Cubic critical

Percentage Accurate: 51.3% → 84.6%

Time: 4.6s

Alternatives: 18

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: 51.3% 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: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-139}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e+135)
   (fma 0.5 (/ c b) (* (/ b a) -0.6666666666666666))
   (if (<= b 8e-139)
     (/ (/ (+ (sqrt (fma (* -3.0 a) c (* b b))) (- b)) 3.0) a)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e+135) {
		tmp = fma(0.5, (c / b), ((b / a) * -0.6666666666666666));
	} else if (b <= 8e-139) {
		tmp = ((sqrt(fma((-3.0 * a), c, (b * b))) + -b) / 3.0) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e+135)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b / a) * -0.6666666666666666));
	elseif (b <= 8e-139)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) + Float64(-b)) / 3.0) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e+135], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-139], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.19999999999999999e135

   1. Initial program 51.3%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 34.1

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c}{b} + \frac{-2}{3} \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 34.9

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

   7. Applied rewrites 34.9%

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

   if -1.19999999999999999e135 < b < 8.00000000000000024e-139

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   if 8.00000000000000024e-139 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 2: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-139}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e+146)
   (fma 0.5 (/ c b) (* (/ b a) -0.6666666666666666))
   (if (<= b 8e-139)
     (/ (* (- (sqrt (fma (* -3.0 a) c (* b b))) b) 0.3333333333333333) a)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e+146) {
		tmp = fma(0.5, (c / b), ((b / a) * -0.6666666666666666));
	} else if (b <= 8e-139) {
		tmp = ((sqrt(fma((-3.0 * a), c, (b * b))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e+146)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b / a) * -0.6666666666666666));
	elseif (b <= 8e-139)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e+146], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-139], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7999999999999999e146

   1. Initial program 51.3%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 34.1

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c}{b} + \frac{-2}{3} \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 34.9

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

   7. Applied rewrites 34.9%

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

   if -1.7999999999999999e146 < b < 8.00000000000000024e-139

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   5. Applied rewrites 51.2%

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

      1. lift-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 51.2

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

   7. Applied rewrites 51.2%

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

   if 8.00000000000000024e-139 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 3: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-139}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e+146)
   (fma 0.5 (/ c b) (* (/ b a) -0.6666666666666666))
   (if (<= b 8e-139)
     (/ (* (- (sqrt (fma b b (* (* a -3.0) c))) b) 0.3333333333333333) a)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e+146) {
		tmp = fma(0.5, (c / b), ((b / a) * -0.6666666666666666));
	} else if (b <= 8e-139) {
		tmp = ((sqrt(fma(b, b, ((a * -3.0) * c))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e+146)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b / a) * -0.6666666666666666));
	elseif (b <= 8e-139)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -3.0) * c))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e+146], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-139], N[(N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7999999999999999e146

   1. Initial program 51.3%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 34.1

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c}{b} + \frac{-2}{3} \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 34.9

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

   7. Applied rewrites 34.9%

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

   if -1.7999999999999999e146 < b < 8.00000000000000024e-139

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   5. Applied rewrites 51.2%

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

   if 8.00000000000000024e-139 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 4: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-139}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e+146)
   (fma 0.5 (/ c b) (* (/ b a) -0.6666666666666666))
   (if (<= b 8e-139)
     (* (- (sqrt (fma (* -3.0 a) c (* b b))) b) (/ 0.3333333333333333 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e+146) {
		tmp = fma(0.5, (c / b), ((b / a) * -0.6666666666666666));
	} else if (b <= 8e-139) {
		tmp = (sqrt(fma((-3.0 * a), c, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e+146)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b / a) * -0.6666666666666666));
	elseif (b <= 8e-139)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e+146], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-139], N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.6e146

   1. Initial program 51.3%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 34.1

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c}{b} + \frac{-2}{3} \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 34.9

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

   7. Applied rewrites 34.9%

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

   if -1.6e146 < b < 8.00000000000000024e-139

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   5. Applied rewrites 51.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. mult-flip-rev N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 51.2%

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

   if 8.00000000000000024e-139 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 5: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -0.6666666666666666, \frac{c}{b} \cdot 0.5\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-139}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(a \cdot -3\right) \cdot c} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-75)
   (fma (/ b a) -0.6666666666666666 (* (/ c b) 0.5))
   (if (<= b 8e-139)
     (/ (/ (+ (sqrt (* (* a -3.0) c)) (- b)) 3.0) a)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-75) {
		tmp = fma((b / a), -0.6666666666666666, ((c / b) * 0.5));
	} else if (b <= 8e-139) {
		tmp = ((sqrt(((a * -3.0) * c)) + -b) / 3.0) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-75)
		tmp = fma(Float64(b / a), -0.6666666666666666, Float64(Float64(c / b) * 0.5));
	elseif (b <= 8e-139)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(a * -3.0) * c)) + Float64(-b)) / 3.0) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-75], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666 + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-139], N[(N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.79999999999999994e-75

   1. Initial program 51.3%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 34.1

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 11.0

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

   7. Applied rewrites 11.0%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{b}{a} \cdot \frac{-2}{3} + \frac{1}{2} \cdot \frac{\color{blue}{c}}{b} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 34.9

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

   10. Applied rewrites 34.9%

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

   if -3.79999999999999994e-75 < b < 8.00000000000000024e-139

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   4. Taylor expanded in a around inf

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 32.6

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

   6. Applied rewrites 32.6%

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

   if 8.00000000000000024e-139 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 6: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -0.6666666666666666, \frac{c}{b} \cdot 0.5\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-139}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-75)
   (fma (/ b a) -0.6666666666666666 (* (/ c b) 0.5))
   (if (<= b 8e-139)
     (/ (+ (- b) (sqrt (* -3.0 (* c a)))) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-75) {
		tmp = fma((b / a), -0.6666666666666666, ((c / b) * 0.5));
	} else if (b <= 8e-139) {
		tmp = (-b + sqrt((-3.0 * (c * a)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-75)
		tmp = fma(Float64(b / a), -0.6666666666666666, Float64(Float64(c / b) * 0.5));
	elseif (b <= 8e-139)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(-3.0 * Float64(c * a)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-75], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666 + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-139], N[(N[((-b) + N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.79999999999999994e-75

   1. Initial program 51.3%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 34.1

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 11.0

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

   7. Applied rewrites 11.0%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{b}{a} \cdot \frac{-2}{3} + \frac{1}{2} \cdot \frac{\color{blue}{c}}{b} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 34.9

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

   10. Applied rewrites 34.9%

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

   if -3.79999999999999994e-75 < b < 8.00000000000000024e-139

   1. Initial program 51.3%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 32.6

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

   4. Applied rewrites 32.6%

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

   if 8.00000000000000024e-139 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 7: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -0.6666666666666666, \frac{c}{b} \cdot 0.5\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-139}:\\ \;\;\;\;\frac{\left(\sqrt{\left(c \cdot a\right) \cdot -3} + \left(-b\right)\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-75)
   (fma (/ b a) -0.6666666666666666 (* (/ c b) 0.5))
   (if (<= b 8e-139)
     (/ (* (+ (sqrt (* (* c a) -3.0)) (- b)) 0.3333333333333333) a)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-75) {
		tmp = fma((b / a), -0.6666666666666666, ((c / b) * 0.5));
	} else if (b <= 8e-139) {
		tmp = ((sqrt(((c * a) * -3.0)) + -b) * 0.3333333333333333) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-75)
		tmp = fma(Float64(b / a), -0.6666666666666666, Float64(Float64(c / b) * 0.5));
	elseif (b <= 8e-139)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(c * a) * -3.0)) + Float64(-b)) * 0.3333333333333333) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-75], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666 + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-139], N[(N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.79999999999999994e-75

   1. Initial program 51.3%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 34.1

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 11.0

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

   7. Applied rewrites 11.0%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{b}{a} \cdot \frac{-2}{3} + \frac{1}{2} \cdot \frac{\color{blue}{c}}{b} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 34.9

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

   10. Applied rewrites 34.9%

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

   if -3.79999999999999994e-75 < b < 8.00000000000000024e-139

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. *-commutative N/A

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

      6. lower-sqrt.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 32.6

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

   8. Applied rewrites 32.6%

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

   if 8.00000000000000024e-139 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 8: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -0.6666666666666666, \frac{c}{b} \cdot 0.5\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-139}:\\ \;\;\;\;\frac{\left(\sqrt{\left(a \cdot -3\right) \cdot c} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-75)
   (fma (/ b a) -0.6666666666666666 (* (/ c b) 0.5))
   (if (<= b 8e-139)
     (/ (* (- (sqrt (* (* a -3.0) c)) b) 0.3333333333333333) a)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-75) {
		tmp = fma((b / a), -0.6666666666666666, ((c / b) * 0.5));
	} else if (b <= 8e-139) {
		tmp = ((sqrt(((a * -3.0) * c)) - b) * 0.3333333333333333) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-75)
		tmp = fma(Float64(b / a), -0.6666666666666666, Float64(Float64(c / b) * 0.5));
	elseif (b <= 8e-139)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(a * -3.0) * c)) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-75], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666 + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-139], N[(N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.79999999999999994e-75

   1. Initial program 51.3%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 34.1

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 11.0

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

   7. Applied rewrites 11.0%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{b}{a} \cdot \frac{-2}{3} + \frac{1}{2} \cdot \frac{\color{blue}{c}}{b} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 34.9

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

   10. Applied rewrites 34.9%

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

   if -3.79999999999999994e-75 < b < 8.00000000000000024e-139

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. *-commutative N/A

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

      6. lower-sqrt.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 32.6

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

   8. Applied rewrites 32.6%

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

   10. Applied rewrites 32.6%

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

   if 8.00000000000000024e-139 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 9: 79.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -0.6666666666666666, \frac{c}{b} \cdot 0.5\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-139}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(a \cdot -3\right) \cdot c}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e-124)
   (fma (/ b a) -0.6666666666666666 (* (/ c b) 0.5))
   (if (<= b 8e-139) (/ (/ (sqrt (* (* a -3.0) c)) 3.0) a) (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-124) {
		tmp = fma((b / a), -0.6666666666666666, ((c / b) * 0.5));
	} else if (b <= 8e-139) {
		tmp = (sqrt(((a * -3.0) * c)) / 3.0) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e-124)
		tmp = fma(Float64(b / a), -0.6666666666666666, Float64(Float64(c / b) * 0.5));
	elseif (b <= 8e-139)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * -3.0) * c)) / 3.0) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e-124], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666 + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-139], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.2000000000000002e-124

   1. Initial program 51.3%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 34.1

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 11.0

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

   7. Applied rewrites 11.0%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{b}{a} \cdot \frac{-2}{3} + \frac{1}{2} \cdot \frac{\color{blue}{c}}{b} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 34.9

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

   10. Applied rewrites 34.9%

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

   if -4.2000000000000002e-124 < b < 8.00000000000000024e-139

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   4. Taylor expanded in a around inf

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{a \cdot c} \cdot \sqrt{-3}}}{3}}{a} \]
   5. Step-by-step derivation

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 28.9

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

   6. Applied rewrites 28.9%

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

   if 8.00000000000000024e-139 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 10: 79.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-139}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(a \cdot -3\right) \cdot c}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e-124)
   (fma 0.5 (/ c b) (* (/ b a) -0.6666666666666666))
   (if (<= b 8e-139) (/ (/ (sqrt (* (* a -3.0) c)) 3.0) a) (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-124) {
		tmp = fma(0.5, (c / b), ((b / a) * -0.6666666666666666));
	} else if (b <= 8e-139) {
		tmp = (sqrt(((a * -3.0) * c)) / 3.0) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e-124)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b / a) * -0.6666666666666666));
	elseif (b <= 8e-139)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * -3.0) * c)) / 3.0) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e-124], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-139], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.2000000000000002e-124

   1. Initial program 51.3%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 34.1

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c}{b} + \frac{-2}{3} \cdot \color{blue}{\frac{b}{a}} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 34.9

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

   7. Applied rewrites 34.9%

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

   if -4.2000000000000002e-124 < b < 8.00000000000000024e-139

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   4. Taylor expanded in a around inf

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{a \cdot c} \cdot \sqrt{-3}}}{3}}{a} \]
   5. Step-by-step derivation

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 28.9

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

   6. Applied rewrites 28.9%

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

   if 8.00000000000000024e-139 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 11: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{-2 \cdot b}{3}}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-139}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(a \cdot -3\right) \cdot c}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e-124)
   (/ (/ (* -2.0 b) 3.0) a)
   (if (<= b 8e-139) (/ (/ (sqrt (* (* a -3.0) c)) 3.0) a) (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-124) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else if (b <= 8e-139) {
		tmp = (sqrt(((a * -3.0) * c)) / 3.0) / 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 <= (-4.2d-124)) then
        tmp = (((-2.0d0) * b) / 3.0d0) / a
    else if (b <= 8d-139) then
        tmp = (sqrt(((a * (-3.0d0)) * c)) / 3.0d0) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-124) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else if (b <= 8e-139) {
		tmp = (Math.sqrt(((a * -3.0) * c)) / 3.0) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.2e-124:
		tmp = ((-2.0 * b) / 3.0) / a
	elif b <= 8e-139:
		tmp = (math.sqrt(((a * -3.0) * c)) / 3.0) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e-124)
		tmp = Float64(Float64(Float64(-2.0 * b) / 3.0) / a);
	elseif (b <= 8e-139)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * -3.0) * c)) / 3.0) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.2e-124)
		tmp = ((-2.0 * b) / 3.0) / a;
	elseif (b <= 8e-139)
		tmp = (sqrt(((a * -3.0) * c)) / 3.0) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e-124], N[(N[(N[(-2.0 * b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 8e-139], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.2000000000000002e-124

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 35.0

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

   6. Applied rewrites 35.0%

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

   if -4.2000000000000002e-124 < b < 8.00000000000000024e-139

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   4. Taylor expanded in a around inf

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{a \cdot c} \cdot \sqrt{-3}}}{3}}{a} \]
   5. Step-by-step derivation

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 28.9

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

   6. Applied rewrites 28.9%

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

   if 8.00000000000000024e-139 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 12: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{-2 \cdot b}{3}}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e-124)
   (/ (/ (* -2.0 b) 3.0) a)
   (if (<= b 8e-139) (/ (sqrt (* -3.0 (* c a))) (* 3.0 a)) (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-124) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else if (b <= 8e-139) {
		tmp = sqrt((-3.0 * (c * a))) / (3.0 * 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 <= (-4.2d-124)) then
        tmp = (((-2.0d0) * b) / 3.0d0) / a
    else if (b <= 8d-139) then
        tmp = sqrt(((-3.0d0) * (c * a))) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-124) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else if (b <= 8e-139) {
		tmp = Math.sqrt((-3.0 * (c * a))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.2e-124:
		tmp = ((-2.0 * b) / 3.0) / a
	elif b <= 8e-139:
		tmp = math.sqrt((-3.0 * (c * a))) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e-124)
		tmp = Float64(Float64(Float64(-2.0 * b) / 3.0) / a);
	elseif (b <= 8e-139)
		tmp = Float64(sqrt(Float64(-3.0 * Float64(c * a))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.2e-124)
		tmp = ((-2.0 * b) / 3.0) / a;
	elseif (b <= 8e-139)
		tmp = sqrt((-3.0 * (c * a))) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e-124], N[(N[(N[(-2.0 * b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 8e-139], N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.2000000000000002e-124

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 35.0

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

   6. Applied rewrites 35.0%

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

   if -4.2000000000000002e-124 < b < 8.00000000000000024e-139

   1. Initial program 51.3%

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

   2. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.9

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

   4. Applied rewrites 28.9%

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

   if 8.00000000000000024e-139 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 13: 71.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{-2 \cdot b}{3}}{a}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -3} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.65e-136)
   (/ (/ (* -2.0 b) 3.0) a)
   (if (<= b 2.6e-151)
     (* (sqrt (* (/ c a) -3.0)) 0.3333333333333333)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.65e-136) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else if (b <= 2.6e-151) {
		tmp = sqrt(((c / a) * -3.0)) * 0.3333333333333333;
	} 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 <= (-2.65d-136)) then
        tmp = (((-2.0d0) * b) / 3.0d0) / a
    else if (b <= 2.6d-151) then
        tmp = sqrt(((c / a) * (-3.0d0))) * 0.3333333333333333d0
    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 <= -2.65e-136) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else if (b <= 2.6e-151) {
		tmp = Math.sqrt(((c / a) * -3.0)) * 0.3333333333333333;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.65e-136:
		tmp = ((-2.0 * b) / 3.0) / a
	elif b <= 2.6e-151:
		tmp = math.sqrt(((c / a) * -3.0)) * 0.3333333333333333
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.65e-136)
		tmp = Float64(Float64(Float64(-2.0 * b) / 3.0) / a);
	elseif (b <= 2.6e-151)
		tmp = Float64(sqrt(Float64(Float64(c / a) * -3.0)) * 0.3333333333333333);
	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 <= -2.65e-136)
		tmp = ((-2.0 * b) / 3.0) / a;
	elseif (b <= 2.6e-151)
		tmp = sqrt(((c / a) * -3.0)) * 0.3333333333333333;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.65e-136], N[(N[(N[(-2.0 * b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.6e-151], N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.65000000000000009e-136

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 35.0

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

   6. Applied rewrites 35.0%

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

   if -2.65000000000000009e-136 < b < 2.6e-151

   1. Initial program 51.3%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-*.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

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

      7. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{1}{3} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{1}{3} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{1}{3} \]

      10. lower-/.f64 16.9

          \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot 0.3333333333333333 \]

   4. Applied rewrites 16.9%

      \[\leadsto \color{blue}{\sqrt{\frac{c}{a} \cdot -3} \cdot 0.3333333333333333} \]

   if 2.6e-151 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 14: 71.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{-2 \cdot b}{3}}{a}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{c \cdot \frac{-3}{a}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.65e-136)
   (/ (/ (* -2.0 b) 3.0) a)
   (if (<= b 2.6e-151)
     (* (sqrt (* c (/ -3.0 a))) 0.3333333333333333)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.65e-136) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else if (b <= 2.6e-151) {
		tmp = sqrt((c * (-3.0 / a))) * 0.3333333333333333;
	} 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 <= (-2.65d-136)) then
        tmp = (((-2.0d0) * b) / 3.0d0) / a
    else if (b <= 2.6d-151) then
        tmp = sqrt((c * ((-3.0d0) / a))) * 0.3333333333333333d0
    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 <= -2.65e-136) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else if (b <= 2.6e-151) {
		tmp = Math.sqrt((c * (-3.0 / a))) * 0.3333333333333333;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.65e-136:
		tmp = ((-2.0 * b) / 3.0) / a
	elif b <= 2.6e-151:
		tmp = math.sqrt((c * (-3.0 / a))) * 0.3333333333333333
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.65e-136)
		tmp = Float64(Float64(Float64(-2.0 * b) / 3.0) / a);
	elseif (b <= 2.6e-151)
		tmp = Float64(sqrt(Float64(c * Float64(-3.0 / a))) * 0.3333333333333333);
	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 <= -2.65e-136)
		tmp = ((-2.0 * b) / 3.0) / a;
	elseif (b <= 2.6e-151)
		tmp = sqrt((c * (-3.0 / a))) * 0.3333333333333333;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.65e-136], N[(N[(N[(-2.0 * b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.6e-151], N[(N[Sqrt[N[(c * N[(-3.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.65000000000000009e-136

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 35.0

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

   6. Applied rewrites 35.0%

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

   if -2.65000000000000009e-136 < b < 2.6e-151

   1. Initial program 51.3%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-*.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

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

      7. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{1}{3} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{1}{3} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{1}{3} \]

      10. lower-/.f64 16.9

          \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot 0.3333333333333333 \]

   4. Applied rewrites 16.9%

      \[\leadsto \color{blue}{\sqrt{\frac{c}{a} \cdot -3} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{1}{3} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{1}{3} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{-3 \cdot \frac{c}{a}} \cdot \frac{1}{3} \]

      4. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{-3 \cdot c}{a}} \cdot \frac{1}{3} \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{-3 \cdot c}{a}} \cdot \frac{1}{3} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{c \cdot -3}{a}} \cdot \frac{1}{3} \]

      7. lower-*.f64 16.9

         \[\leadsto \sqrt{\frac{c \cdot -3}{a}} \cdot 0.3333333333333333 \]

   6. Applied rewrites 16.9%

      \[\leadsto \sqrt{\frac{c \cdot -3}{a}} \cdot 0.3333333333333333 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{c \cdot -3}{a}} \cdot \frac{1}{3} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{c \cdot -3}{a}} \cdot \frac{1}{3} \]

      3. associate-/l* N/A

         \[\leadsto \sqrt{c \cdot \frac{-3}{a}} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{c \cdot \frac{-3}{a}} \cdot \frac{1}{3} \]

      5. lower-/.f64 16.9

         \[\leadsto \sqrt{c \cdot \frac{-3}{a}} \cdot 0.3333333333333333 \]

   8. Applied rewrites 16.9%

      \[\leadsto \sqrt{c \cdot \frac{-3}{a}} \cdot 0.3333333333333333 \]

   if 2.6e-151 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 15: 67.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{-2 \cdot b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ (/ (* -2.0 b) 3.0) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((-2.0 * b) / 3.0) / 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 <= (-5d-310)) then
        tmp = (((-2.0d0) * b) / 3.0d0) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = ((-2.0 * b) / 3.0) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(Float64(-2.0 * b) / 3.0) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = ((-2.0 * b) / 3.0) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(N[(-2.0 * b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.3%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 35.0

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

   6. Applied rewrites 35.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 16: 67.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		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 <= (-5d-310)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(-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 < -4.999999999999985e-310

   1. Initial program 51.3%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.9

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

   4. Applied rewrites 34.9%

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

   if -4.999999999999985e-310 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 35.3

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

   4. Applied rewrites 35.3%

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

Alternative 17: 43.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.3 \cdot 10^{-10}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.3e-10) (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.3e-10) {
		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 <= 7.3d-10) 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 <= 7.3e-10) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.3e-10:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = 0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.3e-10)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7.3e-10)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = 0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.3e-10], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.2999999999999996e-10

   1. Initial program 51.3%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.9

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

   4. Applied rewrites 34.9%

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

   if 7.2999999999999996e-10 < b

   1. Initial program 51.3%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 34.1

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 11.0

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

   7. Applied rewrites 11.0%

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

Alternative 18: 11.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

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 = 0.5d0 * (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.3%

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

2. Taylor expanded in b around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lift-neg.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. mult-flip-rev N/A

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

   11. lower-/.f64 34.1

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

4. Applied rewrites 34.1%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 11.0

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

7. Applied rewrites 11.0%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{c}{b}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025132 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))