Cubic critical

Percentage Accurate: 52.7% → 85.7%

Time: 15.4s

Alternatives: 10

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: 52.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{+153}:\\ \;\;\;\;-2 \cdot \frac{b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.4e+153)
   (* -2.0 (/ b (* 3.0 a)))
   (if (<= b 2.8e-64)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e+153) {
		tmp = -2.0 * (b / (3.0 * a));
	} else if (b <= 2.8e-64) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.4d+153)) then
        tmp = (-2.0d0) * (b / (3.0d0 * a))
    else if (b <= 2.8d-64) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e+153) {
		tmp = -2.0 * (b / (3.0 * a));
	} else if (b <= 2.8e-64) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.4e+153:
		tmp = -2.0 * (b / (3.0 * a))
	elif b <= 2.8e-64:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.4e+153)
		tmp = Float64(-2.0 * Float64(b / Float64(3.0 * a)));
	elseif (b <= 2.8e-64)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.4e+153)
		tmp = -2.0 * (b / (3.0 * a));
	elseif (b <= 2.8e-64)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.4e+153], N[(-2.0 * N[(b / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.8e-64], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.40000000000000067e153

   1. Initial program 23.6%

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

   3. Taylor expanded in b around -inf 97.5%

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

      1. *-commutative 97.5%

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

   5. Simplified 97.5%

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

      1. *-commutative 97.5%

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

      2. associate-/l* 97.5%

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

   7. Applied egg-rr 97.5%

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

   if -8.40000000000000067e153 < b < 2.80000000000000004e-64

   1. Initial program 81.0%

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

   if 2.80000000000000004e-64 < b

   1. Initial program 14.0%

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

   3. Taylor expanded in b around inf 88.9%

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

      1. *-commutative 88.9%

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

   5. Simplified 88.9%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{+153}:\\ \;\;\;\;-2 \cdot \frac{b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{c}{b} \cdot \left(--0.5\right) - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 10^{-64}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{-3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.4e-44)
   (- (* (/ c b) (- -0.5)) (* 0.6666666666666666 (/ b a)))
   (if (<= b 1e-64)
     (* 0.3333333333333333 (/ (+ b (sqrt (* -3.0 (* a c)))) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4e-44) {
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a));
	} else if (b <= 1e-64) {
		tmp = 0.3333333333333333 * ((b + sqrt((-3.0 * (a * c)))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.4d-44)) then
        tmp = ((c / b) * -(-0.5d0)) - (0.6666666666666666d0 * (b / a))
    else if (b <= 1d-64) then
        tmp = 0.3333333333333333d0 * ((b + sqrt(((-3.0d0) * (a * c)))) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4e-44) {
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a));
	} else if (b <= 1e-64) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((-3.0 * (a * c)))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.4e-44:
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a))
	elif b <= 1e-64:
		tmp = 0.3333333333333333 * ((b + math.sqrt((-3.0 * (a * c)))) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.4e-44)
		tmp = Float64(Float64(Float64(c / b) * Float64(-(-0.5))) - Float64(0.6666666666666666 * Float64(b / a)));
	elseif (b <= 1e-64)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(-3.0 * Float64(a * c)))) / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.4e-44)
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a));
	elseif (b <= 1e-64)
		tmp = 0.3333333333333333 * ((b + sqrt((-3.0 * (a * c)))) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.4e-44], N[(N[(N[(c / b), $MachinePrecision] * (--0.5)), $MachinePrecision] - N[(0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-64], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.40000000000000016e-44

   1. Initial program 62.5%

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

   3. Taylor expanded in b around -inf 93.9%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   4. Taylor expanded in c around 0 94.0%

      \[\leadsto -1 \cdot \color{blue}{\left(-0.5 \cdot \frac{c}{b} + 0.6666666666666666 \cdot \frac{b}{a}\right)} \]

   if -3.40000000000000016e-44 < b < 9.99999999999999965e-65

   1. Initial program 73.4%

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

   3. Taylor expanded in b around 0 65.6%

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

      1. *-un-lft-identity 65.6%

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

      2. times-frac 65.5%

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

      3. metadata-eval 65.5%

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

      4. add-sqr-sqrt 34.8%

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

      5. sqrt-unprod 65.0%

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

      6. sqr-neg 65.0%

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

      7. sqrt-prod 31.0%

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

      8. add-sqr-sqrt 63.2%

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

      9. associate-*r* 63.2%

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

      10. *-commutative 63.2%

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

   5. Applied egg-rr 63.2%

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

   6. Taylor expanded in a around 0 63.2%

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

   if 9.99999999999999965e-65 < b

   1. Initial program 14.0%

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

   3. Taylor expanded in b around inf 88.9%

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

      1. *-commutative 88.9%

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

   5. Simplified 88.9%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{c}{b} \cdot \left(--0.5\right) - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 10^{-64}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{-3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{c}{b} \cdot \left(--0.5\right) - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.5e-32)
   (- (* (/ c b) (- -0.5)) (* 0.6666666666666666 (/ b a)))
   (if (<= b 7.5e-64)
     (/ (+ b (sqrt (* c (* a -3.0)))) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-32) {
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a));
	} else if (b <= 7.5e-64) {
		tmp = (b + sqrt((c * (a * -3.0)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.5d-32)) then
        tmp = ((c / b) * -(-0.5d0)) - (0.6666666666666666d0 * (b / a))
    else if (b <= 7.5d-64) then
        tmp = (b + sqrt((c * (a * (-3.0d0))))) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-32) {
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a));
	} else if (b <= 7.5e-64) {
		tmp = (b + Math.sqrt((c * (a * -3.0)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.5e-32:
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a))
	elif b <= 7.5e-64:
		tmp = (b + math.sqrt((c * (a * -3.0)))) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e-32)
		tmp = Float64(Float64(Float64(c / b) * Float64(-(-0.5))) - Float64(0.6666666666666666 * Float64(b / a)));
	elseif (b <= 7.5e-64)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -3.0)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.5e-32)
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a));
	elseif (b <= 7.5e-64)
		tmp = (b + sqrt((c * (a * -3.0)))) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e-32], N[(N[(N[(c / b), $MachinePrecision] * (--0.5)), $MachinePrecision] - N[(0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.5e-64], N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.49999999999999988e-32

   1. Initial program 62.5%

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

   3. Taylor expanded in b around -inf 93.9%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   4. Taylor expanded in c around 0 94.0%

      \[\leadsto -1 \cdot \color{blue}{\left(-0.5 \cdot \frac{c}{b} + 0.6666666666666666 \cdot \frac{b}{a}\right)} \]

   if -6.49999999999999988e-32 < b < 7.49999999999999949e-64

   1. Initial program 73.4%

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

   3. Taylor expanded in b around 0 65.6%

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

      1. div-inv 65.5%

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

      2. add-sqr-sqrt 34.9%

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

      3. sqrt-unprod 65.0%

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

      4. sqr-neg 65.0%

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

      5. sqrt-prod 31.0%

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

      6. add-sqr-sqrt 63.2%

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

      7. associate-*r* 63.2%

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

      8. *-commutative 63.2%

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

      9. *-commutative 63.2%

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

   5. Applied egg-rr 63.2%

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

      1. associate-*r/ 63.4%

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

      2. *-rgt-identity 63.4%

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

      3. *-commutative 63.4%

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

   7. Simplified 63.4%

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

   if 7.49999999999999949e-64 < b

   1. Initial program 14.0%

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

   3. Taylor expanded in b around inf 88.9%

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

      1. *-commutative 88.9%

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

   5. Simplified 88.9%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{c}{b} \cdot \left(--0.5\right) - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{c}{b} \cdot \left(--0.5\right) - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.6e-42)
   (- (* (/ c b) (- -0.5)) (* 0.6666666666666666 (/ b a)))
   (if (<= b 3.2e-62)
     (/ (- (sqrt (* -3.0 (* a c))) b) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e-42) {
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a));
	} else if (b <= 3.2e-62) {
		tmp = (sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.6d-42)) then
        tmp = ((c / b) * -(-0.5d0)) - (0.6666666666666666d0 * (b / a))
    else if (b <= 3.2d-62) then
        tmp = (sqrt(((-3.0d0) * (a * c))) - b) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e-42) {
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a));
	} else if (b <= 3.2e-62) {
		tmp = (Math.sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.6e-42:
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a))
	elif b <= 3.2e-62:
		tmp = (math.sqrt((-3.0 * (a * c))) - b) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.6e-42)
		tmp = Float64(Float64(Float64(c / b) * Float64(-(-0.5))) - Float64(0.6666666666666666 * Float64(b / a)));
	elseif (b <= 3.2e-62)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.6e-42)
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a));
	elseif (b <= 3.2e-62)
		tmp = (sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.6e-42], N[(N[(N[(c / b), $MachinePrecision] * (--0.5)), $MachinePrecision] - N[(0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.2e-62], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.60000000000000035e-42

   1. Initial program 62.5%

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

   3. Taylor expanded in b around -inf 93.9%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   4. Taylor expanded in c around 0 94.0%

      \[\leadsto -1 \cdot \color{blue}{\left(-0.5 \cdot \frac{c}{b} + 0.6666666666666666 \cdot \frac{b}{a}\right)} \]

   if -7.60000000000000035e-42 < b < 3.20000000000000021e-62

   1. Initial program 73.4%

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

   3. Taylor expanded in b around 0 65.6%

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

   if 3.20000000000000021e-62 < b

   1. Initial program 14.0%

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

   3. Taylor expanded in b around inf 88.9%

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

      1. *-commutative 88.9%

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

   5. Simplified 88.9%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{c}{b} \cdot \left(--0.5\right) - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-36}:\\ \;\;\;\;\frac{c}{b} \cdot \left(--0.5\right) - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-36)
   (- (* (/ c b) (- -0.5)) (* 0.6666666666666666 (/ b a)))
   (if (<= b 1.35e-63)
     (/ (- (sqrt (* c (* a -3.0))) b) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-36) {
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a));
	} else if (b <= 1.35e-63) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.15d-36)) then
        tmp = ((c / b) * -(-0.5d0)) - (0.6666666666666666d0 * (b / a))
    else if (b <= 1.35d-63) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-36) {
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a));
	} else if (b <= 1.35e-63) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e-36:
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a))
	elif b <= 1.35e-63:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e-36)
		tmp = Float64(Float64(Float64(c / b) * Float64(-(-0.5))) - Float64(0.6666666666666666 * Float64(b / a)));
	elseif (b <= 1.35e-63)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.15e-36)
		tmp = ((c / b) * -(-0.5)) - (0.6666666666666666 * (b / a));
	elseif (b <= 1.35e-63)
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e-36], N[(N[(N[(c / b), $MachinePrecision] * (--0.5)), $MachinePrecision] - N[(0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-63], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14999999999999998e-36

   1. Initial program 62.5%

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

   3. Taylor expanded in b around -inf 93.9%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   4. Taylor expanded in c around 0 94.0%

      \[\leadsto -1 \cdot \color{blue}{\left(-0.5 \cdot \frac{c}{b} + 0.6666666666666666 \cdot \frac{b}{a}\right)} \]

   if -1.14999999999999998e-36 < b < 1.3500000000000001e-63

   1. Initial program 73.4%

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

   3. Taylor expanded in b around 0 65.6%

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

      1. *-commutative 65.6%

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

      2. *-commutative 65.6%

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

      3. associate-*r* 65.7%

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

   5. Simplified 65.7%

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

   if 1.3500000000000001e-63 < b

   1. Initial program 14.0%

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

   3. Taylor expanded in b around inf 88.9%

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

      1. *-commutative 88.9%

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

   5. Simplified 88.9%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-36}:\\ \;\;\;\;\frac{c}{b} \cdot \left(--0.5\right) - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.0% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 69.0%

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

   3. Taylor expanded in b around -inf 68.1%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   4. Taylor expanded in c around 0 69.9%

      \[\leadsto -1 \cdot \color{blue}{\left(-0.5 \cdot \frac{c}{b} + 0.6666666666666666 \cdot \frac{b}{a}\right)} \]

   if -4.999999999999985e-310 < b

   1. Initial program 31.0%

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

   3. Taylor expanded in b around inf 65.6%

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

      1. *-commutative 65.6%

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

   5. Simplified 65.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} \cdot \left(--0.5\right) - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2e-311) (* -2.0 (/ b (* 3.0 a))) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e-311) {
		tmp = -2.0 * (b / (3.0 * a));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2d-311) then
        tmp = (-2.0d0) * (b / (3.0d0 * a))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e-311) {
		tmp = -2.0 * (b / (3.0 * a));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2e-311:
		tmp = -2.0 * (b / (3.0 * a))
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2e-311)
		tmp = Float64(-2.0 * Float64(b / Float64(3.0 * a)));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2e-311)
		tmp = -2.0 * (b / (3.0 * a));
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2e-311], N[(-2.0 * N[(b / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9999999999999e-311

   1. Initial program 69.0%

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

   3. Taylor expanded in b around -inf 69.8%

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

      1. *-commutative 69.8%

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

   5. Simplified 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-/l* 69.8%

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

   7. Applied egg-rr 69.8%

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

   if 1.9999999999999e-311 < b

   1. Initial program 31.0%

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

   3. Taylor expanded in b around inf 65.6%

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

      1. *-commutative 65.6%

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

   5. Simplified 65.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.8% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 69.0%

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

   3. Taylor expanded in b around -inf 69.6%

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

      1. *-commutative 69.6%

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

   5. Simplified 69.6%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.0%

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

   3. Taylor expanded in b around inf 65.6%

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

      1. *-commutative 65.6%

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

   5. Simplified 65.6%

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

4. Final simplification 67.6%

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

Alternative 9: 35.9% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / 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 * ((-0.6666666666666666d0) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

   1. frac-2neg 50.3%

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

   2. div-inv 50.2%

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

4. Applied egg-rr 50.3%

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

   1. associate-*r/ 50.4%

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

   2. *-rgt-identity 50.4%

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

   3. *-commutative 50.4%

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

   4. metadata-eval 50.4%

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

   5. distribute-lft-neg-in 50.4%

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

   6. distribute-neg-frac2 50.4%

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

   7. distribute-neg-frac 50.4%

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

   8. neg-mul-1 50.4%

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

   9. times-frac 49.9%

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

   10. metadata-eval 49.9%

       \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}}{a} \]

6. Simplified 49.9%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}}{a}} \]

7. Taylor expanded in b around -inf 36.6%

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

   1. associate-*r/ 36.5%

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

   2. *-commutative 36.5%

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

   3. associate-/l* 36.5%

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

9. Simplified 36.5%

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

10. Final simplification 36.5%

    \[\leadsto b \cdot \frac{-0.6666666666666666}{a} \]
11. Add Preprocessing

Alternative 10: 35.9% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in b around -inf 36.6%

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

   1. *-commutative 36.6%

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

5. Simplified 36.6%

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

6. Final simplification 36.6%

   \[\leadsto \frac{b}{a} \cdot -0.6666666666666666 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024082 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))