Cubic critical

Percentage Accurate: 51.9% → 85.4%

Time: 13.8s

Alternatives: 12

Speedup: 16.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{+101}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.05e+101)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 1.8e-126)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e+101) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.8e-126) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.05d+101)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 1.8d-126) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e+101) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.8e-126) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.05e+101:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 1.8e-126:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.05e+101)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.8e-126)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.05e+101)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 1.8e-126)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.05e+101], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-126], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.05e101

   1. Initial program 46.0%

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

   2. Taylor expanded in b around -inf 96.8%

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

   if -2.05e101 < b < 1.8e-126

   1. Initial program 86.0%

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

   if 1.8e-126 < b

   1. Initial program 21.4%

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

   2. Taylor expanded in b around inf 84.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{+101}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 2: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+33}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-126}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.4e+33)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 1.9e-126)
     (* -0.3333333333333333 (/ (- b (sqrt (- (* b b) (* 3.0 (* a c))))) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.4e+33) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.9e-126) {
		tmp = -0.3333333333333333 * ((b - sqrt(((b * b) - (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 <= (-1.4d+33)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 1.9d-126) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt(((b * b) - (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 <= -1.4e+33) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.9e-126) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt(((b * b) - (3.0 * (a * c))))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.4e+33:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 1.9e-126:
		tmp = -0.3333333333333333 * ((b - math.sqrt(((b * b) - (3.0 * (a * c))))) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.4e+33)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 1.9e-126)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(Float64(b * b) - 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 <= -1.4e+33)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 1.9e-126)
		tmp = -0.3333333333333333 * ((b - sqrt(((b * b) - (3.0 * (a * c))))) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.4e+33], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.9e-126], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $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 < -1.4e33

   1. Initial program 57.1%

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

   2. Taylor expanded in b around -inf 96.3%

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

      1. *-commutative 96.3%

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

   4. Simplified 96.3%

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

      1. associate-*l/ 96.3%

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

   6. Applied egg-rr 96.3%

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

   if -1.4e33 < b < 1.8999999999999999e-126

   1. Initial program 84.2%

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

      1. /-rgt-identity 84.2%

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

      2. metadata-eval 84.2%

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

      3. associate-/l* 84.2%

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

      4. associate-*r/ 84.2%

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

      5. *-commutative 84.2%

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

      6. associate-*l/ 84.2%

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

      7. associate-*r/ 84.2%

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

      8. metadata-eval 84.2%

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

      9. metadata-eval 84.2%

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

      10. times-frac 84.2%

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

      11. neg-mul-1 84.2%

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

      12. distribute-rgt-neg-in 84.2%

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

      13. times-frac 84.1%

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

      14. metadata-eval 84.1%

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

      15. neg-mul-1 84.1%

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

   3. Simplified 84.1%

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

      1. fma-udef 84.1%

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

      2. associate-*r* 84.2%

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

      3. *-commutative 84.2%

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

      4. metadata-eval 84.2%

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

      5. cancel-sign-sub-inv 84.2%

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

   5. Applied egg-rr 84.2%

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

   if 1.8999999999999999e-126 < b

   1. Initial program 21.4%

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

   2. Taylor expanded in b around inf 84.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+33}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-126}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 3: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-83}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-127}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.25e-83)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 9.5e-127)
     (* 0.3333333333333333 (/ (+ b (sqrt (* c (* a -3.0)))) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e-83) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 9.5e-127) {
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -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 <= (-1.25d-83)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 9.5d-127) then
        tmp = 0.3333333333333333d0 * ((b + sqrt((c * (a * (-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 <= -1.25e-83) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 9.5e-127) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.25e-83:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 9.5e-127:
		tmp = 0.3333333333333333 * ((b + math.sqrt((c * (a * -3.0)))) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.25e-83)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 9.5e-127)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(c * Float64(a * -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 <= -1.25e-83)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 9.5e-127)
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -3.0)))) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.25e-83], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 9.5e-127], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $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 < -1.25e-83

   1. Initial program 65.5%

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

   2. Taylor expanded in b around -inf 90.2%

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

      1. *-commutative 90.2%

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

   4. Simplified 90.2%

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

      1. associate-*l/ 90.3%

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

   6. Applied egg-rr 90.3%

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

   if -1.25e-83 < b < 9.4999999999999997e-127

   1. Initial program 81.0%

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

   2. Taylor expanded in b around 0 76.0%

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

      1. clear-num 75.9%

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

      2. inv-pow 75.9%

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

      3. *-commutative 75.9%

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

      4. add-sqr-sqrt 46.2%

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

      5. sqrt-unprod 75.6%

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

      6. sqr-neg 75.6%

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

      7. sqrt-prod 29.9%

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

      8. add-sqr-sqrt 75.0%

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

      9. *-commutative 75.0%

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

      10. *-commutative 75.0%

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

      11. associate-*r* 75.1%

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

   4. Applied egg-rr 75.1%

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

      1. unpow-1 75.1%

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

   6. Simplified 75.1%

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

      1. div-inv 74.8%

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

      2. *-commutative 74.8%

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

   8. Applied egg-rr 74.8%

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

      1. *-un-lft-identity 74.8%

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

      2. un-div-inv 75.1%

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

   10. Applied egg-rr 75.1%

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

       1. *-lft-identity 75.1%

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

       2. associate-/l* 74.8%

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

       3. associate-/r/ 75.0%

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

       4. metadata-eval 75.0%

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

       5. associate-*r* 75.1%

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

       6. *-commutative 75.1%

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

       7. associate-*r* 74.9%

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

   12. Simplified 74.9%

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

   if 9.4999999999999997e-127 < b

   1. Initial program 21.4%

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

   2. Taylor expanded in b around inf 84.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-83}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-127}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 4: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\left(b + \sqrt{a \cdot \left(c \cdot -3\right)}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.2e-84)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 4e-127)
     (* (+ b (sqrt (* a (* c -3.0)))) (/ 0.3333333333333333 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.2e-84) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 4e-127) {
		tmp = (b + sqrt((a * (c * -3.0)))) * (0.3333333333333333 / 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 <= (-6.2d-84)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 4d-127) then
        tmp = (b + sqrt((a * (c * (-3.0d0))))) * (0.3333333333333333d0 / 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 <= -6.2e-84) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 4e-127) {
		tmp = (b + Math.sqrt((a * (c * -3.0)))) * (0.3333333333333333 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.2e-84:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 4e-127:
		tmp = (b + math.sqrt((a * (c * -3.0)))) * (0.3333333333333333 / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.2e-84)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 4e-127)
		tmp = Float64(Float64(b + sqrt(Float64(a * Float64(c * -3.0)))) * Float64(0.3333333333333333 / 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 <= -6.2e-84)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 4e-127)
		tmp = (b + sqrt((a * (c * -3.0)))) * (0.3333333333333333 / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.2e-84], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4e-127], N[(N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.20000000000000003e-84

   1. Initial program 65.5%

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

   2. Taylor expanded in b around -inf 90.2%

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

      1. *-commutative 90.2%

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

   4. Simplified 90.2%

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

      1. associate-*l/ 90.3%

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

   6. Applied egg-rr 90.3%

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

   if -6.20000000000000003e-84 < b < 4.0000000000000001e-127

   1. Initial program 81.0%

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

   2. Taylor expanded in b around 0 76.0%

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

      1. expm1-log1p-u 59.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\right)\right)} \]

      2. expm1-udef 18.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\right)} - 1} \]

      3. div-inv 18.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}\right) \cdot \frac{1}{3 \cdot a}}\right)} - 1 \]

      4. add-sqr-sqrt 14.1%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      5. sqrt-unprod 19.0%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      6. sqr-neg 19.0%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sqrt{\color{blue}{b \cdot b}} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      7. sqrt-prod 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      8. add-sqr-sqrt 18.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{b} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      9. *-commutative 18.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(b + \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      10. *-commutative 18.9%

          \[\leadsto e^{\mathsf{log1p}\left(\left(b + \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -3}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      11. associate-*r* 18.9%

          \[\leadsto e^{\mathsf{log1p}\left(\left(b + \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      12. associate-/r* 18.9%

          \[\leadsto e^{\mathsf{log1p}\left(\left(b + \sqrt{a \cdot \left(c \cdot -3\right)}\right) \cdot \color{blue}{\frac{\frac{1}{3}}{a}}\right)} - 1 \]

      13. metadata-eval 18.9%

          \[\leadsto e^{\mathsf{log1p}\left(\left(b + \sqrt{a \cdot \left(c \cdot -3\right)}\right) \cdot \frac{\color{blue}{0.3333333333333333}}{a}\right)} - 1 \]

   4. Applied egg-rr 18.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(b + \sqrt{a \cdot \left(c \cdot -3\right)}\right) \cdot \frac{0.3333333333333333}{a}\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 59.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(b + \sqrt{a \cdot \left(c \cdot -3\right)}\right) \cdot \frac{0.3333333333333333}{a}\right)\right)} \]

      2. expm1-log1p 75.0%

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

      3. *-commutative 75.0%

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

   6. Simplified 75.0%

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

   if 4.0000000000000001e-127 < b

   1. Initial program 21.4%

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

   2. Taylor expanded in b around inf 84.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\left(b + \sqrt{a \cdot \left(c \cdot -3\right)}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 5: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-86}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.25e-86)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 1.9e-126)
     (/ (+ b (sqrt (* a (* c -3.0)))) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e-86) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.9e-126) {
		tmp = (b + sqrt((a * (c * -3.0)))) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.25d-86)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 1.9d-126) then
        tmp = (b + sqrt((a * (c * (-3.0d0))))) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e-86) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.9e-126) {
		tmp = (b + Math.sqrt((a * (c * -3.0)))) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.25e-86:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 1.9e-126:
		tmp = (b + math.sqrt((a * (c * -3.0)))) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.25e-86)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 1.9e-126)
		tmp = Float64(Float64(b + sqrt(Float64(a * Float64(c * -3.0)))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.25e-86)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 1.9e-126)
		tmp = (b + sqrt((a * (c * -3.0)))) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.25e-86], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.9e-126], N[(N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.25e-86

   1. Initial program 65.5%

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

   2. Taylor expanded in b around -inf 90.2%

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

      1. *-commutative 90.2%

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

   4. Simplified 90.2%

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

      1. associate-*l/ 90.3%

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

   6. Applied egg-rr 90.3%

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

   if -1.25e-86 < b < 1.8999999999999999e-126

   1. Initial program 81.0%

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

   2. Taylor expanded in b around 0 76.0%

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

      1. expm1-log1p-u 74.5%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)\right)}}{3 \cdot a} \]

      2. expm1-udef 34.4%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)} - 1}}{3 \cdot a} \]

      3. add-sqr-sqrt 22.8%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)} - 1}{3 \cdot a} \]

      4. sqrt-unprod 34.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)} - 1}{3 \cdot a} \]

      5. sqr-neg 34.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\sqrt{\color{blue}{b \cdot b}} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)} - 1}{3 \cdot a} \]

      6. sqrt-prod 11.6%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)} - 1}{3 \cdot a} \]

      7. add-sqr-sqrt 34.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{b} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)} - 1}{3 \cdot a} \]

      8. *-commutative 34.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(b + \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}\right)} - 1}{3 \cdot a} \]

      9. *-commutative 34.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(b + \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -3}\right)} - 1}{3 \cdot a} \]

      10. associate-*r* 34.4%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(b + \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}\right)} - 1}{3 \cdot a} \]

   4. Applied egg-rr 34.4%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(b + \sqrt{a \cdot \left(c \cdot -3\right)}\right)} - 1}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. expm1-def 73.6%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b + \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right)}}{3 \cdot a} \]

      2. expm1-log1p 75.1%

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

   6. Simplified 75.1%

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

   if 1.8999999999999999e-126 < b

   1. Initial program 21.4%

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

   2. Taylor expanded in b around inf 84.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-86}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 6: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-87}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-87)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 1.9e-126)
     (/ (+ b (sqrt (* c (* a -3.0)))) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-87) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.9e-126) {
		tmp = (b + sqrt((c * (a * -3.0)))) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.15d-87)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 1.9d-126) then
        tmp = (b + sqrt((c * (a * (-3.0d0))))) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-87) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.9e-126) {
		tmp = (b + Math.sqrt((c * (a * -3.0)))) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e-87:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 1.9e-126:
		tmp = (b + math.sqrt((c * (a * -3.0)))) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e-87)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 1.9e-126)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -3.0)))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.15e-87)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 1.9e-126)
		tmp = (b + sqrt((c * (a * -3.0)))) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e-87], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.9e-126], N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1500000000000001e-87

   1. Initial program 65.5%

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

   2. Taylor expanded in b around -inf 90.2%

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

      1. *-commutative 90.2%

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

   4. Simplified 90.2%

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

      1. associate-*l/ 90.3%

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

   6. Applied egg-rr 90.3%

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

   if -1.1500000000000001e-87 < b < 1.8999999999999999e-126

   1. Initial program 81.0%

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

   2. Taylor expanded in b around 0 76.0%

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

      1. expm1-log1p-u 74.5%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)\right)}}{3 \cdot a} \]

      2. expm1-udef 34.4%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)} - 1}}{3 \cdot a} \]

      3. add-sqr-sqrt 22.8%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)} - 1}{3 \cdot a} \]

      4. sqrt-unprod 34.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)} - 1}{3 \cdot a} \]

      5. sqr-neg 34.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\sqrt{\color{blue}{b \cdot b}} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)} - 1}{3 \cdot a} \]

      6. sqrt-prod 11.6%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)} - 1}{3 \cdot a} \]

      7. add-sqr-sqrt 34.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{b} + \sqrt{-3 \cdot \left(c \cdot a\right)}\right)} - 1}{3 \cdot a} \]

      8. *-commutative 34.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(b + \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}}\right)} - 1}{3 \cdot a} \]

      9. *-commutative 34.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(b + \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -3}\right)} - 1}{3 \cdot a} \]

      10. associate-*r* 34.4%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(b + \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}\right)} - 1}{3 \cdot a} \]

   4. Applied egg-rr 34.4%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(b + \sqrt{a \cdot \left(c \cdot -3\right)}\right)} - 1}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. expm1-def 73.6%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b + \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right)}}{3 \cdot a} \]

      2. expm1-log1p 75.1%

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

      3. associate-*r* 75.0%

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

      4. *-commutative 75.0%

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

      5. associate-*r* 75.1%

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

   6. Simplified 75.1%

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

   if 1.8999999999999999e-126 < b

   1. Initial program 21.4%

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

   2. Taylor expanded in b around inf 84.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-87}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 7: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-70}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-70)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 1.9e-126)
     (/ (- (sqrt (* (* a c) -3.0)) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-70) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.9e-126) {
		tmp = (sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-70)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 1.9d-126) then
        tmp = (sqrt(((a * c) * (-3.0d0))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-70) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.9e-126) {
		tmp = (Math.sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-70:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 1.9e-126:
		tmp = (math.sqrt(((a * c) * -3.0)) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-70)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 1.9e-126)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) * -3.0)) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-70)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 1.9e-126)
		tmp = (sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-70], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.9e-126], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.99999999999999999e-70

   1. Initial program 64.8%

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

   2. Taylor expanded in b around -inf 90.9%

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

      1. *-commutative 90.9%

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

   4. Simplified 90.9%

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

      1. associate-*l/ 91.0%

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

   6. Applied egg-rr 91.0%

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

   if -1.99999999999999999e-70 < b < 1.8999999999999999e-126

   1. Initial program 81.5%

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

   2. Taylor expanded in b around 0 75.5%

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

   if 1.8999999999999999e-126 < b

   1. Initial program 21.4%

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

   2. Taylor expanded in b around inf 84.7%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-70}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 8: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.8e-71)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 1.85e-126)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.8e-71) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.85e-126) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.8d-71)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 1.85d-126) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.8e-71) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.85e-126) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.8e-71:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 1.85e-126:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.8e-71)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 1.85e-126)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.8e-71)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 1.85e-126)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.8e-71], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.85e-126], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.80000000000000007e-71

   1. Initial program 64.8%

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

   2. Taylor expanded in b around -inf 90.9%

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

      1. *-commutative 90.9%

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

   4. Simplified 90.9%

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

      1. associate-*l/ 91.0%

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

   6. Applied egg-rr 91.0%

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

   if -6.80000000000000007e-71 < b < 1.85e-126

   1. Initial program 81.5%

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

   2. Taylor expanded in b around 0 75.5%

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

      1. *-commutative 75.5%

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

      2. *-commutative 75.5%

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

      3. *-commutative 75.5%

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

      4. associate-*l* 75.6%

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

   4. Simplified 75.6%

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

   if 1.85e-126 < b

   1. Initial program 21.4%

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

   2. Taylor expanded in b around inf 84.7%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 9: 68.4% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 70.5%

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

   2. Taylor expanded in b around -inf 70.8%

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

   if -1.999999999999994e-310 < b

   1. Initial program 34.3%

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

   2. Taylor expanded in b around inf 66.3%

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

4. Final simplification 68.8%

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

Alternative 10: 68.3% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 70.5%

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

   2. Taylor expanded in b around -inf 70.5%

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

      1. *-commutative 70.5%

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

   4. Simplified 70.5%

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

   if -1.999999999999994e-310 < b

   1. Initial program 34.3%

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

   2. Taylor expanded in b around inf 66.3%

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

4. Final simplification 68.7%

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

Alternative 11: 68.3% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 70.5%

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

   2. Taylor expanded in b around -inf 70.5%

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

      1. *-commutative 70.5%

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

   4. Simplified 70.5%

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

      1. associate-*l/ 70.5%

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

   6. Applied egg-rr 70.5%

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

   if -1.999999999999994e-310 < b

   1. Initial program 34.3%

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

   2. Taylor expanded in b around inf 66.3%

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

4. Final simplification 68.7%

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

Alternative 12: 35.1% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot -0.5 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ c b) -0.5))

C

double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c / b) * (-0.5d0)
end function

Java

public static double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

Python

def code(a, b, c):
	return (c / b) * -0.5

Julia

function code(a, b, c)
	return Float64(Float64(c / b) * -0.5)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c / b) * -0.5;
end

Wolfram

code[a_, b_, c_] := N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 55.0%

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

2. Taylor expanded in b around inf 29.6%

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

3. Final simplification 29.6%

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

Reproduce

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