Cubic critical

Percentage Accurate: 51.6% → 86.1%

Time: 16.9s

Alternatives: 14

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.6% 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: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.2e+109)
   (/ (* b (/ 2.0 a)) -3.0)
   (if (<= b 2.15e-83)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e+109) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 2.15e-83) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.2d+109)) then
        tmp = (b * (2.0d0 / a)) / (-3.0d0)
    else if (b <= 2.15d-83) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e+109) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 2.15e-83) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.2e+109:
		tmp = (b * (2.0 / a)) / -3.0
	elif b <= 2.15e-83:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.2e+109)
		tmp = Float64(Float64(b * Float64(2.0 / a)) / -3.0);
	elseif (b <= 2.15e-83)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.2e+109)
		tmp = (b * (2.0 / a)) / -3.0;
	elseif (b <= 2.15e-83)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.2e+109], N[(N[(b * N[(2.0 / a), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 2.15e-83], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.2e109

   1. Initial program 58.2%

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

   4. Applied egg-rr 54.6%

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

   5. Taylor expanded in b around -inf 96.3%

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

      1. *-commutative 96.3%

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

   7. Simplified 96.3%

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

      1. associate-/r* 96.3%

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

      2. associate-*r/ 96.4%

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

   9. Applied egg-rr 96.4%

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

   10. Taylor expanded in b around 0 96.4%

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

       1. associate-*r/ 96.4%

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

       2. *-commutative 96.4%

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

       3. associate-*r/ 96.4%

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

   12. Simplified 96.4%

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

   if -7.2e109 < b < 2.15000000000000017e-83

   1. Initial program 84.7%

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

   if 2.15000000000000017e-83 < b

   1. Initial program 15.0%

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

   3. Taylor expanded in b around inf 85.1%

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

      1. associate-*r/ 85.1%

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

   5. Simplified 85.1%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-73}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-83}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{-3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.25e-73)
   (/ (* b (/ 2.0 a)) -3.0)
   (if (<= b 1.95e-83)
     (* 0.3333333333333333 (/ (+ b (sqrt (* -3.0 (* a c)))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e-73) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 1.95e-83) {
		tmp = 0.3333333333333333 * ((b + sqrt((-3.0 * (a * c)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.25d-73)) then
        tmp = (b * (2.0d0 / a)) / (-3.0d0)
    else if (b <= 1.95d-83) then
        tmp = 0.3333333333333333d0 * ((b + sqrt(((-3.0d0) * (a * c)))) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e-73) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 1.95e-83) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((-3.0 * (a * c)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.25e-73:
		tmp = (b * (2.0 / a)) / -3.0
	elif b <= 1.95e-83:
		tmp = 0.3333333333333333 * ((b + math.sqrt((-3.0 * (a * c)))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.25e-73)
		tmp = Float64(Float64(b * Float64(2.0 / a)) / -3.0);
	elseif (b <= 1.95e-83)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(-3.0 * Float64(a * c)))) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.25e-73)
		tmp = (b * (2.0 / a)) / -3.0;
	elseif (b <= 1.95e-83)
		tmp = 0.3333333333333333 * ((b + sqrt((-3.0 * (a * c)))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.25e-73], N[(N[(b * N[(2.0 / a), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 1.95e-83], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.25e-73

   1. Initial program 75.2%

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

   4. Applied egg-rr 62.3%

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

   5. Taylor expanded in b around -inf 87.8%

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

      1. *-commutative 87.8%

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

   7. Simplified 87.8%

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

      1. associate-/r* 87.7%

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

      2. associate-*r/ 87.9%

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

   9. Applied egg-rr 87.9%

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

   10. Taylor expanded in b around 0 87.9%

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

       1. associate-*r/ 87.9%

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

       2. *-commutative 87.9%

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

       3. associate-*r/ 87.9%

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

   12. Simplified 87.9%

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

   if -1.25e-73 < b < 1.95e-83

   1. Initial program 77.9%

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

   3. Taylor expanded in b around 0 74.6%

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

      1. associate-*r* 74.8%

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

      2. *-commutative 74.8%

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

      3. *-commutative 74.8%

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

   5. Simplified 74.8%

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

      1. *-un-lft-identity 74.8%

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

      2. times-frac 74.4%

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

      3. metadata-eval 74.4%

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

      4. +-commutative 74.4%

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

      5. add-sqr-sqrt 44.5%

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

      6. sqrt-unprod 74.0%

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

      7. sqr-neg 74.0%

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

      8. sqrt-unprod 29.9%

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

      9. add-sqr-sqrt 72.8%

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

   7. Applied egg-rr 72.8%

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

   8. Taylor expanded in c around 0 73.0%

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

   if 1.95e-83 < b

   1. Initial program 15.0%

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

   3. Taylor expanded in b around inf 85.1%

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

      1. associate-*r/ 85.1%

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

   5. Simplified 85.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-73}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-83}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{-3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.92 \cdot 10^{-60}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{a \cdot \left(-3 \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.92e-60)
   (/ (* b (/ 2.0 a)) -3.0)
   (if (<= b 2.8e-83)
     (* (/ -0.3333333333333333 a) (- b (sqrt (* a (* -3.0 c)))))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.92e-60) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 2.8e-83) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt((a * (-3.0 * c))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.92d-60)) then
        tmp = (b * (2.0d0 / a)) / (-3.0d0)
    else if (b <= 2.8d-83) then
        tmp = ((-0.3333333333333333d0) / a) * (b - sqrt((a * ((-3.0d0) * c))))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.92e-60) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 2.8e-83) {
		tmp = (-0.3333333333333333 / a) * (b - Math.sqrt((a * (-3.0 * c))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.92e-60:
		tmp = (b * (2.0 / a)) / -3.0
	elif b <= 2.8e-83:
		tmp = (-0.3333333333333333 / a) * (b - math.sqrt((a * (-3.0 * c))))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.92e-60)
		tmp = Float64(Float64(b * Float64(2.0 / a)) / -3.0);
	elseif (b <= 2.8e-83)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(Float64(a * Float64(-3.0 * c)))));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.92e-60)
		tmp = (b * (2.0 / a)) / -3.0;
	elseif (b <= 2.8e-83)
		tmp = (-0.3333333333333333 / a) * (b - sqrt((a * (-3.0 * c))));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.92e-60], N[(N[(b * N[(2.0 / a), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 2.8e-83], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9200000000000001e-60

   1. Initial program 74.7%

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

   4. Applied egg-rr 61.6%

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

   5. Taylor expanded in b around -inf 88.5%

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

      1. *-commutative 88.5%

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

   7. Simplified 88.5%

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

      1. associate-/r* 88.4%

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

      2. associate-*r/ 88.6%

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

   9. Applied egg-rr 88.6%

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

   10. Taylor expanded in b around 0 88.6%

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

       1. associate-*r/ 88.6%

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

       2. *-commutative 88.6%

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

       3. associate-*r/ 88.6%

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

   12. Simplified 88.6%

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

   if -1.9200000000000001e-60 < b < 2.8000000000000001e-83

   1. Initial program 78.5%

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

   3. Taylor expanded in b around 0 74.2%

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

      1. associate-*r* 74.4%

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

      2. *-commutative 74.4%

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

      3. *-commutative 74.4%

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

   5. Simplified 74.4%

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

      1. frac-2neg 74.4%

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

      2. div-inv 74.3%

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

      3. neg-sub0 74.3%

         \[\leadsto \color{blue}{\left(0 - \left(\left(-b\right) + \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right)} \cdot \frac{1}{-3 \cdot a} \]

      4. add-sqr-sqrt 45.1%

         \[\leadsto \left(0 - \left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      5. sqrt-unprod 73.9%

         \[\leadsto \left(0 - \left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      6. sqr-neg 73.9%

         \[\leadsto \left(0 - \left(\sqrt{\color{blue}{b \cdot b}} + \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      7. sqrt-unprod 29.2%

         \[\leadsto \left(0 - \left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      8. add-sqr-sqrt 72.5%

         \[\leadsto \left(0 - \left(\color{blue}{b} + \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{1}{-3 \cdot a} \]

      9. associate--l- 72.5%

         \[\leadsto \color{blue}{\left(\left(0 - b\right) - \sqrt{c \cdot \left(a \cdot -3\right)}\right)} \cdot \frac{1}{-3 \cdot a} \]

      10. neg-sub0 72.5%

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

      11. add-sqr-sqrt 43.3%

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

      12. sqrt-unprod 72.5%

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

      13. sqr-neg 72.5%

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

      14. sqrt-unprod 29.1%

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

      15. add-sqr-sqrt 74.3%

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

      16. distribute-lft-neg-in 74.3%

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

      17. metadata-eval 74.3%

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

      18. *-commutative 74.3%

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

      19. associate-/r* 73.9%

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

      20. div-inv 73.7%

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

      21. metadata-eval 73.7%

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

   7. Applied egg-rr 73.7%

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

      1. *-commutative 73.7%

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

      2. associate-*l/ 73.8%

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

      3. metadata-eval 73.8%

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

      4. associate-*r* 74.1%

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

      5. *-commutative 74.1%

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

      6. associate-*r* 74.0%

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

   9. Simplified 74.0%

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

   if 2.8000000000000001e-83 < b

   1. Initial program 15.0%

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

   3. Taylor expanded in b around inf 85.1%

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

      1. associate-*r/ 85.1%

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

   5. Simplified 85.1%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.92 \cdot 10^{-60}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{a \cdot \left(-3 \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-59}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-3 \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.45e-59)
   (/ (* b (/ 2.0 a)) -3.0)
   (if (<= b 8.4e-84)
     (/ (- (sqrt (* a (* -3.0 c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.45e-59) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 8.4e-84) {
		tmp = (sqrt((a * (-3.0 * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.45d-59)) then
        tmp = (b * (2.0d0 / a)) / (-3.0d0)
    else if (b <= 8.4d-84) then
        tmp = (sqrt((a * ((-3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.45e-59) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 8.4e-84) {
		tmp = (Math.sqrt((a * (-3.0 * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.45e-59:
		tmp = (b * (2.0 / a)) / -3.0
	elif b <= 8.4e-84:
		tmp = (math.sqrt((a * (-3.0 * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.45e-59)
		tmp = Float64(Float64(b * Float64(2.0 / a)) / -3.0);
	elseif (b <= 8.4e-84)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-3.0 * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.45e-59)
		tmp = (b * (2.0 / a)) / -3.0;
	elseif (b <= 8.4e-84)
		tmp = (sqrt((a * (-3.0 * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.45e-59], N[(N[(b * N[(2.0 / a), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 8.4e-84], N[(N[(N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.45000000000000008e-59

   1. Initial program 74.7%

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

   4. Applied egg-rr 61.6%

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

   5. Taylor expanded in b around -inf 88.5%

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

      1. *-commutative 88.5%

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

   7. Simplified 88.5%

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

      1. associate-/r* 88.4%

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

      2. associate-*r/ 88.6%

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

   9. Applied egg-rr 88.6%

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

   10. Taylor expanded in b around 0 88.6%

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

       1. associate-*r/ 88.6%

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

       2. *-commutative 88.6%

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

       3. associate-*r/ 88.6%

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

   12. Simplified 88.6%

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

   if -1.45000000000000008e-59 < b < 8.39999999999999992e-84

   1. Initial program 78.5%

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

   3. Taylor expanded in b around 0 74.2%

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

      1. associate-*r* 74.4%

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

      2. *-commutative 74.4%

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

      3. *-commutative 74.4%

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

   5. Simplified 74.4%

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

      1. +-commutative 74.4%

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

      2. unsub-neg 74.4%

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

   7. Applied egg-rr 74.4%

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

      1. associate-*r* 74.2%

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

      2. *-commutative 74.2%

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

      3. associate-*r* 74.2%

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

   9. Simplified 74.2%

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

   if 8.39999999999999992e-84 < b

   1. Initial program 15.0%

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

   3. Taylor expanded in b around inf 85.1%

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

      1. associate-*r/ 85.1%

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

   5. Simplified 85.1%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-59}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-3 \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-63}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 6.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.95e-63)
   (/ (* b (/ 2.0 a)) -3.0)
   (if (<= b 6.7e-84)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e-63) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 6.7e-84) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.95d-63)) then
        tmp = (b * (2.0d0 / a)) / (-3.0d0)
    else if (b <= 6.7d-84) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e-63) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 6.7e-84) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.95e-63:
		tmp = (b * (2.0 / a)) / -3.0
	elif b <= 6.7e-84:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.95e-63)
		tmp = Float64(Float64(b * Float64(2.0 / a)) / -3.0);
	elseif (b <= 6.7e-84)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.95e-63)
		tmp = (b * (2.0 / a)) / -3.0;
	elseif (b <= 6.7e-84)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.95e-63], N[(N[(b * N[(2.0 / a), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 6.7e-84], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.95000000000000011e-63

   1. Initial program 74.7%

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

   4. Applied egg-rr 61.6%

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

   5. Taylor expanded in b around -inf 88.5%

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

      1. *-commutative 88.5%

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

   7. Simplified 88.5%

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

      1. associate-/r* 88.4%

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

      2. associate-*r/ 88.6%

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

   9. Applied egg-rr 88.6%

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

   10. Taylor expanded in b around 0 88.6%

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

       1. associate-*r/ 88.6%

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

       2. *-commutative 88.6%

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

       3. associate-*r/ 88.6%

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

   12. Simplified 88.6%

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

   if -1.95000000000000011e-63 < b < 6.7e-84

   1. Initial program 78.5%

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

   3. Taylor expanded in b around 0 74.2%

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

      1. associate-*r* 74.4%

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

      2. *-commutative 74.4%

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

      3. *-commutative 74.4%

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

   5. Simplified 74.4%

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

      1. +-commutative 74.4%

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

      2. unsub-neg 74.4%

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

   7. Applied egg-rr 74.4%

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

   if 6.7e-84 < b

   1. Initial program 15.0%

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

   3. Taylor expanded in b around inf 85.1%

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

      1. associate-*r/ 85.1%

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

   5. Simplified 85.1%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-63}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 6.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-275}:\\ \;\;\;\;b \cdot \frac{2}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.05e-275) (* b (/ 2.0 (* a -3.0))) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.05e-275) {
		tmp = b * (2.0 / (a * -3.0));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.05d-275) then
        tmp = b * (2.0d0 / (a * (-3.0d0)))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.05e-275) {
		tmp = b * (2.0 / (a * -3.0));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.05e-275:
		tmp = b * (2.0 / (a * -3.0))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.05e-275)
		tmp = Float64(b * Float64(2.0 / Float64(a * -3.0)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.05e-275)
		tmp = b * (2.0 / (a * -3.0));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.05e-275], N[(b * N[(2.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.04999999999999994e-275

   1. Initial program 76.9%

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

   3. Taylor expanded in b around -inf 62.5%

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

      1. *-commutative 62.5%

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

   5. Simplified 62.5%

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

      1. associate-*l/ 62.5%

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

      2. clear-num 62.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{b \cdot -0.6666666666666666}}} \]

   7. Applied egg-rr 62.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{b \cdot -0.6666666666666666}}} \]
   8. Step-by-step derivation

      1. clear-num 62.5%

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

      2. *-commutative 62.5%

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

      3. *-un-lft-identity 62.5%

         \[\leadsto \frac{-0.6666666666666666 \cdot b}{\color{blue}{1 \cdot a}} \]

      4. times-frac 62.5%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{1} \cdot \frac{b}{a}} \]

      5. metadata-eval 62.5%

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

      6. metadata-eval 62.5%

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

      7. times-frac 62.5%

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

      8. *-commutative 62.5%

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

      9. *-commutative 62.5%

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

      10. un-div-inv 62.6%

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

      11. associate-*l* 62.6%

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

      12. un-div-inv 62.6%

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

   9. Applied egg-rr 62.6%

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

   if 1.04999999999999994e-275 < b

   1. Initial program 28.6%

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

   3. Taylor expanded in b around inf 69.9%

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

      1. associate-*r/ 69.9%

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

   5. Simplified 69.9%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-275}:\\ \;\;\;\;b \cdot \frac{2}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{-271}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.35e-271) (/ (* b (/ 2.0 a)) -3.0) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.35e-271) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.35d-271) then
        tmp = (b * (2.0d0 / a)) / (-3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.35e-271) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.35e-271:
		tmp = (b * (2.0 / a)) / -3.0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.35e-271)
		tmp = Float64(Float64(b * Float64(2.0 / a)) / -3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.35e-271)
		tmp = (b * (2.0 / a)) / -3.0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.35e-271], N[(N[(b * N[(2.0 / a), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.35000000000000002e-271

   1. Initial program 76.9%

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

   4. Applied egg-rr 68.8%

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

   5. Taylor expanded in b around -inf 62.6%

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

      1. *-commutative 62.6%

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

   7. Simplified 62.6%

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

      1. associate-/r* 62.5%

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

      2. associate-*r/ 62.6%

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

   9. Applied egg-rr 62.6%

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

   10. Taylor expanded in b around 0 62.6%

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

       1. associate-*r/ 62.6%

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

       2. *-commutative 62.6%

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

       3. associate-*r/ 62.6%

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

   12. Simplified 62.6%

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

   if 2.35000000000000002e-271 < b

   1. Initial program 28.6%

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

   3. Taylor expanded in b around inf 69.9%

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

      1. associate-*r/ 69.9%

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

   5. Simplified 69.9%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{-271}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 24.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-221}:\\ \;\;\;\;b \cdot \frac{-1.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-221) (* b (/ -1.3333333333333333 a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-221) {
		tmp = b * (-1.3333333333333333 / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3d-221) then
        tmp = b * ((-1.3333333333333333d0) / a)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-221) {
		tmp = b * (-1.3333333333333333 / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-221:
		tmp = b * (-1.3333333333333333 / a)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-221)
		tmp = Float64(b * Float64(-1.3333333333333333 / a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3e-221)
		tmp = b * (-1.3333333333333333 / a);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-221], N[(b * N[(-1.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < 3.0000000000000002e-221

   1. Initial program 76.3%

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

      1. +-commutative 76.3%

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

      2. sqr-neg 76.3%

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

      3. unsub-neg 76.3%

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

      4. div-sub 76.3%

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

      5. --rgt-identity 76.3%

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

      6. div-sub 76.3%

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

   3. Simplified 76.2%

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

      1. *-un-lft-identity 76.2%

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

      2. *-un-lft-identity 76.2%

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

      3. prod-diff 76.2%

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

   6. Applied egg-rr 30.4%

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

      1. +-commutative 30.4%

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

      2. associate-+l+ 30.4%

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

      3. associate-*r* 30.3%

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

      4. *-commutative 30.3%

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

      5. associate-*l* 30.3%

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

      6. fma-udef 30.3%

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

      7. *-rgt-identity 30.3%

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

   8. Simplified 30.3%

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

   9. Taylor expanded in b around inf 1.8%

      \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{b}{a}} \]
   10. Step-by-step derivation

       1. *-commutative 1.8%

          \[\leadsto \color{blue}{\frac{b}{a} \cdot 1.3333333333333333} \]

   11. Simplified 1.8%

       \[\leadsto \color{blue}{\frac{b}{a} \cdot 1.3333333333333333} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 1.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)\right)} \]

       2. expm1-udef 2.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)} - 1} \]

   13. Applied egg-rr 2.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)} - 1} \]
   14. Step-by-step derivation

       1. expm1-def 1.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)\right)} \]

       2. expm1-log1p-u 1.8%

          \[\leadsto \color{blue}{\frac{b}{a} \cdot 1.3333333333333333} \]

       3. frac-2neg 1.8%

          \[\leadsto \color{blue}{\frac{-b}{-a}} \cdot 1.3333333333333333 \]

       4. associate-*l/ 1.8%

          \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot 1.3333333333333333}{-a}} \]

       5. add-sqr-sqrt 1.5%

          \[\leadsto \frac{\color{blue}{\left(\sqrt{-b} \cdot \sqrt{-b}\right)} \cdot 1.3333333333333333}{-a} \]

       6. sqrt-unprod 1.8%

          \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} \cdot 1.3333333333333333}{-a} \]

       7. sqr-neg 1.8%

          \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b}} \cdot 1.3333333333333333}{-a} \]

       8. sqrt-unprod 0.2%

          \[\leadsto \frac{\color{blue}{\left(\sqrt{b} \cdot \sqrt{b}\right)} \cdot 1.3333333333333333}{-a} \]

       9. add-sqr-sqrt 25.9%

          \[\leadsto \frac{\color{blue}{b} \cdot 1.3333333333333333}{-a} \]

   15. Applied egg-rr 25.9%

       \[\leadsto \color{blue}{\frac{b \cdot 1.3333333333333333}{-a}} \]
   16. Step-by-step derivation

       1. *-commutative 25.9%

          \[\leadsto \frac{\color{blue}{1.3333333333333333 \cdot b}}{-a} \]

       2. neg-mul-1 25.9%

          \[\leadsto \frac{1.3333333333333333 \cdot b}{\color{blue}{-1 \cdot a}} \]

       3. times-frac 25.9%

          \[\leadsto \color{blue}{\frac{1.3333333333333333}{-1} \cdot \frac{b}{a}} \]

       4. metadata-eval 25.9%

          \[\leadsto \color{blue}{-1.3333333333333333} \cdot \frac{b}{a} \]

       5. metadata-eval 25.9%

          \[\leadsto \color{blue}{\left(-1.3333333333333333\right)} \cdot \frac{b}{a} \]

       6. distribute-lft-neg-in 25.9%

          \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{b}{a}} \]

       7. *-commutative 25.9%

          \[\leadsto -\color{blue}{\frac{b}{a} \cdot 1.3333333333333333} \]

       8. associate-*l/ 25.9%

          \[\leadsto -\color{blue}{\frac{b \cdot 1.3333333333333333}{a}} \]

       9. associate-*r/ 25.9%

          \[\leadsto -\color{blue}{b \cdot \frac{1.3333333333333333}{a}} \]

       10. metadata-eval 25.9%

           \[\leadsto -b \cdot \frac{\color{blue}{1.3333333333333333 \cdot 1}}{a} \]

       11. associate-*r/ 25.9%

           \[\leadsto -b \cdot \color{blue}{\left(1.3333333333333333 \cdot \frac{1}{a}\right)} \]

       12. distribute-rgt-neg-in 25.9%

           \[\leadsto \color{blue}{b \cdot \left(-1.3333333333333333 \cdot \frac{1}{a}\right)} \]

       13. associate-*r/ 25.9%

           \[\leadsto b \cdot \left(-\color{blue}{\frac{1.3333333333333333 \cdot 1}{a}}\right) \]

       14. metadata-eval 25.9%

           \[\leadsto b \cdot \left(-\frac{\color{blue}{1.3333333333333333}}{a}\right) \]

       15. distribute-neg-frac 25.9%

           \[\leadsto b \cdot \color{blue}{\frac{-1.3333333333333333}{a}} \]

       16. metadata-eval 25.9%

           \[\leadsto b \cdot \frac{\color{blue}{-1.3333333333333333}}{a} \]

   17. Simplified 25.9%

       \[\leadsto \color{blue}{b \cdot \frac{-1.3333333333333333}{a}} \]

   if 3.0000000000000002e-221 < b

   1. Initial program 27.7%

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

      1. +-commutative 27.7%

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

      2. sqr-neg 27.7%

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

      3. unsub-neg 27.7%

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

      4. div-sub 27.4%

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

      5. --rgt-identity 27.4%

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

      6. div-sub 27.7%

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

   3. Simplified 27.6%

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

      1. *-un-lft-identity 27.6%

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

      2. *-un-lft-identity 27.6%

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

      3. prod-diff 27.6%

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

   6. Applied egg-rr 23.1%

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

      1. +-commutative 23.1%

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

      2. associate-+l+ 23.1%

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

      3. associate-*r* 23.0%

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

      4. *-commutative 23.0%

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

      5. associate-*l* 23.0%

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

      6. fma-udef 23.0%

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

      7. *-rgt-identity 23.0%

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

   8. Simplified 23.0%

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

   9. Taylor expanded in b around inf 3.6%

      \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{b}{a}} \]
   10. Step-by-step derivation

       1. *-commutative 3.6%

          \[\leadsto \color{blue}{\frac{b}{a} \cdot 1.3333333333333333} \]

   11. Simplified 3.6%

       \[\leadsto \color{blue}{\frac{b}{a} \cdot 1.3333333333333333} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 2.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)\right)} \]

       2. expm1-udef 2.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)} - 1} \]

   13. Applied egg-rr 2.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)} - 1} \]

   14. Taylor expanded in b around 0 20.1%

       \[\leadsto \color{blue}{1} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-221}:\\ \;\;\;\;b \cdot \frac{-1.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-216}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.7e-216) (* -0.6666666666666666 (/ b a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.7e-216) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.7d-216) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.7e-216) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.7e-216:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.7e-216)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.7e-216)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.7e-216], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < 1.6999999999999999e-216

   1. Initial program 76.3%

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

   3. Taylor expanded in b around -inf 60.9%

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

      1. *-commutative 60.9%

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

   5. Simplified 60.9%

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

   if 1.6999999999999999e-216 < b

   1. Initial program 27.7%

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

      1. +-commutative 27.7%

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

      2. sqr-neg 27.7%

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

      3. unsub-neg 27.7%

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

      4. div-sub 27.4%

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

      5. --rgt-identity 27.4%

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

      6. div-sub 27.7%

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

   3. Simplified 27.6%

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

      1. *-un-lft-identity 27.6%

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

      2. *-un-lft-identity 27.6%

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

      3. prod-diff 27.6%

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

   6. Applied egg-rr 23.1%

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

      1. +-commutative 23.1%

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

      2. associate-+l+ 23.1%

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

      3. associate-*r* 23.0%

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

      4. *-commutative 23.0%

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

      5. associate-*l* 23.0%

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

      6. fma-udef 23.0%

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

      7. *-rgt-identity 23.0%

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

   8. Simplified 23.0%

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

   9. Taylor expanded in b around inf 3.6%

      \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{b}{a}} \]
   10. Step-by-step derivation

       1. *-commutative 3.6%

          \[\leadsto \color{blue}{\frac{b}{a} \cdot 1.3333333333333333} \]

   11. Simplified 3.6%

       \[\leadsto \color{blue}{\frac{b}{a} \cdot 1.3333333333333333} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 2.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)\right)} \]

       2. expm1-udef 2.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)} - 1} \]

   13. Applied egg-rr 2.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)} - 1} \]

   14. Taylor expanded in b around 0 20.1%

       \[\leadsto \color{blue}{1} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-216}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{-221}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.9e-221) (/ -0.6666666666666666 (/ a b)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.9e-221) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = 0.0;
	}
	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 <= 3.9d-221) then
        tmp = (-0.6666666666666666d0) / (a / b)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.9e-221) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.9e-221:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.9e-221)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.9e-221)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.9e-221], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < 3.8999999999999998e-221

   1. Initial program 76.3%

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

   3. Taylor expanded in b around -inf 60.9%

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

      1. *-commutative 60.9%

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

   5. Simplified 60.9%

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

      1. *-commutative 60.9%

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

      2. clear-num 60.9%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 61.0%

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

   7. Applied egg-rr 61.0%

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

   if 3.8999999999999998e-221 < b

   1. Initial program 27.7%

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

      1. +-commutative 27.7%

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

      2. sqr-neg 27.7%

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

      3. unsub-neg 27.7%

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

      4. div-sub 27.4%

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

      5. --rgt-identity 27.4%

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

      6. div-sub 27.7%

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

   3. Simplified 27.6%

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

      1. *-un-lft-identity 27.6%

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

      2. *-un-lft-identity 27.6%

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

      3. prod-diff 27.6%

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

   6. Applied egg-rr 23.1%

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

      1. +-commutative 23.1%

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

      2. associate-+l+ 23.1%

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

      3. associate-*r* 23.0%

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

      4. *-commutative 23.0%

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

      5. associate-*l* 23.0%

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

      6. fma-udef 23.0%

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

      7. *-rgt-identity 23.0%

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

   8. Simplified 23.0%

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

   9. Taylor expanded in b around inf 3.6%

      \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{b}{a}} \]
   10. Step-by-step derivation

       1. *-commutative 3.6%

          \[\leadsto \color{blue}{\frac{b}{a} \cdot 1.3333333333333333} \]

   11. Simplified 3.6%

       \[\leadsto \color{blue}{\frac{b}{a} \cdot 1.3333333333333333} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 2.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)\right)} \]

       2. expm1-udef 2.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)} - 1} \]

   13. Applied egg-rr 2.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)} - 1} \]

   14. Taylor expanded in b around 0 20.1%

       \[\leadsto \color{blue}{1} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{-221}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-275}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.05e-275) (/ -0.6666666666666666 (/ a b)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.05e-275) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.05d-275) then
        tmp = (-0.6666666666666666d0) / (a / b)
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.05e-275) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.05e-275:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.05e-275)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.05e-275)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.05e-275], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.04999999999999994e-275

   1. Initial program 76.9%

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

   3. Taylor expanded in b around -inf 62.5%

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

      1. *-commutative 62.5%

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

   5. Simplified 62.5%

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

      1. *-commutative 62.5%

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

      2. clear-num 62.4%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 62.5%

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

   7. Applied egg-rr 62.5%

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

   if 1.04999999999999994e-275 < b

   1. Initial program 28.6%

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

   3. Taylor expanded in b around inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. associate-/l* 69.6%

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

   5. Simplified 69.6%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-275}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-275}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.05e-275) (/ b (* a -1.5)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.05e-275) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.05d-275) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.05e-275) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.05e-275:
		tmp = b / (a * -1.5)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.05e-275)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.05e-275)
		tmp = b / (a * -1.5);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.05e-275], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.04999999999999994e-275

   1. Initial program 76.9%

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

   4. Applied egg-rr 68.8%

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

   5. Taylor expanded in b around -inf 62.6%

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

      1. *-commutative 62.6%

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

   7. Simplified 62.6%

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

      1. associate-/r* 62.5%

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

      2. associate-*r/ 62.6%

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

   9. Applied egg-rr 62.6%

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

   10. Taylor expanded in b around 0 62.5%

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

       1. metadata-eval 62.5%

          \[\leadsto \color{blue}{\frac{1}{-1.5}} \cdot \frac{b}{a} \]

       2. times-frac 62.5%

          \[\leadsto \color{blue}{\frac{1 \cdot b}{-1.5 \cdot a}} \]

       3. *-commutative 62.5%

          \[\leadsto \frac{1 \cdot b}{\color{blue}{a \cdot -1.5}} \]

       4. *-lft-identity 62.5%

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

   12. Simplified 62.5%

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

   if 1.04999999999999994e-275 < b

   1. Initial program 28.6%

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

   3. Taylor expanded in b around inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. associate-/l* 69.6%

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

   5. Simplified 69.6%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-275}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-275}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.05e-275) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.05e-275) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.05d-275) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.05e-275) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.05e-275:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.05e-275)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.05e-275)
		tmp = b / (a * -1.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.05e-275], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.04999999999999994e-275

   1. Initial program 76.9%

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

   4. Applied egg-rr 68.8%

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

   5. Taylor expanded in b around -inf 62.6%

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

      1. *-commutative 62.6%

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

   7. Simplified 62.6%

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

      1. associate-/r* 62.5%

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

      2. associate-*r/ 62.6%

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

   9. Applied egg-rr 62.6%

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

   10. Taylor expanded in b around 0 62.5%

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

       1. metadata-eval 62.5%

          \[\leadsto \color{blue}{\frac{1}{-1.5}} \cdot \frac{b}{a} \]

       2. times-frac 62.5%

          \[\leadsto \color{blue}{\frac{1 \cdot b}{-1.5 \cdot a}} \]

       3. *-commutative 62.5%

          \[\leadsto \frac{1 \cdot b}{\color{blue}{a \cdot -1.5}} \]

       4. *-lft-identity 62.5%

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

   12. Simplified 62.5%

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

   if 1.04999999999999994e-275 < b

   1. Initial program 28.6%

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

   3. Taylor expanded in b around inf 69.9%

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

      1. associate-*r/ 69.9%

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

   5. Simplified 69.9%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-275}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 11.7% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 56.9%

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

   1. +-commutative 56.9%

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

   2. sqr-neg 56.9%

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

   3. unsub-neg 56.9%

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

   4. div-sub 56.8%

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

   5. --rgt-identity 56.8%

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

   6. div-sub 56.9%

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

3. Simplified 56.8%

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

   1. *-un-lft-identity 56.8%

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

   2. *-un-lft-identity 56.8%

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

   3. prod-diff 56.8%

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

6. Applied egg-rr 27.5%

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

   1. +-commutative 27.5%

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

   2. associate-+l+ 27.5%

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

   3. associate-*r* 27.4%

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

   4. *-commutative 27.4%

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

   5. associate-*l* 27.4%

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

   6. fma-udef 27.4%

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

   7. *-rgt-identity 27.4%

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

8. Simplified 27.4%

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

9. Taylor expanded in b around inf 2.5%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{b}{a}} \]
10. Step-by-step derivation

    1. *-commutative 2.5%

       \[\leadsto \color{blue}{\frac{b}{a} \cdot 1.3333333333333333} \]

11. Simplified 2.5%

    \[\leadsto \color{blue}{\frac{b}{a} \cdot 1.3333333333333333} \]
12. Step-by-step derivation

    1. expm1-log1p-u 2.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)\right)} \]

    2. expm1-udef 2.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)} - 1} \]

13. Applied egg-rr 2.1%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b}{a} \cdot 1.3333333333333333\right)} - 1} \]

14. Taylor expanded in b around 0 9.8%

    \[\leadsto \color{blue}{1} - 1 \]

15. Final simplification 9.8%

    \[\leadsto 0 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024029 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))