Cubic critical

Percentage Accurate: 52.3% → 85.1%

Time: 8.0s

Alternatives: 8

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.3% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-97}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+154)
   (fma 0.5 (/ c b) (* (/ b a) -0.6666666666666666))
   (if (<= b 2.8e-97)
     (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+154) {
		tmp = fma(0.5, (c / b), ((b / a) * -0.6666666666666666));
	} else if (b <= 2.8e-97) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+154)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b / a) * -0.6666666666666666));
	elseif (b <= 2.8e-97)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+154], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.8e-97], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.50000000000000013e154

   1. Initial program 28.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{-2}{3} \cdot \frac{b}{a} + \color{blue}{\frac{1}{2} \cdot \frac{c}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.8%

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

   if -1.50000000000000013e154 < b < 2.8000000000000002e-97

   1. Initial program 88.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 88.2

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

   4. Applied rewrites 88.2%

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

   if 2.8000000000000002e-97 < b

   1. Initial program 21.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 21.9%

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

   6. Applied rewrites 21.9%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 89.8

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

   9. Applied rewrites 89.8%

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

   11. Applied rewrites 89.8%

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

Alternative 2: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-97}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+154)
   (fma 0.5 (/ c b) (* (/ b a) -0.6666666666666666))
   (if (<= b 2.8e-97)
     (* 0.3333333333333333 (/ (- (sqrt (fma (* c -3.0) a (* b b))) b) a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+154) {
		tmp = fma(0.5, (c / b), ((b / a) * -0.6666666666666666));
	} else if (b <= 2.8e-97) {
		tmp = 0.3333333333333333 * ((sqrt(fma((c * -3.0), a, (b * b))) - b) / a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+154)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b / a) * -0.6666666666666666));
	elseif (b <= 2.8e-97)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) / a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+154], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.8e-97], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.50000000000000013e154

   1. Initial program 28.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{-2}{3} \cdot \frac{b}{a} + \color{blue}{\frac{1}{2} \cdot \frac{c}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.8%

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

   if -1.50000000000000013e154 < b < 2.8000000000000002e-97

   1. Initial program 88.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 88.2

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

   4. Applied rewrites 88.2%

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

   6. Applied rewrites 88.1%

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

   if 2.8000000000000002e-97 < b

   1. Initial program 21.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 21.9%

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

   6. Applied rewrites 21.9%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 89.8

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

   9. Applied rewrites 89.8%

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

   11. Applied rewrites 89.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-97}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.25 \cdot 10^{-83}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-97}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(c \cdot a\right) \cdot -3}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.25e-83)
   (fma 0.5 (/ c b) (* (/ b a) -0.6666666666666666))
   (if (<= b 2.8e-97)
     (/ (+ (- b) (sqrt (* (* c a) -3.0))) (* 3.0 a))
     (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.25e-83) {
		tmp = fma(0.5, (c / b), ((b / a) * -0.6666666666666666));
	} else if (b <= 2.8e-97) {
		tmp = (-b + sqrt(((c * a) * -3.0))) / (3.0 * a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.25e-83)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b / a) * -0.6666666666666666));
	elseif (b <= 2.8e-97)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(c * a) * -3.0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.25e-83], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.8e-97], N[(N[((-b) + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.25e-83

   1. Initial program 65.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{-2}{3} \cdot \frac{b}{a} + \color{blue}{\frac{1}{2} \cdot \frac{c}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.6%

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

   if -3.25e-83 < b < 2.8000000000000002e-97

   1. Initial program 84.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 83.4

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

   5. Applied rewrites 83.4%

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

   if 2.8000000000000002e-97 < b

   1. Initial program 21.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 21.9%

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

   6. Applied rewrites 21.9%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 89.8

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

   9. Applied rewrites 89.8%

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

   11. Applied rewrites 89.8%

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

Alternative 4: 67.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 71.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 64.0

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{-2}{3} \cdot \frac{b}{a} + \color{blue}{\frac{1}{2} \cdot \frac{c}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.1%

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

   if -1.999999999999994e-310 < b

   1. Initial program 33.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 33.5%

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

   6. Applied rewrites 33.4%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 76.0

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

   9. Applied rewrites 76.0%

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

   11. Applied rewrites 76.0%

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

Alternative 5: 67.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.85 \cdot 10^{-282}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.85e-282) (* -0.6666666666666666 (/ b a)) (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.85e-282) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.85d-282) then
        tmp = (-0.6666666666666666d0) * (b / 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 <= 2.85e-282) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.85e-282:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.85e-282)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.85e-282)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.85e-282], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8500000000000001e-282

   1. Initial program 72.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-*.f64 N/A

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

      2. lower-/.f64 62.5

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

   5. Applied rewrites 62.5%

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

   if 2.8500000000000001e-282 < b

   1. Initial program 32.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 32.4%

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

   6. Applied rewrites 32.4%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 77.1

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

   9. Applied rewrites 77.1%

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

   11. Applied rewrites 77.2%

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

Alternative 6: 67.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.85 \cdot 10^{-282}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.85e-282) (* -0.6666666666666666 (/ b a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.85e-282) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.85d-282) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.85e-282) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.85e-282:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.85e-282)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.85e-282)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.85e-282], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8500000000000001e-282

   1. Initial program 72.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-*.f64 N/A

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

      2. lower-/.f64 62.5

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

   5. Applied rewrites 62.5%

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

   if 2.8500000000000001e-282 < b

   1. Initial program 32.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 77.1

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

   5. Applied rewrites 77.1%

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

Alternative 7: 42.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-284}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.8e-284) (* -0.6666666666666666 (/ b a)) (/ 0.0 a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.8e-284) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.8d-284) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.8e-284) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.8e-284:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.8e-284)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.8e-284)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.8e-284], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7999999999999999e-284

   1. Initial program 72.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lower-*.f64 N/A

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

      2. lower-/.f64 62.9

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

   5. Applied rewrites 62.9%

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

   if 3.7999999999999999e-284 < b

   1. Initial program 32.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 32.9%

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

   6. Applied rewrites 25.9%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)}, 3, 3 \cdot \left(-b\right)\right)}{9}}}{a} \]

   7. Taylor expanded in a around 0

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

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \color{blue}{\left(-3 \cdot b - \left(\mathsf{neg}\left(3\right)\right) \cdot b\right)}}{a} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \left(-3 \cdot b - \color{blue}{-3} \cdot b\right)}{a} \]

      3. +-inverses N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \color{blue}{0}}{a} \]

      4. metadata-eval 21.7

         \[\leadsto \frac{\color{blue}{0}}{a} \]

   9. Applied rewrites 21.7%

      \[\leadsto \frac{\color{blue}{0}}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 10.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 53.0%

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

6. Applied rewrites 49.5%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)}, 3, 3 \cdot \left(-b\right)\right)}{9}}}{a} \]

7. Taylor expanded in a around 0

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

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\frac{1}{9} \cdot \color{blue}{\left(-3 \cdot b - \left(\mathsf{neg}\left(3\right)\right) \cdot b\right)}}{a} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{9} \cdot \left(-3 \cdot b - \color{blue}{-3} \cdot b\right)}{a} \]

   3. +-inverses N/A

      \[\leadsto \frac{\frac{1}{9} \cdot \color{blue}{0}}{a} \]

   4. metadata-eval 12.1

      \[\leadsto \frac{\color{blue}{0}}{a} \]

9. Applied rewrites 12.1%

   \[\leadsto \frac{\color{blue}{0}}{a} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024357 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))