Cubic critical

Percentage Accurate: 51.9% → 84.6%

Time: 11.4s

Alternatives: 10

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: 51.9% 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: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-127}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} - b\right) \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+33)
   (+ (/ (* b -0.6666666666666666) a) (* (/ c b) 0.5))
   (if (<= b 8e-127)
     (* (- (pow (fma b b (* c (* a -3.0))) 0.5) b) (/ 1.0 (* a 3.0)))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+33) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 8e-127) {
		tmp = (pow(fma(b, b, (c * (a * -3.0))), 0.5) - 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 <= -1.5e+33)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 8e-127)
		tmp = Float64(Float64((fma(b, b, Float64(c * Float64(a * -3.0))) ^ 0.5) - b) * Float64(1.0 / Float64(a * 3.0)));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+33], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-127], N[(N[(N[Power[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $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 < -1.49999999999999992e33

   1. Initial program 56.6%

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

      1. add-sqr-sqrt 56.6%

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

      2. pow2 56.6%

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

      3. pow1/2 56.6%

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

      4. sqrt-pow1 56.6%

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

      5. fma-neg 56.8%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      6. *-commutative 56.8%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      7. distribute-rgt-neg-in 56.8%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      8. *-commutative 56.8%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      9. distribute-rgt-neg-in 56.8%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      10. metadata-eval 56.8%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      11. metadata-eval 56.8%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{3 \cdot a} \]

   3. Applied egg-rr 56.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.25}\right)}^{2}}}{3 \cdot a} \]

   4. Taylor expanded in b around -inf 96.3%

      \[\leadsto \color{blue}{2 \cdot \left(-0.3333333333333333 \cdot \frac{b}{a}\right) + 0.5 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. associate-*r* 96.3%

         \[\leadsto \color{blue}{\left(2 \cdot -0.3333333333333333\right) \cdot \frac{b}{a}} + 0.5 \cdot \frac{c}{b} \]

      2. metadata-eval 96.3%

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

      3. *-commutative 96.3%

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

   6. Simplified 96.3%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5} \]
   7. Step-by-step derivation

      1. associate-*r/ 96.3%

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

   8. Applied egg-rr 96.3%

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

   if -1.49999999999999992e33 < b < 8.0000000000000002e-127

   1. Initial program 84.4%

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

      1. add-sqr-sqrt 84.0%

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

      2. pow2 84.0%

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

      3. pow1/2 84.0%

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

      4. sqrt-pow1 84.1%

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

      5. fma-neg 84.1%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      6. *-commutative 84.1%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      7. distribute-rgt-neg-in 84.1%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      8. *-commutative 84.1%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      9. distribute-rgt-neg-in 84.1%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      10. metadata-eval 84.1%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      11. metadata-eval 84.1%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{3 \cdot a} \]

   3. Applied egg-rr 84.1%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.25}\right)}^{2}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. div-inv 84.1%

         \[\leadsto \color{blue}{\left(\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{3 \cdot a}} \]

      2. pow-pow 84.4%

         \[\leadsto \left(\left(-b\right) + \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(0.25 \cdot 2\right)}}\right) \cdot \frac{1}{3 \cdot a} \]

      3. metadata-eval 84.4%

         \[\leadsto \left(\left(-b\right) + {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.5}}\right) \cdot \frac{1}{3 \cdot a} \]

      4. *-commutative 84.4%

         \[\leadsto \left(\left(-b\right) + {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5}\right) \cdot \frac{1}{\color{blue}{a \cdot 3}} \]

   5. Applied egg-rr 84.4%

      \[\leadsto \color{blue}{\left(\left(-b\right) + {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5}\right) \cdot \frac{1}{a \cdot 3}} \]

   if 8.0000000000000002e-127 < b

   1. Initial program 21.4%

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

   2. Taylor expanded in b around inf 84.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-127}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} - b\right) \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 2: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+33)
   (+ (/ (* b -0.6666666666666666) a) (* (/ c b) 0.5))
   (if (<= b 1.35e-126)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+33) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 1.35e-126) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.5d+33)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + ((c / b) * 0.5d0)
    else if (b <= 1.35d-126) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+33) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 1.35e-126) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.5e+33:
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5)
	elif b <= 1.35e-126:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+33)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 1.35e-126)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.5e+33)
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	elseif (b <= 1.35e-126)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+33], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-126], 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 / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.49999999999999992e33

   1. Initial program 56.6%

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

      1. add-sqr-sqrt 56.6%

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

      2. pow2 56.6%

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

      3. pow1/2 56.6%

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

      4. sqrt-pow1 56.6%

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

      5. fma-neg 56.8%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      6. *-commutative 56.8%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      7. distribute-rgt-neg-in 56.8%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      8. *-commutative 56.8%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      9. distribute-rgt-neg-in 56.8%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      10. metadata-eval 56.8%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      11. metadata-eval 56.8%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{3 \cdot a} \]

   3. Applied egg-rr 56.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.25}\right)}^{2}}}{3 \cdot a} \]

   4. Taylor expanded in b around -inf 96.3%

      \[\leadsto \color{blue}{2 \cdot \left(-0.3333333333333333 \cdot \frac{b}{a}\right) + 0.5 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. associate-*r* 96.3%

         \[\leadsto \color{blue}{\left(2 \cdot -0.3333333333333333\right) \cdot \frac{b}{a}} + 0.5 \cdot \frac{c}{b} \]

      2. metadata-eval 96.3%

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

      3. *-commutative 96.3%

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

   6. Simplified 96.3%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5} \]
   7. Step-by-step derivation

      1. associate-*r/ 96.3%

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

   8. Applied egg-rr 96.3%

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

   if -1.49999999999999992e33 < b < 1.34999999999999998e-126

   1. Initial program 84.4%

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

   if 1.34999999999999998e-126 < b

   1. Initial program 21.4%

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

   2. Taylor expanded in b around inf 84.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 3: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-126}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e-71)
   (+ (/ (* b -0.6666666666666666) a) (* (/ c b) 0.5))
   (if (<= b 1.35e-126)
     (* 0.3333333333333333 (/ (- (sqrt (* c (* a -3.0))) b) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e-71) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 1.35e-126) {
		tmp = 0.3333333333333333 * ((sqrt((c * (a * -3.0))) - b) / 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 <= (-2.2d-71)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + ((c / b) * 0.5d0)
    else if (b <= 1.35d-126) then
        tmp = 0.3333333333333333d0 * ((sqrt((c * (a * (-3.0d0)))) - b) / 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 <= -2.2e-71) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 1.35e-126) {
		tmp = 0.3333333333333333 * ((Math.sqrt((c * (a * -3.0))) - b) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.2e-71:
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5)
	elif b <= 1.35e-126:
		tmp = 0.3333333333333333 * ((math.sqrt((c * (a * -3.0))) - b) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e-71)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 1.35e-126)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / 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 <= -2.2e-71)
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	elseif (b <= 1.35e-126)
		tmp = 0.3333333333333333 * ((sqrt((c * (a * -3.0))) - b) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e-71], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-126], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.19999999999999997e-71

   1. Initial program 64.8%

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

      1. add-sqr-sqrt 64.7%

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

      2. pow2 64.7%

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

      3. pow1/2 64.7%

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

      4. sqrt-pow1 64.7%

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

      5. fma-neg 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      6. *-commutative 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      7. distribute-rgt-neg-in 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      8. *-commutative 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      9. distribute-rgt-neg-in 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      10. metadata-eval 64.9%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      11. metadata-eval 64.9%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{3 \cdot a} \]

   3. Applied egg-rr 64.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.25}\right)}^{2}}}{3 \cdot a} \]

   4. Taylor expanded in b around -inf 90.9%

      \[\leadsto \color{blue}{2 \cdot \left(-0.3333333333333333 \cdot \frac{b}{a}\right) + 0.5 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. associate-*r* 90.9%

         \[\leadsto \color{blue}{\left(2 \cdot -0.3333333333333333\right) \cdot \frac{b}{a}} + 0.5 \cdot \frac{c}{b} \]

      2. metadata-eval 90.9%

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

      3. *-commutative 90.9%

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

   6. Simplified 90.9%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5} \]
   7. Step-by-step derivation

      1. associate-*r/ 91.0%

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

   8. Applied egg-rr 91.0%

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

   if -2.19999999999999997e-71 < b < 1.34999999999999998e-126

   1. Initial program 81.5%

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

      1. add-sqr-sqrt 81.2%

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

      2. pow2 81.2%

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

      3. pow1/2 81.2%

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

      4. sqrt-pow1 81.3%

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

      5. fma-neg 81.3%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      6. *-commutative 81.3%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      7. distribute-rgt-neg-in 81.3%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      8. *-commutative 81.3%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      9. distribute-rgt-neg-in 81.3%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      10. metadata-eval 81.3%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      11. metadata-eval 81.3%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{3 \cdot a} \]

   3. Applied egg-rr 81.3%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.25}\right)}^{2}}}{3 \cdot a} \]

   4. Taylor expanded in b around 0 75.3%

      \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(-3 \cdot \left(c \cdot a\right)\right)}^{0.25}\right)}}^{2}}{3 \cdot a} \]

   5. Taylor expanded in c around 0 45.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{\left(e^{0.25 \cdot \left(\log c + \log \left(-3 \cdot a\right)\right)}\right)}^{2} - b}{a}} \]
   6. Step-by-step derivation

   7. Simplified 75.4%

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

   if 1.34999999999999998e-126 < b

   1. Initial program 21.4%

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

   2. Taylor expanded in b around inf 84.7%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-126}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 4: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-127}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e-70)
   (+ (/ (* b -0.6666666666666666) a) (* (/ c b) 0.5))
   (if (<= b 2.5e-127)
     (* 0.3333333333333333 (/ (- (sqrt (* -3.0 (* a c))) b) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-70) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 2.5e-127) {
		tmp = 0.3333333333333333 * ((sqrt((-3.0 * (a * c))) - b) / 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 <= (-1.5d-70)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + ((c / b) * 0.5d0)
    else if (b <= 2.5d-127) then
        tmp = 0.3333333333333333d0 * ((sqrt(((-3.0d0) * (a * c))) - b) / 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 <= -1.5e-70) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 2.5e-127) {
		tmp = 0.3333333333333333 * ((Math.sqrt((-3.0 * (a * c))) - b) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.5e-70:
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5)
	elif b <= 2.5e-127:
		tmp = 0.3333333333333333 * ((math.sqrt((-3.0 * (a * c))) - b) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e-70)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 2.5e-127)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / 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 <= -1.5e-70)
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	elseif (b <= 2.5e-127)
		tmp = 0.3333333333333333 * ((sqrt((-3.0 * (a * c))) - b) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e-70], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5e-127], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5000000000000001e-70

   1. Initial program 64.8%

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

      1. add-sqr-sqrt 64.7%

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

      2. pow2 64.7%

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

      3. pow1/2 64.7%

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

      4. sqrt-pow1 64.7%

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

      5. fma-neg 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      6. *-commutative 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      7. distribute-rgt-neg-in 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      8. *-commutative 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      9. distribute-rgt-neg-in 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      10. metadata-eval 64.9%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      11. metadata-eval 64.9%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{3 \cdot a} \]

   3. Applied egg-rr 64.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.25}\right)}^{2}}}{3 \cdot a} \]

   4. Taylor expanded in b around -inf 90.9%

      \[\leadsto \color{blue}{2 \cdot \left(-0.3333333333333333 \cdot \frac{b}{a}\right) + 0.5 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. associate-*r* 90.9%

         \[\leadsto \color{blue}{\left(2 \cdot -0.3333333333333333\right) \cdot \frac{b}{a}} + 0.5 \cdot \frac{c}{b} \]

      2. metadata-eval 90.9%

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

      3. *-commutative 90.9%

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

   6. Simplified 90.9%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5} \]
   7. Step-by-step derivation

      1. associate-*r/ 91.0%

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

   8. Applied egg-rr 91.0%

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

   if -1.5000000000000001e-70 < b < 2.4999999999999999e-127

   1. Initial program 81.5%

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

      1. add-sqr-sqrt 81.2%

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

      2. pow2 81.2%

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

      3. pow1/2 81.2%

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

      4. sqrt-pow1 81.3%

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

      5. fma-neg 81.3%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      6. *-commutative 81.3%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      7. distribute-rgt-neg-in 81.3%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      8. *-commutative 81.3%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      9. distribute-rgt-neg-in 81.3%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      10. metadata-eval 81.3%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      11. metadata-eval 81.3%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{3 \cdot a} \]

   3. Applied egg-rr 81.3%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.25}\right)}^{2}}}{3 \cdot a} \]

   4. Taylor expanded in b around 0 75.3%

      \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(-3 \cdot \left(c \cdot a\right)\right)}^{0.25}\right)}}^{2}}{3 \cdot a} \]

   5. Taylor expanded in c around 0 45.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{\left(e^{0.25 \cdot \left(\log c + \log \left(-3 \cdot a\right)\right)}\right)}^{2} - b}{a}} \]
   6. Step-by-step derivation

   7. Simplified 75.4%

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

   8. Taylor expanded in c around 0 75.6%

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

      1. *-commutative 75.6%

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

   10. Simplified 75.6%

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

   if 2.4999999999999999e-127 < b

   1. Initial program 21.4%

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

   2. Taylor expanded in b around inf 84.7%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-127}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 5: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.14 \cdot 10^{-69}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.14e-69)
   (+ (/ (* b -0.6666666666666666) a) (* (/ c b) 0.5))
   (if (<= b 1.9e-126)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.14e-69) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 1.9e-126) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.14d-69)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + ((c / b) * 0.5d0)
    else if (b <= 1.9d-126) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.14e-69) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 1.9e-126) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.14e-69:
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5)
	elif b <= 1.9e-126:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.14e-69)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 1.9e-126)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.14e-69)
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	elseif (b <= 1.9e-126)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.14e-69], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.9e-126], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14000000000000006e-69

   1. Initial program 64.8%

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

      1. add-sqr-sqrt 64.7%

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

      2. pow2 64.7%

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

      3. pow1/2 64.7%

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

      4. sqrt-pow1 64.7%

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

      5. fma-neg 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      6. *-commutative 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      7. distribute-rgt-neg-in 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      8. *-commutative 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      9. distribute-rgt-neg-in 64.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      10. metadata-eval 64.9%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      11. metadata-eval 64.9%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{3 \cdot a} \]

   3. Applied egg-rr 64.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.25}\right)}^{2}}}{3 \cdot a} \]

   4. Taylor expanded in b around -inf 90.9%

      \[\leadsto \color{blue}{2 \cdot \left(-0.3333333333333333 \cdot \frac{b}{a}\right) + 0.5 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. associate-*r* 90.9%

         \[\leadsto \color{blue}{\left(2 \cdot -0.3333333333333333\right) \cdot \frac{b}{a}} + 0.5 \cdot \frac{c}{b} \]

      2. metadata-eval 90.9%

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

      3. *-commutative 90.9%

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

   6. Simplified 90.9%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5} \]
   7. Step-by-step derivation

      1. associate-*r/ 91.0%

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

   8. Applied egg-rr 91.0%

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

   if -1.14000000000000006e-69 < b < 1.8999999999999999e-126

   1. Initial program 81.5%

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

   2. Taylor expanded in b around 0 75.5%

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

      1. *-commutative 75.5%

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

      2. *-commutative 75.5%

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

      3. *-commutative 75.5%

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

      4. associate-*r* 75.6%

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

   4. Simplified 75.6%

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

   if 1.8999999999999999e-126 < b

   1. Initial program 21.4%

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

   2. Taylor expanded in b around inf 84.7%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.14 \cdot 10^{-69}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 6: 68.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} 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 <= (-2d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    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 <= -2e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	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 <= -2e-310)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 70.5%

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

   2. Taylor expanded in b around -inf 70.8%

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

   if -1.999999999999994e-310 < b

   1. Initial program 34.3%

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

   2. Taylor expanded in b around inf 66.3%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 7: 68.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310)
   (+ (/ (* b -0.6666666666666666) a) (* (/ c b) 0.5))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} 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 <= (-2d-310)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + ((c / b) * 0.5d0)
    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 <= -2e-310) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(Float64(c / b) * 0.5));
	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 <= -2e-310)
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 70.5%

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

      1. add-sqr-sqrt 70.3%

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

      2. pow2 70.3%

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

      3. pow1/2 70.3%

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

      4. sqrt-pow1 70.3%

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

      5. fma-neg 70.5%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      6. *-commutative 70.5%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      7. distribute-rgt-neg-in 70.5%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      8. *-commutative 70.5%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      9. distribute-rgt-neg-in 70.5%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      10. metadata-eval 70.5%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      11. metadata-eval 70.5%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{3 \cdot a} \]

   3. Applied egg-rr 70.5%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.25}\right)}^{2}}}{3 \cdot a} \]

   4. Taylor expanded in b around -inf 70.8%

      \[\leadsto \color{blue}{2 \cdot \left(-0.3333333333333333 \cdot \frac{b}{a}\right) + 0.5 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. associate-*r* 70.8%

         \[\leadsto \color{blue}{\left(2 \cdot -0.3333333333333333\right) \cdot \frac{b}{a}} + 0.5 \cdot \frac{c}{b} \]

      2. metadata-eval 70.8%

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

      3. *-commutative 70.8%

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

   6. Simplified 70.8%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5} \]
   7. Step-by-step derivation

      1. associate-*r/ 70.8%

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

   8. Applied egg-rr 70.8%

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

   if -1.999999999999994e-310 < b

   1. Initial program 34.3%

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

   2. Taylor expanded in b around inf 66.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 8: 68.3% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (* -0.6666666666666666 (/ b a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = -0.6666666666666666 * (b / 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 <= (-2d-310)) then
        tmp = (-0.6666666666666666d0) * (b / 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 <= -2e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(-0.6666666666666666 * Float64(b / 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 <= -2e-310)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 70.5%

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

   2. Taylor expanded in b around -inf 70.5%

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

      1. *-commutative 70.5%

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

   4. Simplified 70.5%

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

   if -1.999999999999994e-310 < b

   1. Initial program 34.3%

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

   2. Taylor expanded in b around inf 66.3%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 9: 68.3% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (* b -0.6666666666666666) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		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 <= (-2d-310)) 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 <= -2e-310) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(b * -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 <= -2e-310)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 70.5%

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

   2. Taylor expanded in b around -inf 70.5%

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

      1. *-commutative 70.5%

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

   4. Simplified 70.5%

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

      1. associate-*l/ 70.5%

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

   6. Applied egg-rr 70.5%

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

   if -1.999999999999994e-310 < b

   1. Initial program 34.3%

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

   2. Taylor expanded in b around inf 66.3%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 10: 35.1% 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 55.0%

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

2. Taylor expanded in b around inf 29.6%

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

3. Final simplification 29.6%

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

Reproduce

herbie shell --seed 2023187 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))