Cubic critical

Percentage Accurate: 52.4% → 84.8%

Time: 4.1s

Alternatives: 14

Speedup: 2.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.4% 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: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e+80)
   (fma 0.5 (/ c b) (* (/ b a) -0.6666666666666666))
   (if (<= b 3.2e-48)
     (/ (+ (sqrt (fma (* -3.0 a) c (* b b))) (- b)) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e+80) {
		tmp = fma(0.5, (c / b), ((b / a) * -0.6666666666666666));
	} else if (b <= 3.2e-48) {
		tmp = (sqrt(fma((-3.0 * a), c, (b * b))) + -b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e+80)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b / a) * -0.6666666666666666));
	elseif (b <= 3.2e-48)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) + Float64(-b)) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e+80], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.2e-48], N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.79999999999999997e80

   1. Initial program 57.8%

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

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   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. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 94.3

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

   4. Applied rewrites 94.3%

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

   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. lift-/.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. lower-/.f64 94.4

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

   7. Applied rewrites 94.4%

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

   if -1.79999999999999997e80 < b < 3.1999999999999998e-48

   1. Initial program 77.5%

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

   3. Applied rewrites 77.5%

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

   if 3.1999999999999998e-48 < b

   1. Initial program 16.4%

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

   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 87.2

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

   4. Applied rewrites 87.2%

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

Alternative 2: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e+80)
   (fma 0.5 (/ c b) (* (/ b a) -0.6666666666666666))
   (if (<= b 3.2e-48)
     (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e+80) {
		tmp = fma(0.5, (c / b), ((b / a) * -0.6666666666666666));
	} else if (b <= 3.2e-48) {
		tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e+80)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b / a) * -0.6666666666666666));
	elseif (b <= 3.2e-48)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e+80], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.2e-48], 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], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.79999999999999997e80

   1. Initial program 57.8%

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

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   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. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 94.3

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

   4. Applied rewrites 94.3%

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

   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. lift-/.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. lower-/.f64 94.4

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

   7. Applied rewrites 94.4%

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

   if -1.79999999999999997e80 < b < 3.1999999999999998e-48

   1. Initial program 77.5%

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

   if 3.1999999999999998e-48 < b

   1. Initial program 16.4%

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

   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 87.2

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

   4. Applied rewrites 87.2%

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

Alternative 3: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e+80)
   (fma 0.5 (/ c b) (* (/ b a) -0.6666666666666666))
   (if (<= b 3.2e-48)
     (/ (+ (- b) (sqrt (fma (* c a) -3.0 (* b b)))) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e+80) {
		tmp = fma(0.5, (c / b), ((b / a) * -0.6666666666666666));
	} else if (b <= 3.2e-48) {
		tmp = (-b + sqrt(fma((c * a), -3.0, (b * b)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e+80)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b / a) * -0.6666666666666666));
	elseif (b <= 3.2e-48)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(Float64(c * a), -3.0, Float64(b * b)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e+80], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.2e-48], N[(N[((-b) + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $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 < -1.79999999999999997e80

   1. Initial program 57.8%

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

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   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. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 94.3

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

   4. Applied rewrites 94.3%

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

   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. lift-/.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. lower-/.f64 94.4

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

   7. Applied rewrites 94.4%

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

   if -1.79999999999999997e80 < b < 3.1999999999999998e-48

   1. Initial program 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 77.4

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

   3. Applied rewrites 77.4%

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

   if 3.1999999999999998e-48 < b

   1. Initial program 16.4%

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

   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 87.2

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

   4. Applied rewrites 87.2%

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

Alternative 4: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -11:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(a \cdot -3\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -11.0)
   (/ (* -0.6666666666666666 b) a)
   (if (<= b 3.2e-48)
     (/ (+ (- b) (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 <= -11.0) {
		tmp = (-0.6666666666666666 * b) / a;
	} else if (b <= 3.2e-48) {
		tmp = (-b + 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 <= (-11.0d0)) then
        tmp = ((-0.6666666666666666d0) * b) / a
    else if (b <= 3.2d-48) then
        tmp = (-b + 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 <= -11.0) {
		tmp = (-0.6666666666666666 * b) / a;
	} else if (b <= 3.2e-48) {
		tmp = (-b + 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 <= -11.0:
		tmp = (-0.6666666666666666 * b) / a
	elif b <= 3.2e-48:
		tmp = (-b + 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 <= -11.0)
		tmp = Float64(Float64(-0.6666666666666666 * b) / a);
	elseif (b <= 3.2e-48)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(a * -3.0) * c))) / 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 <= -11.0)
		tmp = (-0.6666666666666666 * b) / a;
	elseif (b <= 3.2e-48)
		tmp = (-b + 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, -11.0], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.2e-48], N[(N[((-b) + N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c), $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 < -11

   1. Initial program 65.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 65.6%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 90.3

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

   6. Applied rewrites 90.3%

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

   if -11 < b < 3.1999999999999998e-48

   1. Initial program 74.6%

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

   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 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 62.6

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

   6. Applied rewrites 62.6%

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

   if 3.1999999999999998e-48 < b

   1. Initial program 16.4%

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

   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 87.2

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

   4. Applied rewrites 87.2%

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

Alternative 5: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -11:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -11.0)
   (/ (* -0.6666666666666666 b) a)
   (if (<= b 3.2e-48)
     (/ (+ (- 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 <= -11.0) {
		tmp = (-0.6666666666666666 * b) / a;
	} else if (b <= 3.2e-48) {
		tmp = (-b + 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 <= (-11.0d0)) then
        tmp = ((-0.6666666666666666d0) * b) / a
    else if (b <= 3.2d-48) then
        tmp = (-b + 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 <= -11.0) {
		tmp = (-0.6666666666666666 * b) / a;
	} else if (b <= 3.2e-48) {
		tmp = (-b + 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 <= -11.0:
		tmp = (-0.6666666666666666 * b) / a
	elif b <= 3.2e-48:
		tmp = (-b + 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 <= -11.0)
		tmp = Float64(Float64(-0.6666666666666666 * b) / a);
	elseif (b <= 3.2e-48)
		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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -11.0)
		tmp = (-0.6666666666666666 * b) / a;
	elseif (b <= 3.2e-48)
		tmp = (-b + 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, -11.0], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.2e-48], 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 < -11

   1. Initial program 65.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 65.6%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 90.3

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

   6. Applied rewrites 90.3%

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

   if -11 < b < 3.1999999999999998e-48

   1. Initial program 74.6%

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

   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 62.5

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

   4. Applied rewrites 62.5%

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

   if 3.1999999999999998e-48 < b

   1. Initial program 16.4%

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

   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 87.2

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

   4. Applied rewrites 87.2%

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

Alternative 6: 79.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot -3\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e-108)
   (/ (* -0.6666666666666666 b) a)
   (if (<= b 3.2e-48) (/ (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 <= -3.6e-108) {
		tmp = (-0.6666666666666666 * b) / a;
	} else if (b <= 3.2e-48) {
		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 <= (-3.6d-108)) then
        tmp = ((-0.6666666666666666d0) * b) / a
    else if (b <= 3.2d-48) 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 <= -3.6e-108) {
		tmp = (-0.6666666666666666 * b) / a;
	} else if (b <= 3.2e-48) {
		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 <= -3.6e-108:
		tmp = (-0.6666666666666666 * b) / a
	elif b <= 3.2e-48:
		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 <= -3.6e-108)
		tmp = Float64(Float64(-0.6666666666666666 * b) / a);
	elseif (b <= 3.2e-48)
		tmp = Float64(sqrt(Float64(Float64(a * -3.0) * c)) / 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 <= -3.6e-108)
		tmp = (-0.6666666666666666 * b) / a;
	elseif (b <= 3.2e-48)
		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, -3.6e-108], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.2e-48], N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c), $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 < -3.6000000000000001e-108

   1. Initial program 70.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 70.5%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 82.3

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

   6. Applied rewrites 82.3%

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

   if -3.6000000000000001e-108 < b < 3.1999999999999998e-48

   1. Initial program 70.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 3.4%

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

   5. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{\sqrt{a \cdot c} \cdot \sqrt{-3}}}{3 \cdot a} \]
   6. 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. associate-*r* N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 66.6

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

   7. Applied rewrites 66.6%

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

   if 3.1999999999999998e-48 < b

   1. Initial program 16.4%

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

   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 87.2

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

   4. Applied rewrites 87.2%

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

Alternative 7: 79.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e-108)
   (/ (* -0.6666666666666666 b) a)
   (if (<= b 3.2e-48) (/ (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 <= -3.6e-108) {
		tmp = (-0.6666666666666666 * b) / a;
	} else if (b <= 3.2e-48) {
		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 <= (-3.6d-108)) then
        tmp = ((-0.6666666666666666d0) * b) / a
    else if (b <= 3.2d-48) 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 <= -3.6e-108) {
		tmp = (-0.6666666666666666 * b) / a;
	} else if (b <= 3.2e-48) {
		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 <= -3.6e-108:
		tmp = (-0.6666666666666666 * b) / a
	elif b <= 3.2e-48:
		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 <= -3.6e-108)
		tmp = Float64(Float64(-0.6666666666666666 * b) / a);
	elseif (b <= 3.2e-48)
		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 <= -3.6e-108)
		tmp = (-0.6666666666666666 * b) / a;
	elseif (b <= 3.2e-48)
		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, -3.6e-108], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.2e-48], 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 < -3.6000000000000001e-108

   1. Initial program 70.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 70.5%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 82.3

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

   6. Applied rewrites 82.3%

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

   if -3.6000000000000001e-108 < b < 3.1999999999999998e-48

   1. Initial program 70.5%

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

   2. Taylor expanded in 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 66.5

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

   4. Applied rewrites 66.5%

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

   if 3.1999999999999998e-48 < b

   1. Initial program 16.4%

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

   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 87.2

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

   4. Applied rewrites 87.2%

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

Alternative 8: 71.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-135}:\\ \;\;\;\;\frac{\sqrt{\frac{c}{a} \cdot -3}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e-114)
   (/ (* -0.6666666666666666 b) a)
   (if (<= b 3.4e-135) (/ (sqrt (* (/ c a) -3.0)) 3.0) (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-114) {
		tmp = (-0.6666666666666666 * b) / a;
	} else if (b <= 3.4e-135) {
		tmp = sqrt(((c / a) * -3.0)) / 3.0;
	} 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 <= (-1.5d-114)) then
        tmp = ((-0.6666666666666666d0) * b) / a
    else if (b <= 3.4d-135) then
        tmp = sqrt(((c / a) * (-3.0d0))) / 3.0d0
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-114) {
		tmp = (-0.6666666666666666 * b) / a;
	} else if (b <= 3.4e-135) {
		tmp = Math.sqrt(((c / a) * -3.0)) / 3.0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.5e-114:
		tmp = (-0.6666666666666666 * b) / a
	elif b <= 3.4e-135:
		tmp = math.sqrt(((c / a) * -3.0)) / 3.0
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e-114)
		tmp = Float64(Float64(-0.6666666666666666 * b) / a);
	elseif (b <= 3.4e-135)
		tmp = Float64(sqrt(Float64(Float64(c / a) * -3.0)) / 3.0);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.5e-114)
		tmp = (-0.6666666666666666 * b) / a;
	elseif (b <= 3.4e-135)
		tmp = sqrt(((c / a) * -3.0)) / 3.0;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e-114], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.4e-135], N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.50000000000000008e-114

   1. Initial program 70.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 70.7%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 81.6

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

   6. Applied rewrites 81.6%

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

   if -1.50000000000000008e-114 < b < 3.39999999999999989e-135

   1. Initial program 75.6%

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

   3. Applied rewrites 75.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 35.6

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

   8. Applied rewrites 35.6%

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

   if 3.39999999999999989e-135 < b

   1. Initial program 21.5%

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

   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 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 9: 71.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-135}:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -3} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e-114)
   (/ (* -0.6666666666666666 b) a)
   (if (<= b 3.4e-135)
     (* (sqrt (* (/ c a) -3.0)) 0.3333333333333333)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-114) {
		tmp = (-0.6666666666666666 * b) / a;
	} else if (b <= 3.4e-135) {
		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 <= (-1.5d-114)) then
        tmp = ((-0.6666666666666666d0) * b) / a
    else if (b <= 3.4d-135) 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 <= -1.5e-114) {
		tmp = (-0.6666666666666666 * b) / a;
	} else if (b <= 3.4e-135) {
		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 <= -1.5e-114:
		tmp = (-0.6666666666666666 * b) / a
	elif b <= 3.4e-135:
		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 <= -1.5e-114)
		tmp = Float64(Float64(-0.6666666666666666 * b) / a);
	elseif (b <= 3.4e-135)
		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 <= -1.5e-114)
		tmp = (-0.6666666666666666 * b) / a;
	elseif (b <= 3.4e-135)
		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, -1.5e-114], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.4e-135], 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 < -1.50000000000000008e-114

   1. Initial program 70.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 70.7%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 81.6

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

   6. Applied rewrites 81.6%

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

   if -1.50000000000000008e-114 < b < 3.39999999999999989e-135

   1. Initial program 75.6%

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

   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 35.6

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

   4. Applied rewrites 35.6%

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

   if 3.39999999999999989e-135 < b

   1. Initial program 21.5%

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

   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 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 10: 70.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-152}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -3} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e-152)
   (/ (* -0.6666666666666666 b) a)
   (if (<= b 2.1e-215)
     (* (sqrt (* (/ c a) -3.0)) -0.3333333333333333)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e-152) {
		tmp = (-0.6666666666666666 * b) / a;
	} else if (b <= 2.1e-215) {
		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 <= (-1.2d-152)) then
        tmp = ((-0.6666666666666666d0) * b) / a
    else if (b <= 2.1d-215) 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 <= -1.2e-152) {
		tmp = (-0.6666666666666666 * b) / a;
	} else if (b <= 2.1e-215) {
		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 <= -1.2e-152:
		tmp = (-0.6666666666666666 * b) / a
	elif b <= 2.1e-215:
		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 <= -1.2e-152)
		tmp = Float64(Float64(-0.6666666666666666 * b) / a);
	elseif (b <= 2.1e-215)
		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 <= -1.2e-152)
		tmp = (-0.6666666666666666 * b) / a;
	elseif (b <= 2.1e-215)
		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, -1.2e-152], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.1e-215], 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 < -1.2e-152

   1. Initial program 71.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 71.9%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 78.3

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

   6. Applied rewrites 78.3%

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

   if -1.2e-152 < b < 2.1e-215

   1. Initial program 76.3%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 36.5

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

   4. Applied rewrites 36.5%

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

   if 2.1e-215 < b

   1. Initial program 26.4%

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

   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 74.7

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

   4. Applied rewrites 74.7%

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

Alternative 11: 66.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ (* -0.6666666666666666 b) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-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 <= (-4d-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 <= -4e-310) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-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 <= -4e-310)
		tmp = Float64(Float64(-0.6666666666666666 * 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 <= -4e-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, -4e-310], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 72.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 72.8%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 66.3

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

   6. Applied rewrites 66.3%

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

   if -3.999999999999988e-310 < b

   1. Initial program 31.7%

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

   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 67.3

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

   4. Applied rewrites 67.3%

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

Alternative 12: 66.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (* -0.6666666666666666 (/ b a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-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 <= (-4d-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 <= -4e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-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 <= -4e-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 <= -4e-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, -4e-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 < -3.999999999999988e-310

   1. Initial program 72.8%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.2

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

   4. Applied rewrites 66.2%

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

   if -3.999999999999988e-310 < b

   1. Initial program 31.7%

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

   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 67.3

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

   4. Applied rewrites 67.3%

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

Alternative 13: 41.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+49}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5e+49) (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5e+49) {
		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 <= 5d+49) 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 <= 5e+49) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5e+49:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = 0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5e+49)
		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 <= 5e+49)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = 0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5e+49], 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 < 5.0000000000000004e49

   1. Initial program 66.4%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 45.8

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

   4. Applied rewrites 45.8%

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

   if 5.0000000000000004e49 < b

   1. Initial program 11.9%

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

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   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. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 2.4

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

   4. Applied rewrites 2.4%

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

   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. lift-/.f64 30.3

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

   7. Applied rewrites 30.3%

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

Alternative 14: 10.2% 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 52.4%

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

2. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
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. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 34.0

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

4. Applied rewrites 34.0%

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

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. lift-/.f64 10.2

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

7. Applied rewrites 10.2%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{c}{b}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))