Cubic critical

Percentage Accurate: 53.0% → 84.7%

Time: 11.0s

Alternatives: 10

Speedup: 2.2×

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: 53.0% 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: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+169}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.1e+169)
   (/ b (* a -1.5))
   (if (<= b 1.25e-29)
     (/ (/ (- b (sqrt (fma a (* c -3.0) (* b b)))) a) -3.0)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.1e+169) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.25e-29) {
		tmp = ((b - sqrt(fma(a, (c * -3.0), (b * b)))) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.1e+169)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.25e-29)
		tmp = Float64(Float64(Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))) / a) / -3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.1e+169], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-29], N[(N[(N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.1e169

   1. Initial program 51.2%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.9%

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

   if -3.1e169 < b < 1.24999999999999996e-29

   1. Initial program 83.8%

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

   4. Applied rewrites 83.9%

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

   if 1.24999999999999996e-29 < b

   1. Initial program 13.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 90.5

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

   5. Applied rewrites 90.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+169}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+47)
   (/ b (* a -1.5))
   (if (<= b 1.25e-29)
     (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+47) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.25e-29) {
		tmp = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.5d+47)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 1.25d-29) then
        tmp = (sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+47) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.25e-29) {
		tmp = (Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e+47:
		tmp = b / (a * -1.5)
	elif b <= 1.25e-29:
		tmp = (math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e+47)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.25e-29)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.5e+47)
		tmp = b / (a * -1.5);
	elseif (b <= 1.25e-29)
		tmp = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+47], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-29], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.49999999999999979e47

   1. Initial program 71.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.6%

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

   if -4.49999999999999979e47 < b < 1.24999999999999996e-29

   1. Initial program 79.7%

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

   if 1.24999999999999996e-29 < b

   1. Initial program 13.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 90.5

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

   5. Applied rewrites 90.5%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+47)
   (/ b (* a -1.5))
   (if (<= b 1.25e-29)
     (/ (- (sqrt (fma (* a -3.0) c (* b b))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+47) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.25e-29) {
		tmp = (sqrt(fma((a * -3.0), c, (b * b))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e+47)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.25e-29)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+47], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-29], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.49999999999999979e47

   1. Initial program 71.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.6%

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

   if -4.49999999999999979e47 < b < 1.24999999999999996e-29

   1. Initial program 79.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 79.7

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

   4. Applied rewrites 79.7%

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

   if 1.24999999999999996e-29 < b

   1. Initial program 13.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 90.5

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

   5. Applied rewrites 90.5%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e+47)
   (/ b (* a -1.5))
   (if (<= b 1.25e-29)
     (/ (* (- (sqrt (fma b b (* a (* c -3.0)))) b) 0.3333333333333333) a)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e+47) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.25e-29) {
		tmp = ((sqrt(fma(b, b, (a * (c * -3.0)))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.4e+47)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.25e-29)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.4e+47], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-29], N[(N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.3999999999999999e47

   1. Initial program 71.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.6%

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

   if -4.3999999999999999e47 < b < 1.24999999999999996e-29

   1. Initial program 79.7%

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

   4. Applied rewrites 79.8%

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

   6. Applied rewrites 79.6%

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

   if 1.24999999999999996e-29 < b

   1. Initial program 13.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 90.5

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

   5. Applied rewrites 90.5%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e+47)
   (/ b (* a -1.5))
   (if (<= b 1.25e-29)
     (* (- b (sqrt (fma a (* c -3.0) (* b b)))) (/ -0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e+47) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.25e-29) {
		tmp = (b - sqrt(fma(a, (c * -3.0), (b * b)))) * (-0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.4e+47)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.25e-29)
		tmp = Float64(Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))) * Float64(-0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.4e+47], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-29], N[(N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.3999999999999999e47

   1. Initial program 71.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.6%

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

   if -4.3999999999999999e47 < b < 1.24999999999999996e-29

   1. Initial program 79.7%

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

   4. Applied rewrites 79.6%

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

   if 1.24999999999999996e-29 < b

   1. Initial program 13.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 90.5

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

   5. Applied rewrites 90.5%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-115}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-115)
   (/ b (* a -1.5))
   (if (<= b 8e-65)
     (/ (- (sqrt (* a (* c -3.0))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-115) {
		tmp = b / (a * -1.5);
	} else if (b <= 8e-65) {
		tmp = (sqrt((a * (c * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6d-115)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 8d-65) then
        tmp = (sqrt((a * (c * (-3.0d0)))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-115) {
		tmp = b / (a * -1.5);
	} else if (b <= 8e-65) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e-115:
		tmp = b / (a * -1.5)
	elif b <= 8e-65:
		tmp = (math.sqrt((a * (c * -3.0))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e-115)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 8e-65)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6e-115)
		tmp = b / (a * -1.5);
	elseif (b <= 8e-65)
		tmp = (sqrt((a * (c * -3.0))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e-115], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-65], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.0000000000000003e-115

   1. Initial program 78.2%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 88.7

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

   5. Applied rewrites 88.7%

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

   7. Applied rewrites 88.8%

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

   if -6.0000000000000003e-115 < b < 7.99999999999999939e-65

   1. Initial program 76.6%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if 7.99999999999999939e-65 < b

   1. Initial program 17.8%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 85.4

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

   5. Applied rewrites 85.4%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-115}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-311) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-311) {
		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-311)) 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-311) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-311:
		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-311)
		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-311)
		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-311], 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 < -1.9999999999999e-311

   1. Initial program 79.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 72.6

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

   5. Applied rewrites 72.6%

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

   7. Applied rewrites 72.7%

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

   if -1.9999999999999e-311 < b

   1. Initial program 32.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

Alternative 8: 43.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8600:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8600.0) (/ b (* a -1.5)) (/ (* c 0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8600.0) {
		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 <= 8600.0d0) 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 <= 8600.0) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * 0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8600.0:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * 0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8600.0)
		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 <= 8600.0)
		tmp = b / (a * -1.5);
	else
		tmp = (c * 0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8600.0], 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 < 8600

   1. Initial program 74.3%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 55.3%

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

   if 8600 < b

   1. Initial program 12.8%

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

   4. Applied rewrites 3.7%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot c}{b}} \]

      3. lower-*.f64 30.3

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

   7. Applied rewrites 30.3%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8600:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 10.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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 * 0.5d0) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.0%

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

4. Applied rewrites 34.8%

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

5. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot c}{b}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot c}{b}} \]

   3. lower-*.f64 10.3

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

7. Applied rewrites 10.3%

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

8. Final simplification 10.3%

   \[\leadsto \frac{c \cdot 0.5}{b} \]
9. Add Preprocessing

Alternative 10: 2.5% accurate, 2.9× 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 58.0%

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

4. Applied rewrites 34.8%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 2.2

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

7. Applied rewrites 2.2%

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

9. Applied rewrites 2.2%

   \[\leadsto b \cdot \color{blue}{\frac{0.6666666666666666}{a}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))