Cubic critical

Percentage Accurate: 51.5% → 85.1%

Time: 13.3s

Alternatives: 10

Speedup: 16.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+110}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+110)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 5.4e-94)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+110) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 5.4e-94) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d+110)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 5.4d-94) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+110) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 5.4e-94) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+110:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 5.4e-94:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+110)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 5.4e-94)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e+110)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 5.4e-94)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+110], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.4e-94], 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 * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e110

   1. Initial program 50.0%

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

   2. Taylor expanded in b around -inf 95.7%

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

   if -1e110 < b < 5.4000000000000002e-94

   1. Initial program 84.6%

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

   if 5.4000000000000002e-94 < b

   1. Initial program 18.4%

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

   2. Taylor expanded in b around inf 89.9%

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

      1. *-commutative 89.9%

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

      2. associate-*l/ 89.9%

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

   4. Simplified 89.9%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+110}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-18}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot -1.5 + \frac{b}{a} \cdot 2\right)\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e-18)
   (* -0.3333333333333333 (+ (* (/ c b) -1.5) (* (/ b a) 2.0)))
   (if (<= b 3.2e-93)
     (/ (- (sqrt (* -3.0 (* a c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-18) {
		tmp = -0.3333333333333333 * (((c / b) * -1.5) + ((b / a) * 2.0));
	} else if (b <= 3.2e-93) {
		tmp = (sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8d-18)) then
        tmp = (-0.3333333333333333d0) * (((c / b) * (-1.5d0)) + ((b / a) * 2.0d0))
    else if (b <= 3.2d-93) then
        tmp = (sqrt(((-3.0d0) * (a * c))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-18) {
		tmp = -0.3333333333333333 * (((c / b) * -1.5) + ((b / a) * 2.0));
	} else if (b <= 3.2e-93) {
		tmp = (Math.sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8e-18:
		tmp = -0.3333333333333333 * (((c / b) * -1.5) + ((b / a) * 2.0))
	elif b <= 3.2e-93:
		tmp = (math.sqrt((-3.0 * (a * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e-18)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(c / b) * -1.5) + Float64(Float64(b / a) * 2.0)));
	elseif (b <= 3.2e-93)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8e-18)
		tmp = -0.3333333333333333 * (((c / b) * -1.5) + ((b / a) * 2.0));
	elseif (b <= 3.2e-93)
		tmp = (sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e-18], N[(-0.3333333333333333 * N[(N[(N[(c / b), $MachinePrecision] * -1.5), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.2e-93], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.0000000000000006e-18

   1. Initial program 65.9%

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

   3. Applied egg-rr 23.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 37.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)\right)} \]

      2. expm1-log1p 65.9%

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

      3. *-lft-identity 65.9%

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

      4. *-commutative 65.9%

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

      5. times-frac 65.9%

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

      6. metadata-eval 65.9%

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

   5. Simplified 65.9%

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

   6. Taylor expanded in b around -inf 89.3%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(-1.5 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)} \]

   if -8.0000000000000006e-18 < b < 3.1999999999999999e-93

   1. Initial program 82.9%

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

   2. Taylor expanded in b around 0 74.9%

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

   if 3.1999999999999999e-93 < b

   1. Initial program 18.4%

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

   2. Taylor expanded in b around inf 89.9%

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

      1. *-commutative 89.9%

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

      2. associate-*l/ 89.9%

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

   4. Simplified 89.9%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-18}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot -1.5 + \frac{b}{a} \cdot 2\right)\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-18}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot -1.5 + \frac{b}{a} \cdot 2\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e-18)
   (* -0.3333333333333333 (+ (* (/ c b) -1.5) (* (/ b a) 2.0)))
   (if (<= b 8e-90)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-18) {
		tmp = -0.3333333333333333 * (((c / b) * -1.5) + ((b / a) * 2.0));
	} else if (b <= 8e-90) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.5d-18)) then
        tmp = (-0.3333333333333333d0) * (((c / b) * (-1.5d0)) + ((b / a) * 2.0d0))
    else if (b <= 8d-90) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-18) {
		tmp = -0.3333333333333333 * (((c / b) * -1.5) + ((b / a) * 2.0));
	} else if (b <= 8e-90) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.5e-18:
		tmp = -0.3333333333333333 * (((c / b) * -1.5) + ((b / a) * 2.0))
	elif b <= 8e-90:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e-18)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(c / b) * -1.5) + Float64(Float64(b / a) * 2.0)));
	elseif (b <= 8e-90)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.5e-18)
		tmp = -0.3333333333333333 * (((c / b) * -1.5) + ((b / a) * 2.0));
	elseif (b <= 8e-90)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e-18], N[(-0.3333333333333333 * N[(N[(N[(c / b), $MachinePrecision] * -1.5), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-90], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.49999999999999991e-18

   1. Initial program 65.9%

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

   3. Applied egg-rr 23.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 37.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)\right)} \]

      2. expm1-log1p 65.9%

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

      3. *-lft-identity 65.9%

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

      4. *-commutative 65.9%

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

      5. times-frac 65.9%

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

      6. metadata-eval 65.9%

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

   5. Simplified 65.9%

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

   6. Taylor expanded in b around -inf 89.3%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(-1.5 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)} \]

   if -1.49999999999999991e-18 < b < 7.99999999999999996e-90

   1. Initial program 82.9%

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

   2. Taylor expanded in b around 0 74.9%

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

      1. associate-*r* 75.1%

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

      2. *-commutative 75.1%

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

      3. *-commutative 75.1%

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

   4. Simplified 75.1%

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

   if 7.99999999999999996e-90 < b

   1. Initial program 18.4%

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

   2. Taylor expanded in b around inf 89.9%

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

      1. *-commutative 89.9%

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

      2. associate-*l/ 89.9%

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

   4. Simplified 89.9%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-18}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot -1.5 + \frac{b}{a} \cdot 2\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4: 68.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot -1.5 + \frac{b}{a} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310)
   (* -0.3333333333333333 (+ (* (/ c b) -1.5) (* (/ b a) 2.0)))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = -0.3333333333333333 * (((c / b) * -1.5) + ((b / a) * 2.0));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = (-0.3333333333333333d0) * (((c / b) * (-1.5d0)) + ((b / a) * 2.0d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = -0.3333333333333333 * (((c / b) * -1.5) + ((b / a) * 2.0));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = -0.3333333333333333 * (((c / b) * -1.5) + ((b / a) * 2.0))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(c / b) * -1.5) + Float64(Float64(b / a) * 2.0)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = -0.3333333333333333 * (((c / b) * -1.5) + ((b / a) * 2.0));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(-0.3333333333333333 * N[(N[(N[(c / b), $MachinePrecision] * -1.5), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 74.4%

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

   3. Applied egg-rr 23.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 44.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)\right)} \]

      2. expm1-log1p 74.4%

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

      3. *-lft-identity 74.4%

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

      4. *-commutative 74.4%

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

      5. times-frac 74.3%

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

      6. metadata-eval 74.3%

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

   5. Simplified 74.3%

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

   6. Taylor expanded in b around -inf 66.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(-1.5 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)} \]

   if -1.999999999999994e-310 < b

   1. Initial program 31.9%

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

   2. Taylor expanded in b around inf 70.1%

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

      1. *-commutative 70.1%

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

      2. associate-*l/ 70.1%

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

   4. Simplified 70.1%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot -1.5 + \frac{b}{a} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5: 68.1% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 74.4%

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

   2. Taylor expanded in b around -inf 66.9%

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

   if -1.999999999999994e-310 < b

   1. Initial program 31.9%

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

   2. Taylor expanded in b around inf 70.1%

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

      1. *-commutative 70.1%

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

      2. associate-*l/ 70.1%

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

   4. Simplified 70.1%

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

4. Final simplification 68.6%

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

Alternative 6: 42.8% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-17}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.6e-17) (* b (/ -0.6666666666666666 a)) (* 0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.6e-17) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.6d-17) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = 0.5d0 * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.6e-17) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.6e-17:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = 0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.6e-17)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.6e-17)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = 0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.6e-17], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.60000000000000018e-17

   1. Initial program 71.9%

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

   3. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 44.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)\right)} \]

      2. expm1-log1p 71.9%

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

      3. *-lft-identity 71.9%

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

      4. *-commutative 71.9%

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

      5. times-frac 71.8%

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

      6. metadata-eval 71.8%

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

   5. Simplified 71.8%

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

      1. associate-*r/ 71.9%

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

   7. Applied egg-rr 71.9%

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

   8. Taylor expanded in b around -inf 50.2%

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

      1. associate-*r/ 50.2%

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

      2. *-commutative 50.2%

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

      3. *-rgt-identity 50.2%

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

      4. associate-*r/ 50.2%

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

      5. associate-*l* 50.2%

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

      6. associate-*r/ 50.2%

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

      7. metadata-eval 50.2%

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

   10. Simplified 50.2%

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

   if 4.60000000000000018e-17 < b

   1. Initial program 16.7%

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

   3. Applied egg-rr 10.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 15.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)\right)} \]

      2. expm1-log1p 16.7%

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

      3. *-lft-identity 16.7%

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

      4. *-commutative 16.7%

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

      5. times-frac 16.7%

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

      6. metadata-eval 16.7%

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

   5. Simplified 16.7%

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

      1. associate-*r/ 16.7%

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

   7. Applied egg-rr 16.7%

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

   8. Taylor expanded in b around -inf 2.0%

      \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{\left(-1.5 \cdot \frac{a \cdot c}{b} + 2 \cdot b\right)}}{a} \]

   9. Taylor expanded in a around inf 29.3%

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

       1. *-commutative 29.3%

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

   11. Simplified 29.3%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-17}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7: 67.5% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 74.4%

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

   3. Applied egg-rr 23.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 44.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)\right)} \]

      2. expm1-log1p 74.4%

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

      3. *-lft-identity 74.4%

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

      4. *-commutative 74.4%

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

      5. times-frac 74.3%

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

      6. metadata-eval 74.3%

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

   5. Simplified 74.3%

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

      1. associate-*r/ 74.4%

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

   7. Applied egg-rr 74.4%

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

   8. Taylor expanded in b around -inf 66.3%

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

      1. associate-*r/ 66.3%

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

      2. *-commutative 66.3%

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

      3. *-rgt-identity 66.3%

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

      4. associate-*r/ 66.3%

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

      5. associate-*l* 66.2%

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

      6. associate-*r/ 66.3%

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

      7. metadata-eval 66.3%

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

   10. Simplified 66.3%

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

   if -1.999999999999994e-310 < b

   1. Initial program 31.9%

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

   3. Applied egg-rr 31.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{0.3333333333333333}{a}\right)} \]
   4. Step-by-step derivation

      1. sub-neg 31.2%

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

      2. distribute-rgt-out-- 31.9%

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

   5. Simplified 31.9%

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

   6. Taylor expanded in a around 0 51.9%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(-1.5 \cdot \frac{a \cdot c}{b}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 54.4%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right) \]

      2. associate-*r/ 54.4%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\frac{-1.5 \cdot a}{\frac{b}{c}}} \]

   8. Simplified 54.4%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\frac{-1.5 \cdot a}{\frac{b}{c}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 50.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a} \cdot \frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)} \]

      2. expm1-udef 27.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.3333333333333333}{a} \cdot \frac{-1.5 \cdot a}{\frac{b}{c}}\right)} - 1} \]

      3. *-commutative 27.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1.5 \cdot a}{\frac{b}{c}} \cdot \frac{0.3333333333333333}{a}}\right)} - 1 \]

      4. associate-/l* 27.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1.5}{\frac{\frac{b}{c}}{a}}} \cdot \frac{0.3333333333333333}{a}\right)} - 1 \]

      5. frac-times 27.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1.5 \cdot 0.3333333333333333}{\frac{\frac{b}{c}}{a} \cdot a}}\right)} - 1 \]

      6. metadata-eval 27.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{-0.5}}{\frac{\frac{b}{c}}{a} \cdot a}\right)} - 1 \]

   10. Applied egg-rr 27.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5}{\frac{\frac{b}{c}}{a} \cdot a}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 49.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.5}{\frac{\frac{b}{c}}{a} \cdot a}\right)\right)} \]

       2. expm1-log1p 53.8%

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

       3. *-commutative 53.8%

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

       4. associate-*r/ 61.3%

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

       5. associate-*l/ 68.0%

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

       6. *-inverses 68.0%

          \[\leadsto \frac{-0.5}{\color{blue}{1} \cdot \frac{b}{c}} \]

       7. associate-*r/ 68.0%

          \[\leadsto \frac{-0.5}{\color{blue}{\frac{1 \cdot b}{c}}} \]

       8. *-lft-identity 68.0%

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

   12. Simplified 68.0%

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

4. Final simplification 67.2%

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

Alternative 8: 67.6% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ b (* a -1.5)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = b / (a * -1.5)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = b / (a * -1.5);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 74.4%

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

   3. Applied egg-rr 23.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 44.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)\right)} \]

      2. expm1-log1p 74.4%

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

      3. *-lft-identity 74.4%

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

      4. *-commutative 74.4%

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

      5. times-frac 74.3%

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

      6. metadata-eval 74.3%

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

   5. Simplified 74.3%

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

      1. associate-*r/ 74.4%

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

   7. Applied egg-rr 74.4%

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

   8. Taylor expanded in b around -inf 66.3%

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

      1. associate-*r/ 66.3%

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

      2. *-commutative 66.3%

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

      3. *-rgt-identity 66.3%

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

      4. associate-*r/ 66.3%

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

      5. associate-*l* 66.2%

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

      6. associate-*r/ 66.3%

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

      7. metadata-eval 66.3%

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

   10. Simplified 66.3%

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

       1. clear-num 66.2%

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

       2. un-div-inv 66.3%

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

       3. div-inv 66.4%

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

       4. metadata-eval 66.4%

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

   12. Applied egg-rr 66.4%

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

   if -1.999999999999994e-310 < b

   1. Initial program 31.9%

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

   3. Applied egg-rr 31.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{0.3333333333333333}{a}\right)} \]
   4. Step-by-step derivation

      1. sub-neg 31.2%

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

      2. distribute-rgt-out-- 31.9%

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

   5. Simplified 31.9%

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

   6. Taylor expanded in a around 0 51.9%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(-1.5 \cdot \frac{a \cdot c}{b}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 54.4%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(-1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right) \]

      2. associate-*r/ 54.4%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\frac{-1.5 \cdot a}{\frac{b}{c}}} \]

   8. Simplified 54.4%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\frac{-1.5 \cdot a}{\frac{b}{c}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 50.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a} \cdot \frac{-1.5 \cdot a}{\frac{b}{c}}\right)\right)} \]

      2. expm1-udef 27.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.3333333333333333}{a} \cdot \frac{-1.5 \cdot a}{\frac{b}{c}}\right)} - 1} \]

      3. *-commutative 27.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1.5 \cdot a}{\frac{b}{c}} \cdot \frac{0.3333333333333333}{a}}\right)} - 1 \]

      4. associate-/l* 27.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1.5}{\frac{\frac{b}{c}}{a}}} \cdot \frac{0.3333333333333333}{a}\right)} - 1 \]

      5. frac-times 27.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-1.5 \cdot 0.3333333333333333}{\frac{\frac{b}{c}}{a} \cdot a}}\right)} - 1 \]

      6. metadata-eval 27.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{-0.5}}{\frac{\frac{b}{c}}{a} \cdot a}\right)} - 1 \]

   10. Applied egg-rr 27.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5}{\frac{\frac{b}{c}}{a} \cdot a}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 49.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.5}{\frac{\frac{b}{c}}{a} \cdot a}\right)\right)} \]

       2. expm1-log1p 53.8%

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

       3. *-commutative 53.8%

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

       4. associate-*r/ 61.3%

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

       5. associate-*l/ 68.0%

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

       6. *-inverses 68.0%

          \[\leadsto \frac{-0.5}{\color{blue}{1} \cdot \frac{b}{c}} \]

       7. associate-*r/ 68.0%

          \[\leadsto \frac{-0.5}{\color{blue}{\frac{1 \cdot b}{c}}} \]

       8. *-lft-identity 68.0%

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

   12. Simplified 68.0%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]

Alternative 9: 68.0% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = b / (a * -1.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 74.4%

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

   3. Applied egg-rr 23.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 44.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)\right)} \]

      2. expm1-log1p 74.4%

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

      3. *-lft-identity 74.4%

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

      4. *-commutative 74.4%

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

      5. times-frac 74.3%

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

      6. metadata-eval 74.3%

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

   5. Simplified 74.3%

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

      1. associate-*r/ 74.4%

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

   7. Applied egg-rr 74.4%

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

   8. Taylor expanded in b around -inf 66.3%

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

      1. associate-*r/ 66.3%

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

      2. *-commutative 66.3%

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

      3. *-rgt-identity 66.3%

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

      4. associate-*r/ 66.3%

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

      5. associate-*l* 66.2%

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

      6. associate-*r/ 66.3%

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

      7. metadata-eval 66.3%

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

   10. Simplified 66.3%

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

       1. clear-num 66.2%

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

       2. un-div-inv 66.3%

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

       3. div-inv 66.4%

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

       4. metadata-eval 66.4%

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

   12. Applied egg-rr 66.4%

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

   if -1.999999999999994e-310 < b

   1. Initial program 31.9%

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

   2. Taylor expanded in b around inf 70.1%

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

      1. *-commutative 70.1%

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

      2. associate-*l/ 70.1%

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

   4. Simplified 70.1%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 10: 35.1% 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 52.5%

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

3. Applied egg-rr 17.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)} - 1} \]
4. Step-by-step derivation

   1. expm1-def 34.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}{a \cdot -3}\right)\right)} \]

   2. expm1-log1p 52.5%

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

   3. *-lft-identity 52.5%

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

   4. *-commutative 52.5%

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

   5. times-frac 52.4%

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

   6. metadata-eval 52.4%

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

5. Simplified 52.4%

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

   1. associate-*r/ 52.5%

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

7. Applied egg-rr 52.5%

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

8. Taylor expanded in b around -inf 33.3%

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

   1. associate-*r/ 33.3%

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

   2. *-commutative 33.3%

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

   3. *-rgt-identity 33.3%

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

   4. associate-*r/ 33.3%

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

   5. associate-*l* 33.3%

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

   6. associate-*r/ 33.3%

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

   7. metadata-eval 33.3%

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

10. Simplified 33.3%

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

11. Final simplification 33.3%

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

Reproduce

herbie shell --seed 2023313 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))