7%

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

   6. Simplified 80.7%

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

      1. expm1-log1p-u 57.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -3\right)}}{3 \cdot a}\right)\right)} \]

      2. expm1-udef 22.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -3\right)}}{3 \cdot a}\right)} - 1} \]

      3. div-inv 22.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(-b\right) + \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{1}{3 \cdot a}}\right)} - 1 \]

      4. neg-mul-1 22.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{-1 \cdot b} + \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      5. fma-def 22.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)} \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      6. *-commutative 22.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{1}{\color{blue}{a \cdot 3}}\right)} - 1 \]

      7. metadata-eval 22.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{1}{a \cdot \color{blue}{\frac{1}{0.3333333333333333}}}\right)} - 1 \]

      8. div-inv 22.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{1}{\color{blue}{\frac{a}{0.3333333333333333}}}\right)} - 1 \]

      9. clear-num 22.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \color{blue}{\frac{0.3333333333333333}{a}}\right)} - 1 \]

   8. Applied egg-rr 22.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{0.3333333333333333}{a}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 57.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{0.3333333333333333}{a}\right)\right)} \]

      2. expm1-log1p 80.7%

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

      3. *-commutative 80.7%

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

      4. fma-udef 80.7%

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

      5. neg-mul-1 80.7%

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

      6. +-commutative 80.7%

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

      7. unsub-neg 80.7%

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

   10. Simplified 80.7%

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

   if 5.8000000000000002e-108 < b

   1. Initial program 16.4%

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

      1. sqr-neg 16.4%

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

      2. sqr-neg 16.4%

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

      3. associate-*l* 16.4%

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

   3. Simplified 16.4%

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

   4. Taylor expanded in b around inf 91.3%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-108}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-108}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{c \cdot \left(a \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 4: 67.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} 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 <= (-4d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    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 <= -4e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	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 <= -4e-310)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 75.5%

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

      1. sqr-neg 75.5%

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

      2. sqr-neg 75.5%

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

      3. associate-*l* 75.6%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in b around -inf 69.0%

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

   if -3.999999999999988e-310 < b

   1. Initial program 31.8%

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

      1. sqr-neg 31.8%

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

      2. sqr-neg 31.8%

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

      3. associate-*l* 31.7%

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

   3. Simplified 31.7%

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

   4. Taylor expanded in b around inf 74.2%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 5: 67.6% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.8e-300) (/ (* b -2.0) (* a 3.0)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.8e-300) {
		tmp = (b * -2.0) / (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 <= 1.8d-300) then
        tmp = (b * (-2.0d0)) / (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 <= 1.8e-300) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.8e-300:
		tmp = (b * -2.0) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.8e-300)
		tmp = Float64(Float64(b * -2.0) / 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 <= 1.8e-300)
		tmp = (b * -2.0) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.8e-300], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000008e-300

   1. Initial program 75.9%

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

      1. sqr-neg 75.9%

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

      2. sqr-neg 75.9%

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

      3. associate-*l* 75.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in b around -inf 67.8%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 67.8%

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

   6. Simplified 67.8%

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

   if 1.80000000000000008e-300 < b

   1. Initial program 30.8%

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

      1. sqr-neg 30.8%

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

      2. sqr-neg 30.8%

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

      3. associate-*l* 30.7%

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

   3. Simplified 30.7%

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

   4. Taylor expanded in b around inf 75.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 6: 67.6% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-300}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.8e-300) (* b (/ -0.6666666666666666 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.8e-300) {
		tmp = b * (-0.6666666666666666 / 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 <= 1.8d-300) then
        tmp = b * ((-0.6666666666666666d0) / 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 <= 1.8e-300) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.8e-300:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.8e-300)
		tmp = Float64(b * Float64(-0.6666666666666666 / 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 <= 1.8e-300)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.8e-300], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000008e-300

   1. Initial program 75.9%

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

      1. neg-sub0 75.9%

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

      2. sqr-neg 75.9%

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

      3. associate-+l- 75.9%

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

      4. sub0-neg 75.9%

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

      5. neg-mul-1 75.9%

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

   3. Simplified 75.8%

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

      1. associate-*r* 75.9%

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

      2. metadata-eval 75.9%

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

      3. distribute-rgt-neg-in 75.9%

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

      4. *-commutative 75.9%

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

      5. fma-neg 75.9%

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

      6. associate-*r* 75.8%

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

      7. *-commutative 75.8%

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

      8. associate-*l* 75.8%

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

   5. Applied egg-rr 75.8%

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

      1. associate-*r* 75.8%

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

   7. Simplified 75.8%

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

      1. div-inv 75.8%

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

      2. associate-*l* 75.7%

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

      3. clear-num 75.9%

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

   9. Applied egg-rr 75.9%

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

       1. clear-num 75.7%

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

       2. inv-pow 75.7%

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

       3. div-inv 75.9%

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

       4. metadata-eval 75.9%

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

   11. Applied egg-rr 75.9%

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

   12. Taylor expanded in b around -inf 67.7%

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

       1. associate-*r/ 67.7%

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

       2. associate-*l/ 67.8%

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

       3. *-commutative 67.8%

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

   14. Simplified 67.8%

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

   if 1.80000000000000008e-300 < b

   1. Initial program 30.8%

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

      1. sqr-neg 30.8%

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

      2. sqr-neg 30.8%

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

      3. associate-*l* 30.7%

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

   3. Simplified 30.7%

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

   4. Taylor expanded in b around inf 75.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-300}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 7: 67.6% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.8e-300) (/ b (/ a -0.6666666666666666)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.8e-300) {
		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 <= 1.8d-300) 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 <= 1.8e-300) {
		tmp = b / (a / -0.6666666666666666);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.8e-300:
		tmp = b / (a / -0.6666666666666666)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.8e-300)
		tmp = Float64(b / Float64(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 <= 1.8e-300)
		tmp = b / (a / -0.6666666666666666);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.8e-300], N[(b / N[(a / -0.6666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000008e-300

   1. Initial program 75.9%

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

      1. neg-sub0 75.9%

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

      2. sqr-neg 75.9%

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

      3. associate-+l- 75.9%

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

      4. sub0-neg 75.9%

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

      5. neg-mul-1 75.9%

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

   3. Simplified 75.8%

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

      1. associate-*r* 75.9%

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

      2. metadata-eval 75.9%

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

      3. distribute-rgt-neg-in 75.9%

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

      4. *-commutative 75.9%

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

      5. fma-neg 75.9%

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

      6. associate-*r* 75.8%

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

      7. *-commutative 75.8%

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

      8. associate-*l* 75.8%

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

   5. Applied egg-rr 75.8%

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

      1. associate-*r* 75.8%

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

   7. Simplified 75.8%

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

   8. Taylor expanded in b around -inf 67.7%

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

      1. associate-*r/ 67.7%

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

      2. *-commutative 67.7%

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

      3. associate-/l* 67.8%

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

   10. Simplified 67.8%

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

   if 1.80000000000000008e-300 < b

   1. Initial program 30.8%

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

      1. sqr-neg 30.8%

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

      2. sqr-neg 30.8%

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

      3. associate-*l* 30.7%

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

   3. Simplified 30.7%

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

   4. Taylor expanded in b around inf 75.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 8: 34.8% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot -0.5 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ c b) -0.5))

C

double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c / b) * (-0.5d0)
end function

Java

public static double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

Python

def code(a, b, c):
	return (c / b) * -0.5

Julia

function code(a, b, c)
	return Float64(Float64(c / b) * -0.5)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c / b) * -0.5;
end

Wolfram

code[a_, b_, c_] := N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

   1. sqr-neg 53.3%

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

   2. sqr-neg 53.3%

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

   3. associate-*l* 53.3%

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

3. Simplified 53.3%

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

4. Taylor expanded in b around inf 38.7%

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

5. Final simplification 38.7%

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

Reproduce

herbie shell --seed 2023274 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))