Cubic critical

Percentage Accurate: 52.2% → 84.9%

Time: 13.5s

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: 52.2% 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: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e+55)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 1.15e-40)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e+55) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.15e-40) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.4d+55)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 1.15d-40) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e+55) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.15e-40) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.4e+55:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 1.15e-40:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.4e+55)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 1.15e-40)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.4e+55)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 1.15e-40)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.4e+55], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.15e-40], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.40000000000000021e55

   1. Initial program 65.3%

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

   2. Taylor expanded in b around -inf 95.8%

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

      1. *-commutative 95.8%

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

   4. Simplified 95.8%

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

   if -4.40000000000000021e55 < b < 1.15e-40

   1. Initial program 77.8%

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

   if 1.15e-40 < b

   1. Initial program 15.7%

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

   2. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. associate-*l/ 90.1%

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

   4. Simplified 90.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.3 \cdot 10^{-71}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left({\left(c \cdot \frac{a}{-0.3333333333333333}\right)}^{0.5} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.3e-71)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 1.4e-40)
     (*
      (/ 0.3333333333333333 a)
      (- (pow (* c (/ a -0.3333333333333333)) 0.5) b))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.3e-71) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.4e-40) {
		tmp = (0.3333333333333333 / a) * (pow((c * (a / -0.3333333333333333)), 0.5) - b);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.3d-71)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 1.4d-40) then
        tmp = (0.3333333333333333d0 / a) * (((c * (a / (-0.3333333333333333d0))) ** 0.5d0) - b)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.3e-71) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.4e-40) {
		tmp = (0.3333333333333333 / a) * (Math.pow((c * (a / -0.3333333333333333)), 0.5) - b);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.3e-71:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 1.4e-40:
		tmp = (0.3333333333333333 / a) * (math.pow((c * (a / -0.3333333333333333)), 0.5) - b)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.3e-71)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 1.4e-40)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64((Float64(c * Float64(a / -0.3333333333333333)) ^ 0.5) - b));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.3e-71)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 1.4e-40)
		tmp = (0.3333333333333333 / a) * (((c * (a / -0.3333333333333333)) ^ 0.5) - b);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.3e-71], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.4e-40], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Power[N[(c * N[(a / -0.3333333333333333), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.29999999999999999e-71

   1. Initial program 70.7%

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

   2. Taylor expanded in b around -inf 88.6%

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

      1. *-commutative 88.6%

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

   4. Simplified 88.6%

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

   if -5.29999999999999999e-71 < b < 1.4e-40

   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. add-sqr-sqrt 75.3%

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

      2. pow2 75.3%

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

      3. pow1/2 75.3%

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

      4. sqrt-pow1 75.3%

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

      5. sub-neg 75.3%

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

      6. +-commutative 75.3%

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

      7. distribute-lft-neg-in 75.3%

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

      8. *-commutative 75.3%

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

      9. distribute-rgt-neg-in 75.3%

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

      10. metadata-eval 75.3%

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

      11. associate-*r* 75.2%

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

      12. *-commutative 75.2%

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

      13. fma-udef 75.2%

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

      14. pow2 75.2%

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

      15. metadata-eval 75.2%

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

   3. Applied egg-rr 75.2%

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

   4. Taylor expanded in c around -inf 40.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{\left(e^{0.25 \cdot \left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}\right)}^{2} - b}{a}} \]
   5. Step-by-step derivation

      1. unpow2 40.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{e^{0.25 \cdot \left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} \cdot e^{0.25 \cdot \left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}} - b}{a} \]

      2. exp-prod 39.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}} \cdot e^{0.25 \cdot \left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} - b}{a} \]

      3. exp-prod 38.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{{\left(e^{0.25}\right)}^{\left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} \cdot \color{blue}{{\left(e^{0.25}\right)}^{\left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}} - b}{a} \]

      4. pow-sqr 38.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)\right)}} - b}{a} \]

      5. mul-1-neg 38.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(3 \cdot a\right) + \color{blue}{\left(-\log \left(\frac{-1}{c}\right)\right)}\right)\right)} - b}{a} \]

      6. unsub-neg 38.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \color{blue}{\left(\log \left(3 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right)}\right)} - b}{a} \]

      7. *-commutative 38.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \color{blue}{\left(a \cdot 3\right)} - \log \left(\frac{-1}{c}\right)\right)\right)} - b}{a} \]

   6. Simplified 38.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(a \cdot 3\right) - \log \left(\frac{-1}{c}\right)\right)\right)} - b}{a}} \]

   7. Taylor expanded in a around 0 39.8%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{e^{0.5 \cdot \left(\left(\log 3 + \log a\right) - \log \left(\frac{-1}{c}\right)\right)} - b}{a}} \]

   9. Simplified 66.3%

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

   if 1.4e-40 < b

   1. Initial program 15.7%

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

   2. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. associate-*l/ 90.1%

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

   4. Simplified 90.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.3 \cdot 10^{-71}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left({\left(c \cdot \frac{a}{-0.3333333333333333}\right)}^{0.5} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-68)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 1.35e-40)
     (/ (- (sqrt (* c (* a -3.0))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-68) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.35e-40) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.15d-68)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 1.35d-40) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-68) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.35e-40) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e-68:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 1.35e-40:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e-68)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 1.35e-40)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.15e-68)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 1.35e-40)
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e-68], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-40], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14999999999999998e-68

   1. Initial program 70.7%

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

   2. Taylor expanded in b around -inf 88.6%

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

      1. *-commutative 88.6%

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

   4. Simplified 88.6%

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

   if -1.14999999999999998e-68 < b < 1.35e-40

   1. Initial program 75.5%

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

   2. Taylor expanded in b around 0 66.2%

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

      1. associate-*r* 66.3%

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

      2. *-commutative 66.3%

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

      3. *-commutative 66.3%

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

   4. Simplified 66.3%

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

   if 1.35e-40 < b

   1. Initial program 15.7%

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

   2. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. associate-*l/ 90.1%

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

   4. Simplified 90.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-40}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{\left(c \cdot \frac{a}{-0.3333333333333333}\right)}^{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.6e-119)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 1.5e-40)
     (* 0.3333333333333333 (/ (pow (* c (/ a -0.3333333333333333)) 0.5) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.6e-119) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.5e-40) {
		tmp = 0.3333333333333333 * (pow((c * (a / -0.3333333333333333)), 0.5) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-9.6d-119)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 1.5d-40) then
        tmp = 0.3333333333333333d0 * (((c * (a / (-0.3333333333333333d0))) ** 0.5d0) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.6e-119) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.5e-40) {
		tmp = 0.3333333333333333 * (Math.pow((c * (a / -0.3333333333333333)), 0.5) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.6e-119:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 1.5e-40:
		tmp = 0.3333333333333333 * (math.pow((c * (a / -0.3333333333333333)), 0.5) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.6e-119)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 1.5e-40)
		tmp = Float64(0.3333333333333333 * Float64((Float64(c * Float64(a / -0.3333333333333333)) ^ 0.5) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.6e-119)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 1.5e-40)
		tmp = 0.3333333333333333 * (((c * (a / -0.3333333333333333)) ^ 0.5) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.6e-119], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.5e-40], N[(0.3333333333333333 * N[(N[Power[N[(c * N[(a / -0.3333333333333333), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.60000000000000034e-119

   1. Initial program 72.7%

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

   2. Taylor expanded in b around -inf 84.6%

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

      1. *-commutative 84.6%

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

   4. Simplified 84.6%

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

   if -9.60000000000000034e-119 < b < 1.5000000000000001e-40

   1. Initial program 72.7%

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

      1. add-sqr-sqrt 72.6%

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

      2. pow2 72.6%

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

      3. pow1/2 72.6%

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

      4. sqrt-pow1 72.6%

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

      5. sub-neg 72.6%

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

      6. +-commutative 72.6%

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

      7. distribute-lft-neg-in 72.6%

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

      8. *-commutative 72.6%

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

      9. distribute-rgt-neg-in 72.6%

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

      10. metadata-eval 72.6%

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

      11. associate-*r* 72.5%

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

      12. *-commutative 72.5%

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

      13. fma-udef 72.5%

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

      14. pow2 72.5%

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

      15. metadata-eval 72.5%

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

   3. Applied egg-rr 72.5%

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

   4. Taylor expanded in c around -inf 44.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{\left(e^{0.25 \cdot \left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}\right)}^{2} - b}{a}} \]
   5. Step-by-step derivation

      1. unpow2 44.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{e^{0.25 \cdot \left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} \cdot e^{0.25 \cdot \left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}} - b}{a} \]

      2. exp-prod 43.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}} \cdot e^{0.25 \cdot \left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} - b}{a} \]

      3. exp-prod 42.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{{\left(e^{0.25}\right)}^{\left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} \cdot \color{blue}{{\left(e^{0.25}\right)}^{\left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}} - b}{a} \]

      4. pow-sqr 42.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(3 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)\right)}} - b}{a} \]

      5. mul-1-neg 42.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(3 \cdot a\right) + \color{blue}{\left(-\log \left(\frac{-1}{c}\right)\right)}\right)\right)} - b}{a} \]

      6. unsub-neg 42.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \color{blue}{\left(\log \left(3 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right)}\right)} - b}{a} \]

      7. *-commutative 42.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \color{blue}{\left(a \cdot 3\right)} - \log \left(\frac{-1}{c}\right)\right)\right)} - b}{a} \]

   6. Simplified 42.5%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(a \cdot 3\right) - \log \left(\frac{-1}{c}\right)\right)\right)} - b}{a}} \]

   7. Taylor expanded in b around 0 43.8%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{e^{0.5 \cdot \left(\log \left(3 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right)}}{a}} \]
   8. Step-by-step derivation

      1. *-commutative 43.8%

         \[\leadsto 0.3333333333333333 \cdot \frac{e^{\color{blue}{\left(\log \left(3 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5}}}{a} \]

      2. log-prod 43.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{e^{\left(\color{blue}{\left(\log 3 + \log a\right)} - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5}}{a} \]

      3. +-commutative 43.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{e^{\left(\color{blue}{\left(\log a + \log 3\right)} - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5}}{a} \]

      4. log-prod 43.8%

         \[\leadsto 0.3333333333333333 \cdot \frac{e^{\left(\color{blue}{\log \left(a \cdot 3\right)} - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5}}{a} \]

      5. log-div 63.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{e^{\color{blue}{\log \left(\frac{a \cdot 3}{\frac{-1}{c}}\right)} \cdot 0.5}}{a} \]

      6. exp-to-pow 67.8%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{{\left(\frac{a \cdot 3}{\frac{-1}{c}}\right)}^{0.5}}}{a} \]

      7. associate-/r/ 67.8%

         \[\leadsto 0.3333333333333333 \cdot \frac{{\color{blue}{\left(\frac{a \cdot 3}{-1} \cdot c\right)}}^{0.5}}{a} \]

      8. *-commutative 67.8%

         \[\leadsto 0.3333333333333333 \cdot \frac{{\color{blue}{\left(c \cdot \frac{a \cdot 3}{-1}\right)}}^{0.5}}{a} \]

      9. associate-/l* 67.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{{\left(c \cdot \color{blue}{\frac{a}{\frac{-1}{3}}}\right)}^{0.5}}{a} \]

      10. metadata-eval 67.9%

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

   9. Simplified 67.9%

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

   if 1.5000000000000001e-40 < b

   1. Initial program 15.7%

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

   2. Taylor expanded in b around inf 90.0%

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

      1. *-commutative 90.0%

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

      2. associate-*l/ 90.1%

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

   4. Simplified 90.1%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-40}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{\left(c \cdot \frac{a}{-0.3333333333333333}\right)}^{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5: 67.5% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-278}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4e-278) (/ (* b -2.0) (* 3.0 a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4e-278) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4d-278) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4e-278) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4e-278:
		tmp = (b * -2.0) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4e-278)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4e-278)
		tmp = (b * -2.0) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4e-278], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.99999999999999975e-278

   1. Initial program 75.2%

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

   2. Taylor expanded in b around -inf 70.3%

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

      1. *-commutative 70.3%

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

   4. Simplified 70.3%

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

   if 3.99999999999999975e-278 < b

   1. Initial program 29.1%

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

   2. Taylor expanded in b around inf 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*l/ 70.7%

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

   4. Simplified 70.7%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-278}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6: 43.2% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+46}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8e+46) (* (/ b a) -0.6666666666666666) (/ (* c 0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e+46) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c * 0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 8d+46) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = (c * 0.5d0) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e+46) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c * 0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8e+46:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (c * 0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e+46)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(Float64(c * 0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8e+46)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (c * 0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e+46], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(c * 0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.9999999999999999e46

   1. Initial program 67.5%

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

   2. Taylor expanded in b around -inf 52.6%

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

      1. *-commutative 52.6%

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

   4. Simplified 52.6%

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

   if 7.9999999999999999e46 < b

   1. Initial program 15.5%

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

   2. Taylor expanded in b around -inf 2.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. *-commutative 2.0%

         \[\leadsto \frac{\color{blue}{b \cdot -2} + 1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a} \]

      2. fma-def 2.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      3. *-commutative 2.0%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{a \cdot c}{b} \cdot 1.5}\right)}{3 \cdot a} \]

      4. associate-/l* 2.2%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{a}{\frac{b}{c}}} \cdot 1.5\right)}{3 \cdot a} \]

      5. associate-/r/ 2.2%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\left(\frac{a}{b} \cdot c\right)} \cdot 1.5\right)}{3 \cdot a} \]

   4. Simplified 2.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, \left(\frac{a}{b} \cdot c\right) \cdot 1.5\right)}}{3 \cdot a} \]

   5. Taylor expanded in b around 0 29.5%

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

      1. associate-*r/ 29.5%

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

   7. Simplified 29.5%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+46}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 0.5}{b}\\ \end{array} \]

Alternative 7: 67.5% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{-278}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.6e-278) (* (/ b a) -0.6666666666666666) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.6e-278) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.6d-278) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.6e-278) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.6e-278:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.6e-278)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.6e-278)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.6e-278], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.59999999999999996e-278

   1. Initial program 75.2%

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

   2. Taylor expanded in b around -inf 70.2%

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

      1. *-commutative 70.2%

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

   4. Simplified 70.2%

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

   if 3.59999999999999996e-278 < b

   1. Initial program 29.1%

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

   2. Taylor expanded in b around inf 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*l/ 70.7%

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

   4. Simplified 70.7%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{-278}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 8: 35.4% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (b / a) * (-0.6666666666666666d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

2. Taylor expanded in b around -inf 40.2%

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

   1. *-commutative 40.2%

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

4. Simplified 40.2%

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

5. Final simplification 40.2%

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

Reproduce

herbie shell --seed 2023320 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))