Cubic critical

Percentage Accurate: 51.5% → 85.3%

Time: 14.3s

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: 51.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+145}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 3.25 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+145)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 3.25e-105)
     (/ (- (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 <= -1e+145) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 3.25e-105) {
		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 <= (-1d+145)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 3.25d-105) 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 <= -1e+145) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 3.25e-105) {
		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 <= -1e+145:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 3.25e-105:
		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 <= -1e+145)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 3.25e-105)
		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 <= -1e+145)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 3.25e-105)
		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, -1e+145], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.25e-105], 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 < -9.9999999999999999e144

   1. Initial program 44.0%

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

   2. Taylor expanded in b around -inf 99.9%

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

      1. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   if -9.9999999999999999e144 < b < 3.25000000000000003e-105

   1. Initial program 81.9%

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

   if 3.25000000000000003e-105 < b

   1. Initial program 16.3%

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

   2. Taylor expanded in b around inf 89.3%

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

      1. *-commutative 89.3%

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

      2. associate-*l/ 89.3%

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

   4. Simplified 89.3%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+145}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 3.25 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-76}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a}{\frac{b}{c}} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-106}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.05e-76)
   (/ (- (- (* 1.5 (/ a (/ b c))) b) b) (* 3.0 a))
   (if (<= b 1.05e-106)
     (* (/ 0.3333333333333333 a) (- (sqrt (* a (* c -3.0))) b))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-76) {
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a);
	} else if (b <= 1.05e-106) {
		tmp = (0.3333333333333333 / a) * (sqrt((a * (c * -3.0))) - 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 <= (-2.05d-76)) then
        tmp = (((1.5d0 * (a / (b / c))) - b) - b) / (3.0d0 * a)
    else if (b <= 1.05d-106) then
        tmp = (0.3333333333333333d0 / a) * (sqrt((a * (c * (-3.0d0)))) - 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 <= -2.05e-76) {
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a);
	} else if (b <= 1.05e-106) {
		tmp = (0.3333333333333333 / a) * (Math.sqrt((a * (c * -3.0))) - b);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.05e-76:
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a)
	elif b <= 1.05e-106:
		tmp = (0.3333333333333333 / a) * (math.sqrt((a * (c * -3.0))) - b)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.05e-76)
		tmp = Float64(Float64(Float64(Float64(1.5 * Float64(a / Float64(b / c))) - b) - b) / Float64(3.0 * a));
	elseif (b <= 1.05e-106)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(a * Float64(c * -3.0))) - 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 <= -2.05e-76)
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a);
	elseif (b <= 1.05e-106)
		tmp = (0.3333333333333333 / a) * (sqrt((a * (c * -3.0))) - b);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.05e-76], N[(N[(N[(N[(1.5 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e-106], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0499999999999999e-76

   1. Initial program 73.0%

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

      1. flip-- 38.7%

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

      2. div-inv 38.7%

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

      3. pow2 38.7%

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

      4. pow2 38.7%

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

      5. pow-prod-up 38.7%

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

      6. metadata-eval 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{\color{blue}{4}} - \left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      7. pow2 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - \color{blue}{{\left(\left(3 \cdot a\right) \cdot c\right)}^{2}}\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      8. associate-*l* 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{2}\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      9. fma-def 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}}}}{3 \cdot a} \]

      10. associate-*l* 38.7%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{3 \cdot \left(a \cdot c\right)}\right)}}}{3 \cdot a} \]

   3. Applied egg-rr 38.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-*r* 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\color{blue}{\left(\left(3 \cdot a\right) \cdot c\right)}}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]

      2. *-commutative 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\color{blue}{\left(a \cdot 3\right)} \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]

      3. associate-*r* 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a} \]

      4. *-commutative 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot 3\right)} \cdot c\right)}}}{3 \cdot a} \]

   5. Simplified 38.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \left(a \cdot 3\right) \cdot c\right)}}}}{3 \cdot a} \]

   6. Taylor expanded in b around 0 73.0%

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

      1. fma-def 73.0%

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

   8. Simplified 73.0%

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

   9. Taylor expanded in b around -inf 76.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   10. Step-by-step derivation

       1. +-commutative 76.9%

          \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} + -1 \cdot b\right)}}{3 \cdot a} \]

       2. mul-1-neg 76.9%

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

       3. unsub-neg 76.9%

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

       4. associate-/l* 81.9%

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

   11. Simplified 81.9%

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

   if -2.0499999999999999e-76 < b < 1.05000000000000002e-106

   1. Initial program 73.2%

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

      1. prod-diff 72.9%

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

      2. *-commutative 72.9%

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

      3. fma-def 72.9%

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

      4. associate-+l+ 72.9%

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

      5. pow2 72.9%

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

      6. *-commutative 72.9%

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

      7. distribute-rgt-neg-in 72.9%

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

      8. *-commutative 72.9%

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

      9. distribute-rgt-neg-in 72.9%

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

      10. metadata-eval 72.9%

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

      11. *-commutative 72.9%

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

      12. fma-udef 72.9%

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

      13. distribute-lft-neg-in 72.9%

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

      14. distribute-rgt-neg-in 72.9%

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

      15. *-commutative 72.9%

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

      16. distribute-rgt-neg-in 72.9%

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

      17. metadata-eval 72.9%

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

      18. associate-*l* 73.0%

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

   3. Applied egg-rr 73.0%

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

   4. Taylor expanded in b around 0 70.1%

      \[\leadsto \frac{\color{blue}{\sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)} + -1 \cdot b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. neg-mul-1 70.1%

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

      2. unsub-neg 70.1%

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

      3. distribute-rgt-out 70.4%

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

      4. metadata-eval 70.4%

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

      5. *-commutative 70.4%

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

      6. metadata-eval 70.4%

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

      7. distribute-lft-neg-in 70.4%

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

      8. associate-*r* 70.5%

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

      9. *-commutative 70.5%

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

      10. associate-*r* 70.4%

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

      11. distribute-rgt-neg-in 70.4%

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

      12. distribute-lft-neg-in 70.4%

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

      13. metadata-eval 70.4%

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

      14. *-commutative 70.4%

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

   6. Simplified 70.4%

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

      1. add-sqr-sqrt 70.2%

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

      2. pow2 70.2%

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

      3. pow1/2 70.2%

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

      4. sqrt-pow1 70.3%

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

      5. *-commutative 70.3%

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

      6. metadata-eval 70.3%

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

   8. Applied egg-rr 70.3%

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

   9. Taylor expanded in a around 0 42.7%

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

       1. associate-*r/ 42.7%

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

       2. remove-double-neg 42.7%

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

       3. neg-mul-1 42.7%

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

       4. times-frac 42.7%

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

   11. Simplified 70.4%

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

   if 1.05000000000000002e-106 < b

   1. Initial program 16.3%

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

   2. Taylor expanded in b around inf 89.3%

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

      1. *-commutative 89.3%

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

      2. associate-*l/ 89.3%

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

   4. Simplified 89.3%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-76}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a}{\frac{b}{c}} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-106}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-74}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a}{\frac{b}{c}} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-74)
   (/ (- (- (* 1.5 (/ a (/ b c))) b) b) (* 3.0 a))
   (if (<= b 1.45e-104)
     (/ (- (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 <= -4e-74) {
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a);
	} else if (b <= 1.45e-104) {
		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 <= (-4d-74)) then
        tmp = (((1.5d0 * (a / (b / c))) - b) - b) / (3.0d0 * a)
    else if (b <= 1.45d-104) 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 <= -4e-74) {
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a);
	} else if (b <= 1.45e-104) {
		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 <= -4e-74:
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a)
	elif b <= 1.45e-104:
		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 <= -4e-74)
		tmp = Float64(Float64(Float64(Float64(1.5 * Float64(a / Float64(b / c))) - b) - b) / Float64(3.0 * a));
	elseif (b <= 1.45e-104)
		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 <= -4e-74)
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a);
	elseif (b <= 1.45e-104)
		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, -4e-74], N[(N[(N[(N[(1.5 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.45e-104], 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 < -3.99999999999999983e-74

   1. Initial program 73.0%

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

      1. flip-- 38.7%

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

      2. div-inv 38.7%

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

      3. pow2 38.7%

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

      4. pow2 38.7%

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

      5. pow-prod-up 38.7%

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

      6. metadata-eval 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{\color{blue}{4}} - \left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      7. pow2 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - \color{blue}{{\left(\left(3 \cdot a\right) \cdot c\right)}^{2}}\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      8. associate-*l* 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{2}\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      9. fma-def 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}}}}{3 \cdot a} \]

      10. associate-*l* 38.7%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{3 \cdot \left(a \cdot c\right)}\right)}}}{3 \cdot a} \]

   3. Applied egg-rr 38.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-*r* 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\color{blue}{\left(\left(3 \cdot a\right) \cdot c\right)}}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]

      2. *-commutative 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\color{blue}{\left(a \cdot 3\right)} \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]

      3. associate-*r* 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a} \]

      4. *-commutative 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot 3\right)} \cdot c\right)}}}{3 \cdot a} \]

   5. Simplified 38.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \left(a \cdot 3\right) \cdot c\right)}}}}{3 \cdot a} \]

   6. Taylor expanded in b around 0 73.0%

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

      1. fma-def 73.0%

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

   8. Simplified 73.0%

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

   9. Taylor expanded in b around -inf 76.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   10. Step-by-step derivation

       1. +-commutative 76.9%

          \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} + -1 \cdot b\right)}}{3 \cdot a} \]

       2. mul-1-neg 76.9%

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

       3. unsub-neg 76.9%

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

       4. associate-/l* 81.9%

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

   11. Simplified 81.9%

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

   if -3.99999999999999983e-74 < b < 1.4500000000000001e-104

   1. Initial program 73.2%

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

   2. Taylor expanded in b around 0 70.4%

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

      1. associate-*r* 70.5%

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

      2. *-commutative 70.5%

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

      3. *-commutative 70.5%

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

   4. Simplified 70.5%

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

   if 1.4500000000000001e-104 < b

   1. Initial program 16.3%

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

   2. Taylor expanded in b around inf 89.3%

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

      1. *-commutative 89.3%

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

      2. associate-*l/ 89.3%

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

   4. Simplified 89.3%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-74}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a}{\frac{b}{c}} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a}{\frac{b}{c}} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-108}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e-75)
   (/ (- (- (* 1.5 (/ a (/ b c))) b) b) (* 3.0 a))
   (if (<= b 1.45e-108)
     (* 0.3333333333333333 (/ (sqrt (* a (* c -3.0))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-75) {
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a);
	} else if (b <= 1.45e-108) {
		tmp = 0.3333333333333333 * (sqrt((a * (c * -3.0))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.5d-75)) then
        tmp = (((1.5d0 * (a / (b / c))) - b) - b) / (3.0d0 * a)
    else if (b <= 1.45d-108) then
        tmp = 0.3333333333333333d0 * (sqrt((a * (c * (-3.0d0)))) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-75) {
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a);
	} else if (b <= 1.45e-108) {
		tmp = 0.3333333333333333 * (Math.sqrt((a * (c * -3.0))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e-75:
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a)
	elif b <= 1.45e-108:
		tmp = 0.3333333333333333 * (math.sqrt((a * (c * -3.0))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e-75)
		tmp = Float64(Float64(Float64(Float64(1.5 * Float64(a / Float64(b / c))) - b) - b) / Float64(3.0 * a));
	elseif (b <= 1.45e-108)
		tmp = Float64(0.3333333333333333 * Float64(sqrt(Float64(a * Float64(c * -3.0))) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.5e-75)
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a);
	elseif (b <= 1.45e-108)
		tmp = 0.3333333333333333 * (sqrt((a * (c * -3.0))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e-75], N[(N[(N[(N[(1.5 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.45e-108], N[(0.3333333333333333 * N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.5000000000000003e-75

   1. Initial program 73.0%

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

      1. flip-- 38.7%

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

      2. div-inv 38.7%

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

      3. pow2 38.7%

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

      4. pow2 38.7%

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

      5. pow-prod-up 38.7%

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

      6. metadata-eval 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{\color{blue}{4}} - \left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      7. pow2 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - \color{blue}{{\left(\left(3 \cdot a\right) \cdot c\right)}^{2}}\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      8. associate-*l* 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{2}\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      9. fma-def 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}}}}{3 \cdot a} \]

      10. associate-*l* 38.7%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{3 \cdot \left(a \cdot c\right)}\right)}}}{3 \cdot a} \]

   3. Applied egg-rr 38.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-*r* 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\color{blue}{\left(\left(3 \cdot a\right) \cdot c\right)}}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]

      2. *-commutative 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\color{blue}{\left(a \cdot 3\right)} \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]

      3. associate-*r* 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a} \]

      4. *-commutative 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot 3\right)} \cdot c\right)}}}{3 \cdot a} \]

   5. Simplified 38.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \left(a \cdot 3\right) \cdot c\right)}}}}{3 \cdot a} \]

   6. Taylor expanded in b around 0 73.0%

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

      1. fma-def 73.0%

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

   8. Simplified 73.0%

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

   9. Taylor expanded in b around -inf 76.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   10. Step-by-step derivation

       1. +-commutative 76.9%

          \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} + -1 \cdot b\right)}}{3 \cdot a} \]

       2. mul-1-neg 76.9%

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

       3. unsub-neg 76.9%

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

       4. associate-/l* 81.9%

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

   11. Simplified 81.9%

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

   if -4.5000000000000003e-75 < b < 1.45e-108

   1. Initial program 73.2%

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

      1. prod-diff 72.9%

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

      2. *-commutative 72.9%

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

      3. fma-def 72.9%

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

      4. associate-+l+ 72.9%

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

      5. pow2 72.9%

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

      6. *-commutative 72.9%

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

      7. distribute-rgt-neg-in 72.9%

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

      8. *-commutative 72.9%

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

      9. distribute-rgt-neg-in 72.9%

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

      10. metadata-eval 72.9%

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

      11. *-commutative 72.9%

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

      12. fma-udef 72.9%

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

      13. distribute-lft-neg-in 72.9%

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

      14. distribute-rgt-neg-in 72.9%

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

      15. *-commutative 72.9%

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

      16. distribute-rgt-neg-in 72.9%

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

      17. metadata-eval 72.9%

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

      18. associate-*l* 73.0%

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

   3. Applied egg-rr 73.0%

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

   4. Taylor expanded in b around 0 69.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{a} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 69.1%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}}{a}} \]

      2. *-lft-identity 69.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}}}{a} \]

      3. distribute-rgt-out 69.4%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}}}{a} \]

      4. metadata-eval 69.4%

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

      5. *-commutative 69.4%

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

      6. metadata-eval 69.4%

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

      7. distribute-lft-neg-in 69.4%

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

      8. associate-*r* 69.5%

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

      9. *-commutative 69.5%

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

      10. associate-*r* 69.5%

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

      11. distribute-rgt-neg-in 69.5%

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

      12. distribute-lft-neg-in 69.5%

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

      13. metadata-eval 69.5%

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

      14. *-commutative 69.5%

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

   6. Simplified 69.5%

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

   if 1.45e-108 < b

   1. Initial program 16.3%

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

   2. Taylor expanded in b around inf 89.3%

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

      1. *-commutative 89.3%

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

      2. associate-*l/ 89.3%

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

   4. Simplified 89.3%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a}{\frac{b}{c}} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-108}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a}{\frac{b}{c}} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{c \cdot \left(a \cdot -3\right)} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.5e-76)
   (/ (- (- (* 1.5 (/ a (/ b c))) b) b) (* 3.0 a))
   (if (<= b 2.3e-95)
     (* (sqrt (* c (* a -3.0))) (/ 0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-76) {
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a);
	} else if (b <= 2.3e-95) {
		tmp = sqrt((c * (a * -3.0))) * (0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.5d-76)) then
        tmp = (((1.5d0 * (a / (b / c))) - b) - b) / (3.0d0 * a)
    else if (b <= 2.3d-95) then
        tmp = sqrt((c * (a * (-3.0d0)))) * (0.3333333333333333d0 / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-76) {
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a);
	} else if (b <= 2.3e-95) {
		tmp = Math.sqrt((c * (a * -3.0))) * (0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.5e-76:
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a)
	elif b <= 2.3e-95:
		tmp = math.sqrt((c * (a * -3.0))) * (0.3333333333333333 / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e-76)
		tmp = Float64(Float64(Float64(Float64(1.5 * Float64(a / Float64(b / c))) - b) - b) / Float64(3.0 * a));
	elseif (b <= 2.3e-95)
		tmp = Float64(sqrt(Float64(c * Float64(a * -3.0))) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.5e-76)
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a);
	elseif (b <= 2.3e-95)
		tmp = sqrt((c * (a * -3.0))) * (0.3333333333333333 / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e-76], N[(N[(N[(N[(1.5 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.3e-95], N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5e-76

   1. Initial program 73.0%

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

      1. flip-- 38.7%

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

      2. div-inv 38.7%

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

      3. pow2 38.7%

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

      4. pow2 38.7%

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

      5. pow-prod-up 38.7%

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

      6. metadata-eval 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{\color{blue}{4}} - \left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      7. pow2 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - \color{blue}{{\left(\left(3 \cdot a\right) \cdot c\right)}^{2}}\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      8. associate-*l* 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{2}\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      9. fma-def 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}}}}{3 \cdot a} \]

      10. associate-*l* 38.7%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{3 \cdot \left(a \cdot c\right)}\right)}}}{3 \cdot a} \]

   3. Applied egg-rr 38.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-*r* 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\color{blue}{\left(\left(3 \cdot a\right) \cdot c\right)}}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]

      2. *-commutative 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\color{blue}{\left(a \cdot 3\right)} \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]

      3. associate-*r* 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a} \]

      4. *-commutative 38.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot 3\right)} \cdot c\right)}}}{3 \cdot a} \]

   5. Simplified 38.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \left(a \cdot 3\right) \cdot c\right)}}}}{3 \cdot a} \]

   6. Taylor expanded in b around 0 73.0%

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

      1. fma-def 73.0%

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

   8. Simplified 73.0%

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

   9. Taylor expanded in b around -inf 76.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   10. Step-by-step derivation

       1. +-commutative 76.9%

          \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} + -1 \cdot b\right)}}{3 \cdot a} \]

       2. mul-1-neg 76.9%

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

       3. unsub-neg 76.9%

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

       4. associate-/l* 81.9%

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

   11. Simplified 81.9%

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

   if -6.5e-76 < b < 2.29999999999999999e-95

   1. Initial program 71.9%

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

      1. prod-diff 71.6%

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

      2. *-commutative 71.6%

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

      3. fma-def 71.6%

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

      4. associate-+l+ 71.6%

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

      5. pow2 71.6%

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

      6. *-commutative 71.6%

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

      7. distribute-rgt-neg-in 71.6%

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

      8. *-commutative 71.6%

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

      9. distribute-rgt-neg-in 71.6%

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

      10. metadata-eval 71.6%

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

      11. *-commutative 71.6%

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

      12. fma-udef 71.6%

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

      13. distribute-lft-neg-in 71.6%

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

      14. distribute-rgt-neg-in 71.6%

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

      15. *-commutative 71.6%

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

      16. distribute-rgt-neg-in 71.6%

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

      17. metadata-eval 71.6%

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

      18. associate-*l* 71.6%

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

   3. Applied egg-rr 71.6%

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

   4. Taylor expanded in b around 0 67.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{a} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 68.0%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}} \]

      2. associate-*r/ 68.0%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot 1}{a}} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)} \]

      3. metadata-eval 68.0%

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)} \]

      4. distribute-rgt-out 68.3%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}} \]

      5. metadata-eval 68.3%

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

      6. *-commutative 68.3%

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

      7. associate-*r* 68.4%

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

   6. Simplified 68.4%

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

   if 2.29999999999999999e-95 < b

   1. Initial program 14.8%

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

   2. Taylor expanded in b around inf 90.8%

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

      1. *-commutative 90.8%

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

      2. associate-*l/ 90.8%

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

   4. Simplified 90.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a}{\frac{b}{c}} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{c \cdot \left(a \cdot -3\right)} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6: 68.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a}{\frac{b}{c}} - b\right) - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (/ (- (- (* 1.5 (/ a (/ b c))) b) b) (* 3.0 a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (((1.5 * (a / (b / c))) - b) - 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 <= (-5d-310)) then
        tmp = (((1.5d0 * (a / (b / c))) - b) - 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 <= -5e-310) {
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(Float64(Float64(1.5 * Float64(a / Float64(b / c))) - b) - 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 <= -5e-310)
		tmp = (((1.5 * (a / (b / c))) - b) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(N[(N[(1.5 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $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 < -4.999999999999985e-310

   1. Initial program 74.2%

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

      1. flip-- 41.8%

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

      2. div-inv 41.8%

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

      3. pow2 41.8%

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

      4. pow2 41.8%

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

      5. pow-prod-up 41.8%

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

      6. metadata-eval 41.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{\color{blue}{4}} - \left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      7. pow2 41.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - \color{blue}{{\left(\left(3 \cdot a\right) \cdot c\right)}^{2}}\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      8. associate-*l* 41.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{2}\right) \cdot \frac{1}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      9. fma-def 41.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}}}}{3 \cdot a} \]

      10. associate-*l* 41.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{3 \cdot \left(a \cdot c\right)}\right)}}}{3 \cdot a} \]

   3. Applied egg-rr 41.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-*r* 41.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\color{blue}{\left(\left(3 \cdot a\right) \cdot c\right)}}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]

      2. *-commutative 41.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\color{blue}{\left(a \cdot 3\right)} \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]

      3. associate-*r* 41.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a} \]

      4. *-commutative 41.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot 3\right)} \cdot c\right)}}}{3 \cdot a} \]

   5. Simplified 41.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{4} - {\left(\left(a \cdot 3\right) \cdot c\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, \left(a \cdot 3\right) \cdot c\right)}}}}{3 \cdot a} \]

   6. Taylor expanded in b around 0 74.2%

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

      1. fma-def 74.2%

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

   8. Simplified 74.2%

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

   9. Taylor expanded in b around -inf 62.1%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   10. Step-by-step derivation

       1. +-commutative 62.1%

          \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} + -1 \cdot b\right)}}{3 \cdot a} \]

       2. mul-1-neg 62.1%

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

       3. unsub-neg 62.1%

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

       4. associate-/l* 65.9%

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

   11. Simplified 65.9%

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

   if -4.999999999999985e-310 < b

   1. Initial program 27.6%

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

   2. Taylor expanded in b around inf 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-*l/ 72.6%

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

   4. Simplified 72.6%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a}{\frac{b}{c}} - b\right) - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7: 68.1% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.2%

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

   2. Taylor expanded in b around -inf 65.9%

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

   if -4.999999999999985e-310 < b

   1. Initial program 27.6%

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

   2. Taylor expanded in b around inf 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-*l/ 72.6%

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

   4. Simplified 72.6%

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

4. Final simplification 69.3%

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

Alternative 8: 67.9% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-283}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.1e-283) (/ (* b -2.0) (* 3.0 a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.1e-283) {
		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 <= 3.1d-283) 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 <= 3.1e-283) {
		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 <= 3.1e-283:
		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 <= 3.1e-283)
		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 <= 3.1e-283)
		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, 3.1e-283], 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 < 3.10000000000000004e-283

   1. Initial program 73.5%

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

   2. Taylor expanded in b around -inf 63.9%

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

      1. *-commutative 63.9%

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

   4. Simplified 63.9%

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

   if 3.10000000000000004e-283 < b

   1. Initial program 26.8%

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

   2. Taylor expanded in b around inf 74.8%

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

      1. *-commutative 74.8%

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

      2. associate-*l/ 74.8%

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

   4. Simplified 74.8%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-283}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 9: 67.9% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-283}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.1e-283) (* -0.6666666666666666 (/ b a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.1e-283) {
		tmp = -0.6666666666666666 * (b / 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 <= 3.1d-283) then
        tmp = (-0.6666666666666666d0) * (b / 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 <= 3.1e-283) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.1e-283:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.1e-283)
		tmp = Float64(-0.6666666666666666 * Float64(b / 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 <= 3.1e-283)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.1e-283], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.10000000000000004e-283

   1. Initial program 73.5%

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

   2. Taylor expanded in b around -inf 63.9%

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

      1. *-commutative 63.9%

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

   4. Simplified 63.9%

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

   if 3.10000000000000004e-283 < b

   1. Initial program 26.8%

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

   2. Taylor expanded in b around inf 74.8%

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

      1. *-commutative 74.8%

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

      2. associate-*l/ 74.8%

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

   4. Simplified 74.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-283}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 10: 34.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.7%

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

2. Taylor expanded in b around -inf 34.0%

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

   1. *-commutative 34.0%

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

4. Simplified 34.0%

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

5. Final simplification 34.0%

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

Reproduce

herbie shell --seed 2023318 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))